# Given Information - Prevalence of disease in the population: \( 1\% \) (i.e., \( P(D) = .01 \)) - Probability test detects disease when present (sensitivity): \( 98\% \) (i.e., \( P(+|D) = .98 \)) - Probability of a false positive: \( 3\% \) (i.e., \( P(+|\neg D) = .03 \)) - The person tests positive twice, and tests are independent. # What To Find - Probability that a person actually has the disease given they test positive twice: \( P(D|++) \) # Definition or Concept Used - **Bayes' Theorem** is used to update the probability of a hypothesis given new evidence. - Independence of tests: \( P(\text{both } +|D) = P(+|D) \times P(+|D) \) - Total probability for two positives: \( P(++|D) = .98 \times .98 \) \( P(++|\neg D) = .03 \times .03 \) # Solution Let: - \( D \): Person has the disease. - \( \neg D \): Person does not have the disease. - \( ++ \): Person tests positive twice. Calculate probabilities: - \( P(D) = .01 \) - \( P(\neg D) = .99 \) - \( P(++|D) = .98 \times .98 = .9604 \) - \( P(++|\neg D) = .03 \times .03 = .0009 \) Using the law of total probability: \[ P(++) = P(++|D)P(D) + P(++|\neg D)P(\neg D) \] \[ P(++) = (.9604)(.01) + (.0009)(.99) \] \[ P(++) = .009604 + .000891 = .010495 \] Apply Bayes' Theorem: \[ P(D|++) = \frac{P(++|D)P(D)}{P(++)} \] \[ P(D|++) = \frac{.9604 \times .01}{.010495} \] \[ P(D|++) = \frac{.009604}{.010495} \approx .9154 \] # Summary The probability that a person actually has the disease given they test positive twice independently is approximately \( 91.5\% \).

# Given Information - Prevalence of disease in the population: \( P(D) = 0.01 \) (i.e., \( 1\% \)) - Probability of detecting disease when present (sensitivity): \( P(+|D) = 0.98 \) (i.e., \( 98\% \)) - Probability of a false positive: \( P(+|\neg D) = 0.03 \) (i.e., \( 3\% \)) - The person tests positive twice independently. # What To Find - Probability that a person actually has the disease given they test positive twice: \( P(D|++) \) # Definition or Concept Used - **Bayes' Theorem**: This theorem allows us to update the probability of a hypothesis based on new evidence. - **Independence of Tests**: The probability of two independent events both occurring is the product of their individual probabilities. - **Total Probability**: Used to find the overall probability of a positive result based on the disease status. # Solution Define: - \( D \): Person has the disease. - \( \neg D \): Person does not have the disease. - \( ++ \): Person tests positive twice. Calculate the probabilities: - \( P(D) = 0.01 \) - \( P(\neg D) = 0.99 \) - Probability of testing positive twice if the person has the disease: \[ P(++|D) = P(+|D) \times P(+|D) = 0.98 \times 0.98 = 0.9604 \] - Probability of testing positive twice if the person does not have the disease: \[ P(++|\neg D) = P(+|\neg D) \times P(+|\neg D) = 0.03 \times 0.03 = 0.0009 \] Using the law of total probability to find \( P(++) \): \[ P(++) = P(++|D)P(D) + P(++|\neg D)P(\neg D) \] Substituting the values: \[ P(++) = (0.9604)(0.01) + (0.0009)(0.99) \] \[ P(++) = 0.009604 + 0.000891 = 0.010495 \] Applying Bayes' Theorem to find \( P(D|++) \): \[ P(D|++) = \frac{P(++|D)P(D)}{P(++)} \] Substituting the values: \[ P(D|++) = \frac{0.9604 \times 0.01}{0.010495} \] Calculating: \[ P(D|++) = \frac{0.009604}{0.010495} \approx 0.9154 \] # Summary The probability that a person actually has the disease given they test positive twice independently is approximately \( 91.5\% \).

# Given Information - Disease prevalence: \( P(D) = 0.01 \) (1%) - Test sensitivity: \( P(+|D) = 0.98 \) (98%) - False positive rate: \( P(+|\neg D) = 0.03 \) (3%) - A person tests positive twice independently. # What To Find - Probability that a person actually has the disease after testing positive twice: \( P(D|++) \) # Definition or Concept Used - **Bayes' Theorem**: Used to update the probability of a condition given new evidence. - **Independence of Tests**: For independent events, the joint probability is the product of their individual probabilities. - **Total Probability**: Used to calculate the overall likelihood of a positive test result. # Solution Define: - \( D \): Person has the disease. - \( \neg D \): Person does not have the disease. - \( ++ \): Person tests positive twice. Compute the probabilities: - \( P(D) = 0.01 \) - \( P(\neg D) = 0.99 \) - Probability of testing positive twice if the person has the disease: \[ P(++|D) = P(+|D) \times P(+|D) = 0.98 \times 0.98 = 0.9604 \] - Probability of testing positive twice if the person does not have the disease: \[ P(++|\neg D) = P(+|\neg D) \times P(+|\neg D) = 0.03 \times 0.03 = 0.0009 \] Using the law of total probability to find \( P(++) \): \[ P(++) = P(++|D)P(D) + P(++|\neg D)P(\neg D) \] Substituting the values: \[ P(++) = (0.9604)(0.01) + (0.0009)(0.99) \] Calculating: \[ P(++) = 0.009604 + 0.000891 = 0.010495 \] Applying Bayes' Theorem to calculate \( P(D|++) \): \[ P(D|++) = \frac{P(++|D)P(D)}{P(++)} \] Substituting the known values: \[ P(D|++) = \frac{0.9604 \times 0.01}{0.010495} \] Calculating: \[ P(D|++) = \frac{0.009604}{0.010495} \approx 0.9154 \] # Summary The probability that a person actually has the disease given they test positive twice independently is approximately \( 91.5\% \).

# Given Information - Number of gold coins: \( G = 3 \) - Number of silver coins: \( S = 5 \) - Number of fake coins: \( F = 2 \) - Total coins: \( T = G + S + F = 3 + 5 + 2 = 10 \) # What To Find - Probability that both coins are gold given that at least one is gold: \( P(B|A) \) Where: - \( B \): Event that both coins are gold. - \( A \): Event that at least one coin is gold. # Definitions or Concepts Used - **Conditional Probability**: \[ P(B|A) = \frac{P(B \cap A)}{P(A)} \] - **Combinations**: The number of ways to choose \( k \) items from \( n \) items is given by \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). # Solution Calculate the probabilities: 1. **Total ways to pick 2 coins from 10**: \[ P(\text{Total}) = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \] 2. **Ways to pick 2 gold coins**: \[ P(B) = \binom{3}{2} = \frac{3 \times 2}{2 \times 1} = 3 \] 3. **Ways to pick at least one gold coin**: First calculate the cases where no gold coins are picked: - Picking 2 coins from silver and fake (5 silver + 2 fake = 7 total): \[ P(\text{No gold}) = \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \] - Thus, the ways to pick at least one gold coin: \[ P(A) = P(\text{Total}) - P(\text{No gold}) = 45 - 21 = 24 \] 4. **Probability that both coins are gold given at least one is gold**: \[ P(B|A) = \frac{P(B)}{P(A)} = \frac{3}{24} = \frac{1}{8} \] # Summary The probability that both coins are gold given that at least one is gold is \( \frac{1}{8} \).

# Given Information - Number of gold coins: \( G = 3 \) - Number of silver coins: \( S = 5 \) - Number of fake coins: \( F = 2 \) - Total number of coins: \( T = G + S + F = 3 + 5 + 2 = 10 \) # What To Find - Probability that both coins are gold, given that at least one is gold: \( P(B|A) \) Where: - \( B \): Event that both coins are gold. - \( A \): Event that at least one coin is gold. # Definitions or Concepts Used - **Conditional Probability**: \[ P(B|A) = \frac{P(B \cap A)}{P(A)} \] - **Combinatorial Selection**: The number of ways to choose \( k \) items from \( n \) items is given by \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). # Solution 1. **Total ways to choose 2 coins from 10**: \[ P(\text{Total}) = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \] 2. **Ways to pick 2 gold coins**: \[ P(B) = \binom{3}{2} = \frac{3 \times 2}{2 \times 1} = 3 \] 3. **Ways to pick at least one gold coin**: - First, calculate the number of ways to pick 2 coins that are not gold (from silver and fake, \( S + F = 5 + 2 = 7 \)): \[ P(\text{No gold}) = \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \] - Therefore, the number of ways to pick at least one gold coin: \[ P(A) = P(\text{Total}) - P(\text{No gold}) = 45 - 21 = 24 \] 4. **Calculate the probability that both coins are gold given at least one is gold**: \[ P(B|A) = \frac{P(B)}{P(A)} = \frac{3}{24} = \frac{1}{8} \] # Summary The probability that both coins are gold given that at least one is gold is \( \frac{1}{8} \).

# Given Information - **Gold Coins**: \( G = 3 \) - **Silver Coins**: \( S = 5 \) - **Fake Coins**: \( F = 2 \) - **Total Coins**: \( T = G + S + F = 10 \) # What To Find - We need to find the probability that both coins drawn are gold, given that at least one of them is gold: \( P(B|A) \) Where: - \( B \): Both coins are gold. - \( A \): At least one coin is gold. # Definitions or Concepts Used - **Conditional Probability**: The formula for conditional probability is: \[ P(B|A) = \frac{P(B \cap A)}{P(A)} \] - **Combinations**: The number of ways to choose \( k \) items from \( n \) items is calculated using: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] # Solution 1. **Total Ways to Choose 2 Coins from 10**: Compute the total combinations of selecting 2 coins from the 10 available: \[ P(\text{Total}) = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \] 2. **Ways to Pick 2 Gold Coins**: Calculate how many ways we can select 2 coins from the 3 gold coins: \[ P(B) = \binom{3}{2} = \frac{3 \times 2}{2 \times 1} = 3 \] 3. **Ways to Pick At Least One Gold Coin**: First, we need to determine the ways to select coins that do not include any gold coins: - The non-gold coins consist of 5 silver and 2 fake coins, totaling 7 coins. - Calculate the combinations of picking 2 coins from these 7 non-gold coins: \[ P(\text{No gold}) = \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \] - Now, subtract this from the total combinations to find the number of ways to pick at least one gold coin: \[ P(A) = P(\text{Total}) - P(\text{No gold}) = 45 - 21 = 24 \] 4. **Calculate the Conditional Probability**: Now we can find the probability that both coins are gold, given that at least one is gold: \[ P(B|A) = \frac{P(B)}{P(A)} = \frac{3}{24} = \frac{1}{8} \] # Summary The probability that both coins drawn are gold, given that at least one of them is gold, is \( \frac{1}{8} \).

# Question If two events cannot occur at the same time, they are called: - A) Independent events - B) Complementary events - C) Mutually exclusive events - D) Exhaustive events # Answer The correct option is **C) Mutually exclusive events**. # Explanation - **Independent Events**: Two events are independent if the occurrence of one does not affect the occurrence of the other. - **Complementary Events**: Two events are complementary if one event occurs if and only if the other does not, but they can occur together (i.e., they are not mutually exclusive). - **Mutually Exclusive Events**: Two events are mutually exclusive if they cannot occur at the same time. If one event occurs, the other cannot. - **Exhaustive Events**: A set of events is exhaustive if they cover all possible outcomes of a probability experiment. Thus, events that cannot occur simultaneously are defined as **mutually exclusive events**.

# Given Information - Total television sets in shipment: \( N = 7 \) - Defective television sets: \( D = 2 \) - Non-defective television sets: \( N - D = 5 \) - Sets purchased by the hotel: \( n = 3 \) # What To Find - Probability distribution of \( X \), where \( X \) is the number of defective sets purchased by the hotel. # Definitions or Concepts Used - **Hypergeometric Distribution**: Used when sampling without replacement. The probability of \( k \) successes (defective sets) in \( n \) draws (purchases) from a finite population of size \( N \) containing \( K \) successes is given by: \[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \] Where: - \( K \): Total number of successes in the population (defective sets). - \( N \): Total population size. - \( n \): Number of draws. - \( k \): Number of observed successes. # Solution 1. **Identify Parameters**: - \( N = 7 \) (total sets) - \( K = 2 \) (defective sets) - \( n = 3 \) (sets purchased) 2. **Calculate Probabilities for Each Possible Value of \( X \)**: - Possible values of \( X \) are \( 0, 1, 2 \) (since there are only 2 defective sets). - **For \( X = 0 \) (0 defective sets)**: \[ P(X = 0) = \frac{\binom{2}{0} \binom{5}{3}}{\binom{7}{3}} = \frac{1 \times 10}{35} = \frac{10}{35} = \frac{2}{7} \] - **For \( X = 1 \) (1 defective set)**: \[ P(X = 1) = \frac{\binom{2}{1} \binom{5}{2}}{\binom{7}{3}} = \frac{2 \times 10}{35} = \frac{20}{35} = \frac{4}{7} \] - **For \( X = 2 \) (2 defective sets)**: \[ P(X = 2) = \frac{\binom{2}{2} \binom{5}{1}}{\binom{7}{3}} = \frac{1 \times 5}{35} = \frac{5}{35} = \frac{1}{7} \] 3. **Summary of Probability Distribution**: | \( X \) | Probability \( P(X) \) | |---------|-----------------------| | 0 | \( \frac{2}{7} \) | | 1 | \( \frac{4}{7} \) | | 2 | \( \frac{1}{7} \) | # Graphical Representation ```plaintext Probability Histogram | # | # | # # | # # | # # # |_________#________#_________#_________ 0 1 2 ``` - The height of each bar represents the probability of purchasing \( 0, 1, \) or \( 2 \) defective television sets. # Summary The probability distribution of the number of defective sets \( X \) purchased by the hotel is: - \( P(X = 0) = \frac{2}{7} \) - \( P(X = 1) = \frac{4}{7} \) - \( P(X = 2) = \frac{1}{7} \)

# Given Information - Two fair dice are rolled. - We need to find the probability that at least one die shows a 6 given that the sum of the dice is greater than 8. # What To Find - Probability that at least one die shows a 6 given the condition of the sum being greater than 8: \( P(A|B) \) Where: - \( A \): Event that at least one die shows a 6. - \( B \): Event that the sum of the dice is greater than 8. # Definitions or Concepts Used - **Conditional Probability**: The formula for conditional probability is: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] - **Total Outcomes**: The total number of outcomes when rolling two dice is \( 6 \times 6 = 36 \). # Solution 1. **Determine the outcomes for \( B \)** (sum greater than 8): - Possible sums greater than 8 are 9, 10, 11, and 12. - The outcomes for each sum: - **Sum = 9**: (3,6), (4,5), (5,4), (6,3) → 4 outcomes - **Sum = 10**: (4,6), (5,5), (6,4) → 3 outcomes - **Sum = 11**: (5,6), (6,5) → 2 outcomes - **Sum = 12**: (6,6) → 1 outcome - Total outcomes for \( B \): \[ P(B) = 4 + 3 + 2 + 1 = 10 \] 2. **Determine the outcomes for \( A \cap B \)** (at least one die shows a 6 and sum is greater than 8): - From the outcomes of \( B \): - **Sum = 9**: (3,6), (6,3) → 2 outcomes (both include a 6) - **Sum = 10**: (4,6), (6,4) → 2 outcomes (both include a 6) - **Sum = 11**: (5,6), (6,5) → 2 outcomes (both include a 6) - **Sum = 12**: (6,6) → 1 outcome (includes a 6) - Total outcomes for \( A \cap B \): \[ P(A \cap B) = 2 + 2 + 2 + 1 = 7 \] 3. **Calculate the conditional probability**: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{7}{10} \] # Summary The probability that at least one die shows a 6 given that the sum is greater than 8 is \( \frac{7}{10} \).

# Given Information - **Production Percentages**: - Machine 1: \( P(M_1) = 30\% = 0.30 \) - Machine 2: \( P(M_2) = 45\% = 0.45 \) - Machine 3: \( P(M_3) = 25\% = 0.25 \) - **Defect Rates**: - Machine 1: \( P(D|M_1) = 2\% = 0.02 \) - Machine 2: \( P(D|M_2) = 3\% = 0.03 \) - Machine 3: \( P(D|M_3) = 5\% = 0.05 \) # What To Find - Probability that a randomly selected defective product came from Machine 3: \( P(M_3|D) \) # Definitions or Concepts Used - **Bayes' Theorem**: \[ P(M_k|D) = \frac{P(D|M_k) \cdot P(M_k)}{P(D)} \] - **Total Probability**: \[ P(D) = P(D|M_1)P(M_1) + P(D|M_2)P(M_2) + P(D|M_3)P(M_3) \] # Solution 1. **Calculate the Total Probability of Defect \( P(D) \)**: \[ P(D) = P(D|M_1)P(M_1) + P(D|M_2)P(M_2) + P(D|M_3)P(M_3) \] Substituting the values: \[ P(D) = (0.02)(0.30) + (0.03)(0.45) + (0.05)(0.25) \] Calculating each term: - From Machine 1: \( 0.02 \times 0.30 = 0.006 \) - From Machine 2: \( 0.03 \times 0.45 = 0.0135 \) - From Machine 3: \( 0.05 \times 0.25 = 0.0125 \) Now sum them up: \[ P(D) = 0.006 + 0.0135 + 0.0125 = 0.032 \] 2. **Calculate \( P(M_3|D) \)** using Bayes' Theorem: \[ P(M_3|D) = \frac{P(D|M_3) \cdot P(M_3)}{P(D)} \] Substituting the known values: \[ P(M_3|D) = \frac{(0.05)(0.25)}{0.032} \] Calculate the numerator: \[ = \frac{0.0125}{0.032} \] Finally, perform the division: \[ P(M_3|D) \approx 0.390625 \] # Summary The probability that a randomly selected defective product came from Machine 3 is approximately \( 0.3906 \) or \( 39.06\% \).

# Given Information - Variance of \( X \): \( \text{Var}(X) = 9 \) - Variance of \( Y \): \( \text{Var}(Y) = 16 \) - Covariance between \( X \) and \( Y \): \( \text{Cov}(X, Y) = 6 \) # What To Find 1. Correlation coefficient between \( X \) and \( Y \): \( \rho_{X,Y} \) 2. Variance of the linear combination \( Z = 2X - 3Y \): \( \text{Var}(Z) \) # Definitions or Concepts Used - **Correlation Coefficient**: \[ \rho_{X,Y} = \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \] - **Variance of a Linear Combination**: For any constants \( a \) and \( b \): \[ \text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(X,Y) \] # Solution 1. **Calculate the Correlation Coefficient**: \[ \rho_{X,Y} = \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \] Substitute the known values: \[ \rho_{X,Y} = \frac{6}{\sqrt{9 \cdot 16}} = \frac{6}{\sqrt{144}} = \frac{6}{12} = 0.5 \] 2. **Calculate the Variance of \( Z = 2X - 3Y \)**: Using the variance formula for a linear combination: \[ \text{Var}(Z) = (2^2) \text{Var}(X) + (-3^2) \text{Var}(Y) + 2(2)(-3) \text{Cov}(X,Y) \] Substitute the known values: \[ \text{Var}(Z) = 4 \cdot 9 + 9 \cdot 16 + 2(2)(-3)(6) \] Calculate each term: - \( 4 \cdot 9 = 36 \) - \( 9 \cdot 16 = 144 \) - \( 2(2)(-3)(6) = -72 \) Combine the results: \[ \text{Var}(Z) = 36 + 144 - 72 = 108 \] # Summary - The correlation coefficient between \( X \) and \( Y \) is \( 0.5 \). - The variance of \( Z = 2X - 3Y \) is \( 108 \).

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The correct option is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This option is primarily related to algebra and mathematics rather than statistics. While statistics may involve mathematical equations, its main focus is on data analysis rather than solving purely mathematical problems. - **Option B: Collecting, organizing, analyzing, and interpreting data** This is the essence of statistics. Statistics involves methods for collecting data, summarizing it (organizing), analyzing it to discover patterns or trends, and interpreting the results to make informed decisions or predictions. This option accurately reflects the comprehensive nature of statistics as a field. - **Option C: Drawing geometric figures** This option pertains more to geometry and visual representation in mathematics. Statistics may use graphical representations (like charts and graphs) to present data, but the act of drawing geometric figures is not a core function of statistics. - **Option D: Studying planets** This option relates to astronomy and astrophysics. While statistics can be utilized in astronomical studies (e.g., analyzing data from space missions), the primary focus of statistics is not on celestial bodies but rather on data in various fields, including social sciences, business, and health. # Summary Statistics is mainly concerned with **collecting, organizing, analyzing, and interpreting data**. Other options do not accurately reflect the focus of the field.

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The correct option is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This choice is focused on mathematical problem-solving, typically found in algebra or calculus. While statistics may utilize equations in analysis, its primary role is not centered around solving equations but rather dealing with data. - **Option B: Collecting, organizing, analyzing, and interpreting data** This option encapsulates the core activities of statistics. Statisticians gather data through various methods, organize it for clarity, analyze it to identify trends or relationships, and interpret results to inform decision-making. This holistic approach is what defines the field of statistics. - **Option C: Drawing geometric figures** This option pertains to the study of geometry, which involves creating and understanding shapes and spatial relationships. Although statistics can involve graphical representation of data, the act of drawing geometric figures is not a primary concern of the discipline. - **Option D: Studying planets** This option relates to the field of astronomy, which investigates celestial bodies and their movements. While statistical methods can be applied in astronomical research (for instance, analyzing star patterns), the main focus of statistics is not on celestial studies but on data from various domains. # Summary Statistics is fundamentally concerned with **collecting, organizing, analyzing, and interpreting data**. Other options do not represent the main focus of the field.

# Given Information - **Production Percentages**: - Machine 1: \( P(M_1) = 30\% = 0.30 \) - Machine 2: \( P(M_2) = 45\% = 0.45 \) - Machine 3: \( P(M_3) = 25\% = 0.25 \) - **Defect Rates**: - Machine 1: \( P(D|M_1) = 2\% = 0.02 \) - Machine 2: \( P(D|M_2) = 3\% = 0.03 \) - Machine 3: \( P(D|M_3) = 5\% = 0.05 \) # What To Find - Probability that a randomly selected defective product came from Machine 3: \( P(M_3|D) \) # Definitions or Concepts Used - **Bayes' Theorem**: \[ P(M_k|D) = \frac{P(D|M_k) \cdot P(M_k)}{P(D)} \] - **Total Probability**: \[ P(D) = P(D|M_1)P(M_1) + P(D|M_2)P(M_2) + P(D|M_3)P(M_3) \] # Solution Calculate the total probability of defect \( P(D) \): \[ P(D) = P(D|M_1)P(M_1) + P(D|M_2)P(M_2) + P(D|M_3)P(M_3) \] Substituting the values: \[ P(D) = (0.02)(0.30) + (0.03)(0.45) + (0.05)(0.25) \] Calculating each term: - From Machine 1: \( 0.02 \times 0.30 = 0.006 \) - From Machine 2: \( 0.03 \times 0.45 = 0.0135 \) - From Machine 3: \( 0.05 \times 0.25 = 0.0125 \) Sum these results: \[ P(D) = 0.006 + 0.0135 + 0.0125 = 0.032 \] Now, apply Bayes' Theorem to find \( P(M_3|D) \): \[ P(M_3|D) = \frac{P(D|M_3) \cdot P(M_3)}{P(D)} \] Substituting the known values: \[ P(M_3|D) = \frac{(0.05)(0.25)}{0.032} \] Calculate the numerator: \[ = \frac{0.0125}{0.032} \] Now, perform the division: \[ P(M_3|D) \approx 0.390625 \] # Summary The probability that a randomly selected defective product came from Machine 3 is approximately \( 0.3906 \) or \( 39.06\% \).

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The correct option is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This choice emphasizes algebraic and mathematical problem-solving. While statistics may require the use of equations, its primary focus is not on solving these equations but rather on the analysis and interpretation of data collected from various sources. - **Option B: Collecting, organizing, analyzing, and interpreting data** This option accurately reflects the essence of statistics. The field is dedicated to the methodology of gathering data, structuring it for clarity, analyzing it to derive insights, and interpreting the results to make informed decisions across various domains such as economics, healthcare, and social sciences. - **Option C: Drawing geometric figures** This option is related to geometry, which focuses on shapes, sizes, and the properties of space. While statistics may use graphical representations (like charts or graphs) to visualize data, the act of drawing geometric figures is not a primary focus of statistical study. - **Option D: Studying planets** This choice pertains to astronomy, which investigates celestial bodies and their movements. Although statistical methods can be applied in astronomy (for instance, analyzing data from telescopes), the main focus of statistics lies in data from diverse fields, not specifically in celestial studies. # Summary Statistics is fundamentally concerned with **collecting, organizing, analyzing, and interpreting data**. Other options do not represent the primary focus of the field.

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The accurate choice is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This option focuses on the domain of mathematics, particularly algebra, where the emphasis is on finding values that satisfy mathematical expressions. While statistics may involve some equations, its primary concern is not with algebraic solutions but rather with how data is gathered, processed, and analyzed to derive meaningful insights. Thus, this option inadequately captures the essence of statistical practice. - **Option B: Collecting, organizing, analyzing, and interpreting data** This option encapsulates the core functions of statistics. Statistics is fundamentally about the systematic approach to gathering data from various sources, which may include surveys, experiments, and observational studies. Once the data is collected, it must be organized into a format that facilitates analysis, such as tables or charts. The analysis phase involves employing various statistical methods to identify trends, patterns, or relationships within the data. Finally, interpreting the results is crucial for making informed decisions, predictions, or conclusions based on the analyzed data. This comprehensive view reflects the multifaceted nature of statistics as a discipline that is crucial in numerous fields, including social sciences, business, health, and engineering. - **Option C: Drawing geometric figures** This choice pertains to the field of geometry, which deals with the properties and relations of points, lines, surfaces, and solids. While statistics can utilize graphical representations to visualize data (such as through bar charts or scatter plots), the act of drawing geometric figures itself is not a primary focus of statistics. Therefore, this option does not capture the essence of what statistics is about. - **Option D: Studying planets** This option relates to astronomy, a scientific discipline that explores celestial bodies, their movements, and the universe's structure. Although statistical techniques can be employed in astronomical research for data analysis (like measuring the brightness of stars or the distribution of galaxies), the study of planets is not the central aim of statistics. Statistics serves a broader purpose across various domains, not limited to celestial phenomena. # Summary In conclusion, statistics is primarily concerned with **collecting, organizing, analyzing, and interpreting data**. This holistic approach distinguishes it from the other options, which do not adequately reflect the core activities that define the field of statistics. Understanding this distinction is crucial for recognizing the importance of statistics in making data-driven decisions across multiple disciplines.

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The most appropriate choice is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This option emphasizes the area of mathematics focused on finding solutions to mathematical expressions. While equations can be a part of statistical analysis, statistics itself is not primarily about solving equations but is rather concerned with the processing and interpretation of data gathered from various sources. Thus, this option does not accurately reflect the primary focus of statistics. - **Option B: Collecting, organizing, analyzing, and interpreting data** This choice effectively captures the essence of statistics. The field is dedicated to a systematic methodology that begins with the collection of data through various methods such as surveys, experiments, or observations. Once data is gathered, it must be organized for clarity and efficiency, which often involves structuring it into tables, charts, or graphs. The next phase involves analyzing the data using statistical techniques to identify patterns, correlations, or trends. Finally, interpreting the results is critical; it allows statisticians to draw meaningful conclusions and make informed decisions based on the data analysis. This comprehensive approach is foundational to numerous fields, including healthcare, economics, psychology, and many more, making this option the most accurate representation of what statistics encompasses. - **Option C: Drawing geometric figures** This option pertains to the study of geometry, which focuses on shapes, sizes, and spatial relationships. Although statistics may involve visualizing data through graphs and charts, the act of drawing geometric figures is not central to the field of statistics. Therefore, this option diverges from the primary focus of statistics. - **Option D: Studying planets** This option refers to astronomy, which investigates celestial bodies and their movements. While statistical methods can indeed be applied in astronomy for data analysis (such as studying star distributions or celestial phenomena), the primary aim of statistics is not limited to astronomical studies but extends to a wide range of disciplines that require data analysis. Thus, this option does not accurately reflect the central purpose of statistics. # Summary In summary, statistics is fundamentally concerned with **collecting, organizing, analyzing, and interpreting data**. This comprehensive approach distinguishes it from the other options, which do not adequately represent the core functions of the statistical field. Recognizing this distinction is crucial for understanding the role of statistics in data-driven decision-making across various disciplines.

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The correct choice is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This choice highlights the mathematical aspect of finding solutions to algebraic or numerical problems. While equations play a role in statistical calculations (such as computing means or variances), the primary focus of statistics is not merely the act of solving equations. Instead, statistics is fundamentally about understanding and utilizing data to make informed decisions. Therefore, this option does not encompass the broader scope of what statistics represents in practice. - **Option B: Collecting, organizing, analyzing, and interpreting data** This option accurately reflects the comprehensive nature of statistics. Statistics is a discipline that begins with the **collection of data**—this can involve surveys, experiments, observations, and more. Once data is collected, the next step is to **organize** it, which involves structuring the data in a way that makes it easier to understand and analyze, such as using tables or charts. The **analysis** phase employs statistical methods and techniques to explore the data, uncover patterns, test hypotheses, and identify relationships between variables. Finally, the **interpretation** of the results is crucial for translating the findings into actionable insights that can inform decision-making in various fields, including healthcare, economics, social sciences, and engineering. This holistic view makes option B the most representative of the field of statistics. - **Option C: Drawing geometric figures** This option pertains specifically to geometry, which deals with the properties and relationships of shapes and spaces. Although statistics can utilize graphical representations—such as histograms, pie charts, and scatter plots—to visualize data, the act of drawing geometric figures itself is not a core function of statistics. Therefore, this option fails to capture the essence of statistical work. - **Option D: Studying planets** This choice refers to the field of astronomy, which is focused on the study of celestial bodies, their movements, and the universe as a whole. While statistical methods can be applied in astronomical research (for example, in analyzing the distribution of stars or measuring the properties of planets), the primary focus of statistics is not on celestial phenomena but rather on the analysis and interpretation of data across various domains. Thus, this option does not accurately represent the main objective of statistics. # Summary In conclusion, statistics is primarily concerned with **collecting, organizing, analyzing, and interpreting data**. This comprehensive framework distinguishes it from the other options, which do not adequately capture the core functions that define the field of statistics. Understanding this distinction is essential for recognizing the importance of statistics in data-driven decision-making in numerous disciplines, including business, public health, education, and social research. This recognition underscores how vital statistics is in helping us understand complex data and derive meaningful insights that can guide actions and policies.

# Given Information - The probability density function (PDF) of a random variable \( X \) is defined as: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 20 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: A function that describes the likelihood of a random variable taking on a particular value. The total area under the PDF curve over its range must equal 1. - **Probability Calculation**: For continuous random variables, the probability over an interval can be computed as the integral of the PDF over that interval. # Solution First, to find the value of \( k \), we need to ensure that the total area under the PDF equals 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) = 0 \) outside the interval \( [0, 20] \), we focus on that interval: \[ \int_{0}^{20} k \cdot x \, dx = 1 \] Calculating the integral: \[ \int_{0}^{20} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{20} = k \cdot \left[ \frac{20^2}{2} - 0 \right] = k \cdot \left[ \frac{400}{2} \right] = k \cdot 200 \] Setting this equal to 1: \[ k \cdot 200 = 1 \implies k = \frac{1}{200} \] Next, we compute the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{200} \cdot x\right) \, dx \] Calculating this integral: \[ \int_{1}^{2} \left(\frac{1}{200} \cdot x\right) \, dx = \frac{1}{200} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{200} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Calculating further: \[ = \frac{1}{200} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{200} \cdot \left[ 2 - 0.5 \right] = \frac{1}{200} \cdot \left[ 1.5 \right] = \frac{1.5}{200} = \frac{3}{400} \] # Summary - The value of \( k \) is \( \frac{1}{200} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{400} \).

# Given Information - The probability density function (PDF) of a random variable \( X \) is defined as: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: A function that describes the likelihood of a random variable taking on a particular value, where the total area under the PDF over its range must equal 1. - **Probability Calculation**: For continuous random variables, the probability over an interval can be computed as the integral of the PDF over that interval. # Solution To find the value of \( k \), we ensure that the total area under the PDF equals 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) = 0 \) outside the interval \([0, 2]\), we evaluate the integral over this range: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting this equal to 1 gives: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we calculate the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Calculating this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking it down further: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot \left[ 1.5 \right] = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - The probability density function (PDF) of a random variable \( X \) is defined as: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: A function that represents the likelihood of a random variable taking specific values. The total area under the PDF across its defined range must equal 1. - **Probability Calculation**: For continuous random variables, the probability for an interval is determined by integrating the PDF over that interval. # Solution To determine the value of \( k \), we need the total area under the PDF to equal 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) \) is zero outside \([0, 2]\), we focus on integrating over this range: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting this equal to 1 gives us: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we calculate the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Calculating this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking it down: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - The probability density function (PDF) of a random variable \( X \) is defined as: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: This function denotes the likelihood of a continuous random variable assuming a particular value, where the total area beneath the PDF must equal 1. - **Probability Calculation**: For continuous variables, the probability of falling within a specific range is given by integrating the PDF over that range. # Solution To determine the value of \( k \), we ensure that the area under the PDF sums to 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Given that \( f(x) = 0 \) outside the interval \([0, 2]\), we focus the integration within that range: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting this equation equal to 1 yields: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we calculate the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Calculating this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking down the calculation: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The correct choice is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This option pertains to mathematical problem-solving, which is often associated with algebra and calculus. While statistics may involve using equations for calculations, its primary focus is not on the act of solving equations but rather on the analysis of data gathered from various contexts. Therefore, this choice does not accurately represent the core function of statistics. - **Option B: Collecting, organizing, analyzing, and interpreting data** This option effectively encapsulates the essence of statistics. The discipline is fundamentally about the systematic approach to gathering data from various sources, such as surveys, experiments, or observational studies. Once data is collected, it must be organized in a way that facilitates analysis; this often involves summarizing it into tables, charts, or graphs. The next crucial step is the analysis of the data using various statistical methods to identify patterns, relationships, or trends. Finally, interpreting the results allows statisticians to draw meaningful conclusions and make informed decisions based on the analyzed data. This comprehensive view highlights the significance of statistics across multiple fields, including social sciences, business, healthcare, and engineering. - **Option C: Drawing geometric figures** This option is related to geometry, which focuses on the properties and relationships of shapes and spaces. While statistics can involve graphical representations of data (such as histograms or scatter plots), the act of drawing geometric figures is not a central concern of statistics. Thus, this option does not capture the primary focus of the statistical field. - **Option D: Studying planets** This choice refers to astronomy, which investigates celestial bodies, their movements, and various aspects of the universe. Although statistical techniques can be applied in astronomical research (e.g., analyzing star distributions or the characteristics of planets), the primary focus of statistics extends beyond celestial studies to encompass data analysis in a wide range of disciplines. Therefore, this option does not accurately reflect the main objective of statistics. # Summary In conclusion, statistics is primarily concerned with **collecting, organizing, analyzing, and interpreting data**. This thorough approach distinguishes it from the other options, which do not adequately encapsulate the core activities that define the field of statistics. Understanding this distinction is essential for recognizing the importance of statistics in making data-driven decisions across various disciplines, thereby underscoring its role in enhancing our understanding of complex information.

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The most appropriate answer is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This option focuses on the mathematical process of finding solutions to equations, which is primarily a feature of algebra and calculus. Although statistics may utilize equations in its analyses, its main objective is not centered on solving mathematical problems. Instead, statistics is concerned with the examination and interpretation of data. Therefore, this choice fails to capture the core function of statistics. - **Option B: Collecting, organizing, analyzing, and interpreting data** This option accurately represents the fundamental aspects of statistics. The field is dedicated to systematically gathering data through various means such as surveys, experiments, and observational studies. After collection, the data must be organized to facilitate analysis, often using formats like tables or graphs. The analysis stage involves applying statistical methods to discover trends, relationships, and patterns within the data. Finally, interpreting the results is crucial, as it allows statisticians to extract insights and make informed decisions based on their findings. This holistic approach underscores the importance of statistics across diverse fields, including business, healthcare, social sciences, and engineering. - **Option C: Drawing geometric figures** This choice pertains to the discipline of geometry, which is concerned with the properties and relations of shapes and spaces. While statistics does involve visual representations of data (like histograms and pie charts), the act of drawing geometric figures itself is not a primary focus of statistics. Consequently, this option does not accurately reflect the essence of statistical work. - **Option D: Studying planets** This option relates to the field of astronomy, which investigates celestial bodies and their movements. Although statistical methods can be applied in astronomy for data analysis (such as studying the distribution of stars), the main focus of statistics is not on celestial phenomena but rather on analyzing data across a wide range of disciplines. Thus, this option does not capture the core purpose of statistics. # Summary In summary, statistics is fundamentally concerned with **collecting, organizing, analyzing, and interpreting data**. This comprehensive framework distinguishes it from the other options, which fail to adequately describe the primary functions that define the field of statistics. Recognizing this distinction is vital for understanding the role of statistics in making informed, data-driven decisions across various sectors, emphasizing its significance in enhancing our comprehension of complex datasets.

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The most relevant choice is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This choice pertains to mathematical problem-solving techniques typically associated with algebra and calculus. While statistics may involve the use of equations in computations, its primary focus is not on solving these equations. Instead, statistics deals with the collection and interpretation of data, which does not align with the core purpose of this option. - **Option B: Collecting, organizing, analyzing, and interpreting data** This option accurately reflects the essence of statistics as a discipline. Statistics encompasses a systematic approach to data collection through various methods such as surveys, experiments, and observational studies. Once data is gathered, it must be organized in a way that facilitates analysis, often utilizing charts and tables. The analysis phase involves applying statistical techniques to uncover patterns, relationships, and trends within the data. The final step, interpretation, is crucial for translating findings into actionable insights that can inform decisions across various fields such as health, business, education, and social sciences. This comprehensive understanding highlights the integral role statistics plays in data-driven decision-making. - **Option C: Drawing geometric figures** This option is focused on the geometric discipline, which studies shapes, sizes, and properties of space. Although statistics can utilize visual data representation techniques (like graphs and charts), the actual process of drawing geometric figures is not central to statistics. Therefore, this choice does not accurately represent the primary functions of the field. - **Option D: Studying planets** This option refers to astronomy, which is the scientific study of celestial bodies and their movements. While statistical methods can be applied within astronomy (such as analyzing data from celestial observations), the primary aim of statistics is not limited to astronomical data analysis. Instead, statistics is concerned with data from a variety of disciplines, making this option less representative of its core purpose. # Summary In conclusion, statistics is fundamentally focused on **collecting, organizing, analyzing, and interpreting data**. This holistic approach sets it apart from the other options, which do not effectively capture the primary activities that define the field of statistics. Understanding the importance of this distinction is essential for recognizing the role of statistics in facilitating informed, data-driven decisions across diverse sectors, emphasizing its relevance in navigating complex information landscapes.

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The correct selection is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This option pertains to mathematical practices focused on finding solutions to equations. While statistics may utilize equations in its analyses, solving equations is not the primary function of statistics. Instead, statistics focuses on understanding and interpreting data collected from various sources, which makes this option insufficient to describe the field. - **Option B: Collecting, organizing, analyzing, and interpreting data** This option encapsulates the core functions of statistics. The discipline is dedicated to the systematic collection of data through various means, such as surveys, experiments, and observational studies. Following data collection, the next critical step is organizing the data to facilitate effective analysis, often by summarizing it in tables or graphical formats. The analysis phase employs various statistical techniques to identify trends, correlations, and patterns within the data. Finally, interpreting the results is crucial for deriving meaningful insights and making informed decisions based on the analyzed data. This comprehensive approach is what makes statistics vital across numerous domains, including healthcare, economics, social sciences, and engineering. - **Option C: Drawing geometric figures** This choice relates to geometry, which is primarily concerned with the properties and relationships of shapes and sizes. While statistics may involve visual representations of data (like charts or graphs), the specific act of drawing geometric figures is not a primary concern of the field. Therefore, this option does not accurately represent the scope of statistics. - **Option D: Studying planets** This option refers to the field of astronomy, which examines celestial bodies and their movements. Although statistical methods can be applied in astronomy to analyze data (for example, studying the distribution of stars), the primary focus of statistics is much broader and encompasses data analysis across a variety of disciplines. Hence, this option does not adequately capture the essence of statistics. # Summary In summary, statistics is fundamentally concerned with **collecting, organizing, analyzing, and interpreting data**. This comprehensive framework clearly distinguishes it from the other options, which fail to accurately describe the core activities that define the statistical field. Recognizing this distinction is essential for appreciating the role of statistics in facilitating informed, data-driven decisions across diverse sectors, thereby underscoring its importance in enhancing our understanding of complex datasets.

# Given Information - The probability density function (PDF) of a random variable \( X \) is defined as: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: A function that describes the likelihood of a continuous random variable taking specific values. The total area under the PDF must equal 1. - **Probability Calculation**: For continuous random variables, the probability of falling within a specific interval can be computed as the integral of the PDF over that interval. # Solution To find the value of \( k \), we need to ensure that the total area under the PDF equals 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) = 0 \) outside the interval \([0, 2]\), we focus on integrating over this range: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting this equal to 1 gives: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we compute the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Calculating this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking it down further: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - Number of original data points: \( n = 20 \) - Mean of the original data set: \( \mu = 50 \) - New number added to the data set: \( x = 80 \) # What To Find - The new mean of the data set after adding the new number. # Definitions or Concepts Used - **Mean**: The mean (average) of a data set is calculated as the sum of all data points divided by the number of data points: \[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \] # Solution First, calculate the total sum of the original 20 numbers using the mean: \[ \text{Sum}_{\text{original}} = \mu \times n = 50 \times 20 = 1000 \] Next, add the new number \( x \) to the sum: \[ \text{Sum}_{\text{new}} = \text{Sum}_{\text{original}} + x = 1000 + 80 = 1080 \] Now, calculate the new number of data points: \[ n_{\text{new}} = n + 1 = 20 + 1 = 21 \] Finally, calculate the new mean: \[ \text{Mean}_{\text{new}} = \frac{\text{Sum}_{\text{new}}}{n_{\text{new}}} = \frac{1080}{21} \approx 51.43 \] # Summary The new mean after adding the number \( 80 \) to the data set is approximately \( 51.43 \).

# Given Information - The total number of original data points: \( n = 20 \) - The mean of the original data set: \( \mu = 50 \) - The new number being added to the data set: \( x = 80 \) # What To Find - The updated mean of the data set after including the new number. # Definitions or Concepts Used - **Mean**: The mean (or average) of a set of numbers is defined as the total sum of all the numbers divided by the count of those numbers: \[ \text{Mean} = \frac{\text{Total Sum}}{\text{Count of Numbers}} \] # Solution To start, we will first calculate the total sum of the original 20 numbers using the provided mean: \[ \text{Total Sum}_{\text{original}} = \mu \times n = 50 \times 20 = 1000 \] Next, we will add the new number \( x \) to this sum: \[ \text{Total Sum}_{\text{new}} = \text{Total Sum}_{\text{original}} + x = 1000 + 80 = 1080 \] Now, we need to determine the new total count of the numbers after adding the new value: \[ n_{\text{new}} = n + 1 = 20 + 1 = 21 \] Finally, we can compute the new mean by dividing the updated total sum by the new count of numbers: \[ \text{Mean}_{\text{new}} = \frac{\text{Total Sum}_{\text{new}}}{n_{\text{new}}} = \frac{1080}{21} \approx 51.43 \] # Summary The updated mean after including the new number \( 80 \) into the data set is approximately \( 51.43 \).

# Given Information - Total count of existing numbers: \( n = 20 \) - Original mean of the data set: \( \mu = 50 \) - Newly added number to the data set: \( x = 80 \) # What To Find - The new mean after incorporating the new number. # Definitions or Concepts Used - **Mean**: The mean (or average) of a set of numbers is calculated by dividing the sum of all the values by the total number of values: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of values}} \] # Solution To begin, we first calculate the total sum of the original 20 numbers using the provided mean value: \[ \text{Total Sum}_{\text{original}} = \mu \times n = 50 \times 20 = 1000 \] Next, we need to incorporate the new number \( x \) into this sum: \[ \text{Total Sum}_{\text{new}} = \text{Total Sum}_{\text{original}} + x = 1000 + 80 = 1080 \] Following this, we determine the new total count of numbers after adding the new value: \[ n_{\text{new}} = n + 1 = 20 + 1 = 21 \] Finally, we calculate the new mean by dividing the updated total sum by the new total count: \[ \text{Mean}_{\text{new}} = \frac{\text{Total Sum}_{\text{new}}}{n_{\text{new}}} = \frac{1080}{21} \approx 51.43 \] # Summary The new mean, after adding the number \( 80 \) to the dataset, is approximately \( 51.43 \).

# Given Information - The original number of data points: \( n = 20 \) - The mean of these data points: \( \mu = 50 \) - The new number being added to the dataset: \( x = 80 \) # What To Find - The updated mean of the dataset after including the new number. # Definitions or Concepts Used - **Mean**: The mean, commonly referred to as the average, is calculated by dividing the total sum of all values by the number of values: \[ \text{Mean} = \frac{\text{Total Sum}}{\text{Count of Values}} \] # Solution First, we calculate the total sum of the original data points by using the information about the mean: \[ \text{Total Sum}_{\text{original}} = \mu \times n = 50 \times 20 = 1000 \] Next, we add the new value \( x \) to this total to find the new combined sum: \[ \text{Total Sum}_{\text{new}} = \text{Total Sum}_{\text{original}} + x = 1000 + 80 = 1080 \] To find the new total number of data points, we add 1 to the original count: \[ n_{\text{new}} = n + 1 = 20 + 1 = 21 \] Now we can calculate the new mean by dividing the updated total sum by the new count of data points: \[ \text{Mean}_{\text{new}} = \frac{\text{Total Sum}_{\text{new}}}{n_{\text{new}}} = \frac{1080}{21} \approx 51.43 \] # Summary The updated mean, after incorporating the new number \( 80 \) into the dataset, is approximately \( 51.43 \).

# Given Information - The initial count of data points is \( n = 20 \). - The mean of these data points is \( \mu = 50 \). - A new number added to the dataset is \( x = 80 \). # What To Find - The new mean of the dataset after including the additional number. # Definitions or Concepts Used - **Mean**: The mean (average) of a dataset is defined as the total sum of all values divided by the number of values: \[ \text{Mean} = \frac{\text{Total Sum}}{\text{Number of Values}} \] # Solution To start, we first determine the total sum of the original 20 numbers using the mean: \[ \text{Total Sum}_{\text{original}} = \mu \times n = 50 \times 20 = 1000 \] Next, we will add the new number \( x \) to the existing total sum: \[ \text{Total Sum}_{\text{new}} = \text{Total Sum}_{\text{original}} + x = 1000 + 80 = 1080 \] We then calculate the new count of data points after including the new number: \[ n_{\text{new}} = n + 1 = 20 + 1 = 21 \] Finally, we compute the new mean by dividing the updated total sum by the new count: \[ \text{Mean}_{\text{new}} = \frac{\text{Total Sum}_{\text{new}}}{n_{\text{new}}} = \frac{1080}{21} \approx 51.43 \] # Summary The new mean of the dataset, after adding the number \( 80 \), is approximately \( 51.43 \).

# Given Information - The total number of initial data points: \( n = 20 \) - The original mean of these data points: \( \mu = 50 \) - The new number being added to the dataset: \( x = 80 \) # What To Find - The updated mean of the dataset after including the new number. # Definitions or Concepts Used - **Mean**: The mean, or average, of a dataset is calculated by dividing the sum of all values by the count of those values: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of values}} \] # Solution To begin, we calculate the total sum of the original 20 numbers using the mean: \[ \text{Total Sum}_{\text{original}} = \mu \times n = 50 \times 20 = 1000 \] Next, we add the new number \( x \) to this total sum: \[ \text{Total Sum}_{\text{new}} = \text{Total Sum}_{\text{original}} + x = 1000 + 80 = 1080 \] Now, we need to determine the new total count of numbers after including the new value: \[ n_{\text{new}} = n + 1 = 20 + 1 = 21 \] Finally, we calculate the new mean by dividing the updated total sum by the new total count: \[ \text{Mean}_{\text{new}} = \frac{\text{Total Sum}_{\text{new}}}{n_{\text{new}}} = \frac{1080}{21} \approx 51.43 \] # Summary The updated mean after adding the number \( 80 \) to the dataset is approximately \( 51.43 \).

# Given Information - The initial count of data points: \( n = 20 \) - The original mean of these data points: \( \mu = 50 \) - The new number being added to the dataset: \( x = 80 \) # What To Find - The updated mean of the dataset after including the new number. # Definitions or Concepts Used - **Mean**: The mean, also known as the average, is a measure of central tendency that is calculated by dividing the sum of all values in a dataset by the total number of values: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of values}} \] # Solution To find the updated mean after adding a new number, we will first calculate the total sum of the original dataset using the information provided about the average. The total sum can be calculated as follows: \[ \text{Total Sum}_{\text{original}} = \mu \times n = 50 \times 20 = 1000 \] Next, we need to incorporate the new number \( x \) into this total sum. So, we add \( 80 \) to the original total: \[ \text{Total Sum}_{\text{new}} = \text{Total Sum}_{\text{original}} + x = 1000 + 80 = 1080 \] Now that we have the new total sum, we should also update the count of data points. Since we are adding one new number to the original set, the new total number of data points will be: \[ n_{\text{new}} = n + 1 = 20 + 1 = 21 \] With the new total sum and the updated count of data points, we can now compute the new mean. This is done by dividing the updated total sum by the new count of data points: \[ \text{Mean}_{\text{new}} = \frac{\text{Total Sum}_{\text{new}}}{n_{\text{new}}} = \frac{1080}{21} \] Calculating this division: \[ \text{Mean}_{\text{new}} \approx 51.42857 \] For practical purposes, we can round this value to two decimal places, resulting in: \[ \text{Mean}_{\text{new}} \approx 51.43 \] # Summary The updated mean after including the number \( 80 \) into the dataset is approximately \( 51.43 \). This calculation illustrates how the addition of a new data point can influence the overall average, showing the importance of understanding averages in data analysis.

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The most accurate choice is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This option emphasizes mathematical problem-solving, which is primarily associated with algebra and calculus. While statistics may involve equations in its analyses, its main objective is not focused on solving mathematical equations. Instead, statistics is dedicated to the systematic study of data and its implications. Therefore, this option does not correctly reflect the core purpose of statistics. - **Option B: Collecting, organizing, analyzing, and interpreting data** This option effectively captures the essence of the field of statistics. Statistics encompasses a wide range of activities that begin with the **collection of data**, which can be performed through surveys, experiments, or observational methods. Once the data is collected, the next step is to **organize** it for clarity, often using tables or graphical representations. The subsequent step involves **analyzing** the data using various statistical techniques to identify trends, relationships, or significant patterns. Finally, **interpreting** the data is crucial for drawing meaningful conclusions and making informed decisions based on the findings. This comprehensive and methodical approach underscores the importance of statistics across numerous domains, including healthcare, business, social sciences, and education. - **Option C: Drawing geometric figures** This option pertains to geometry, which focuses on the properties and relationships of shapes and spaces. Although statistics may utilize graphical tools for visualizing data (like bar graphs or scatter plots), the act of drawing geometric figures is not a central concern of statistics itself. As such, this option does not appropriately represent the focus of the field. - **Option D: Studying planets** This choice relates to astronomy, which investigates celestial bodies and their movements. While statistical methods can be applied in astronomical research (for instance, analyzing star distributions), the main focus of statistics is not limited to celestial phenomena but rather encompasses a broad spectrum of data analysis across various disciplines. Hence, this option does not accurately encapsulate the core purpose of statistics. # Summary In conclusion, statistics is fundamentally concerned with **collecting, organizing, analyzing, and interpreting data**. This holistic perspective clearly distinguishes it from the other options, which do not adequately encapsulate the primary functions that define the field of statistics. Recognizing this distinction is vital for understanding the role of statistics in various applications and its significance in informing decisions based on data.

# Question What is Statistics mainly concerned with? - A) Solving equations - B) Collecting, organizing, analyzing, and interpreting data - C) Drawing geometric figures - D) Studying planets # Answer The most fitting answer is **B) Collecting, organizing, analyzing, and interpreting data**. # Explanation - **Option A: Solving equations** This choice highlights the mathematical aspect of finding solutions to equations, which is mainly associated with fields like algebra or calculus. While statistics may involve mathematical formulas, its primary focus is not on solving equations per se. Rather, statistics is concerned with the systematic analysis of data to derive meaningful insights. Therefore, this option does not accurately represent the essence of statistics. - **Option B: Collecting, organizing, analyzing, and interpreting data** This option concisely encapsulates the core functions of statistics. The discipline starts with **data collection**, which can be conducted through various methods such as surveys, experiments, or observational studies. After gathering the data, it must be **organized** to facilitate analysis, often presented in formats like tables or graphs. The next phase entails **analyzing** the data using statistical methods to uncover trends, patterns, or relationships. Finally, the process culminates in **interpreting** the results, which allows statisticians to draw conclusions and make informed decisions based on the data. This comprehensive approach is essential across a multitude of disciplines, including healthcare, economics, and social sciences. - **Option C: Drawing geometric figures** This option pertains to geometry, a field focused on the properties and relationships of shapes and spaces. While statistics may involve visual data representation (like pie charts or histograms), the act of drawing geometric figures is not a central aspect of statistics. Consequently, this option does not adequately capture the primary focus of the field. - **Option D: Studying planets** This choice refers to astronomy, which investigates celestial bodies and their movements. Although statistical methods can be employed in astronomical research (for example, analyzing the distribution of stars), the primary focus of statistics extends beyond celestial phenomena to include data analysis across various fields. Thus, this option does not accurately reflect the main objectives of statistics. # Summary In summary, statistics is fundamentally focused on **collecting, organizing, analyzing, and interpreting data**. This comprehensive understanding differentiates it from the other options, which do not effectively represent the primary activities that define the field of statistics. Recognizing this distinction is crucial for appreciating the importance of statistics in data-driven decision-making across diverse sectors.

# Given Information - Number of participants in Group A: \( n_A = 55 \) - Number of participants in Group B: \( n_B = 65 \) # What To Find - Degrees of freedom for an independent samples t-test. # Definition or Concept Used - **Degrees of Freedom (df)** for an independent samples t-test is calculated using the formula: \[ df = n_A + n_B - 2 \] where \( n_A \) is the number of participants in Group A and \( n_B \) is the number of participants in Group B. # Solution Substituting the values of participants into the formula: \[ df = 55 + 65 - 2 \] Calculating: \[ df = 120 - 2 = 118 \] # Summary The degrees of freedom for the independent samples t-test is \( 118 \).

# Given Information - Number of participants in Group A: \( n_A = 55 \) - Number of participants in Group B: \( n_B = 65 \) # What To Find - The degrees of freedom (df) for an independent samples t-test. # Definition or Concept Used - **Degrees of Freedom (df)** for an independent samples t-test is determined by the formula: \[ df = n_A + n_B - 2 \] where \( n_A \) represents the number of participants in Group A and \( n_B \) represents the number of participants in Group B. # Solution To calculate the degrees of freedom, we substitute the given values into the formula: \[ df = 55 + 65 - 2 \] Now, performing the arithmetic: \[ df = 120 - 2 = 118 \] # Summary The degrees of freedom for the independent samples t-test is \( 118 \).

# Given Information - Size of Group A: \( n_A = 55 \) - Size of Group B: \( n_B = 65 \) # What To Find - The degrees of freedom (df) for the independent samples t-test. # Definition or Concept Used - **Degrees of Freedom (df)** in an independent samples t-test is calculated using the formula: \[ df = n_A + n_B - 2 \] where \( n_A \) is the number of participants in Group A and \( n_B \) is the number of participants in Group B. # Solution To find the degrees of freedom, we will plug the values of the number of participants into the formula: \[ df = 55 + 65 - 2 \] Calculating this gives: \[ df = 120 - 2 = 118 \] # Summary For the independent samples t-test, the degrees of freedom is \( 118 \).

# Given Information - The probability density function (PDF) of a random variable \( X \) is defined as: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: A function that describes the likelihood of a continuous random variable falling within a particular range. The total area under the PDF curve must equal 1. - **Probability Calculation**: For continuous random variables, the probability of an interval can be computed by integrating the PDF over that specified interval. # Solution To find the value of \( k \), we start by ensuring that the total area under the PDF equals 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) = 0 \) outside the interval \([0, 2]\), we will integrate over this range: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting this equal to 1: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we calculate the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Calculating this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking it down: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - The probability density function (PDF) for a random variable \( X \) is expressed as: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: A function that illustrates the likelihood of a continuous random variable falling within a specific range. The integral of the PDF over its entire range must equal 1. - **Probability Calculation**: For continuous variables, the probability of a specific interval can be determined by integrating the PDF over that interval. # Solution To determine \( k \), we need to ensure that the total area under the PDF is equal to 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] As \( f(x) = 0 \) outside the interval \([0, 2]\), we restrict our integration to that range: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting this integral equal to 1 gives us: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we will calculate the probability for the interval \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Calculating the integral over the specified range: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking this down further: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - The probability density function (PDF) of a random variable \( X \) is defined as: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: A function that represents the likelihood of a continuous random variable taking on specific values. The total area under the PDF curve across its defined range must equal 1. - **Probability Calculation**: For continuous random variables, the probability of an interval can be calculated by integrating the PDF over that interval. # Solution To determine the value of \( k \), we must ensure that the total area under the PDF is equal to 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) = 0 \) outside the interval \([0, 2]\), we focus our integration within this range: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral, we have: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting this equal to 1 gives: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we calculate the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Calculating this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking it down further: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - The probability density function (PDF) for the random variable \( X \) is defined as: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: This function describes the likelihood of a continuous random variable taking a value within a specific interval. The integral of the PDF across its defined range must equal 1. - **Probability Calculation**: For continuous random variables, the probability of falling within a certain interval can be computed by integrating the PDF over that interval. # Solution To find the value of \( k \), we need to ensure that the total area under the PDF equals 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) = 0 \) for values outside the interval \([0, 2]\), we can restrict our integration to this range: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral gives: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting the area equal to 1: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, let's calculate the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Performing this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Calculating further: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - The probability density function (PDF) of a random variable \( X \) is defined as: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: This function characterizes the likelihood of a continuous random variable falling within a specific range. The total area under the PDF curve over its defined range must equal 1. - **Probability Calculation**: The probability of a continuous random variable being within a specific interval can be determined by calculating the integral of the PDF over that interval. # Solution To determine \( k \), we need the total area under the PDF to equal 1. Hence, we evaluate the integral: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) = 0 \) outside the interval \([0, 2]\), we can restrict our integration to this range: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral gives: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting this equal to 1 leads to: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we proceed to calculate the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Now we evaluate this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking it down further gives: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - The probability density function (PDF) for the random variable \( X \) is defined as follows: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: This function indicates the likelihood of a continuous random variable taking on specific values. The integral of the PDF over its entire range must equal 1. - **Probability Calculation**: For continuous random variables, the probability of falling within a certain interval is found by integrating the PDF over that interval. # Solution To find the value of \( k \), we need to ensure that the total area under the PDF equals 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) = 0 \) outside the range \([0, 2]\), we can restrict our integration to this interval: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot 2 \] Setting this equal to 1 gives us: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we will compute the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Now, let's evaluate this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking it down further: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - The probability density function (PDF) for the random variable \( X \) is defined as follows: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The value of \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: This function describes the probability of a continuous random variable taking on particular values, where the total area under the PDF curve must equal 1. - **Probability Calculation**: For continuous random variables, the probability of being in a specific interval is obtained by integrating the PDF over that interval. # Solution To determine \( k \), we start with the requirement that the total area under the PDF be equal to 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) \) is zero outside the interval \([0, 2]\), we will focus our integration on this segment: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral, we have: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting this equal to 1 provides: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we will calculate the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Now, we compute this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking it down step by step: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - The probability density function (PDF) of the random variable \( X \) is given by: \[ f(x) = \begin{cases} k \cdot x & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] # What To Find - The constant \( k \). - The probability \( P(1 \leq x \leq 2) \). # Definitions or Concepts Used - **Probability Density Function (PDF)**: This function indicates the likelihood of a continuous random variable taking on a specific value. The total area under the PDF curve must equal 1. - **Probability Calculation**: For continuous random variables, the probability of falling within a specific range can be calculated by integrating the PDF over that range. # Solution To find the value of \( k \), we need the total area under the PDF to equal 1: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Since \( f(x) = 0 \) outside the interval \([0, 2]\), we will focus our integration on this interval: \[ \int_{0}^{2} k \cdot x \, dx = 1 \] Calculating the integral: \[ \int_{0}^{2} k \cdot x \, dx = k \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = k \cdot \left[ \frac{2^2}{2} - 0 \right] = k \cdot \left[ \frac{4}{2} \right] = k \cdot 2 \] Setting this equal to 1 gives: \[ k \cdot 2 = 1 \implies k = \frac{1}{2} \] Next, we will determine the probability \( P(1 \leq x \leq 2) \): \[ P(1 \leq x \leq 2) = \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx \] Now, we compute this integral: \[ \int_{1}^{2} \left(\frac{1}{2} \cdot x\right) \, dx = \frac{1}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{1}{2} \cdot \left[ \frac{2^2}{2} - \frac{1^2}{2} \right] \] Breaking down the calculation: \[ = \frac{1}{2} \cdot \left[ \frac{4}{2} - \frac{1}{2} \right] = \frac{1}{2} \cdot \left[ 2 - 0.5 \right] = \frac{1}{2} \cdot 1.5 = \frac{1.5}{2} = \frac{3}{4} \] # Summary - The value of \( k \) is \( \frac{1}{2} \). - The probability \( P(1 \leq x \leq 2) \) is \( \frac{3}{4} \).

# Given Information - Marks in Subject X: - \( X = [40, 50, 60, 70, 80] \) - Marks in Subject Y: - \( Y = [42, 55, 63, 68, 75] \) # What To Find - The regression equation of \( Y \) on \( X \). # Definitions or Concepts Used - **Linear Regression Equation**: The regression line can be expressed in the form: \[ Y = bX + a \] where \( b \) is the slope and \( a \) is the y-intercept. - **Slope Calculation**: The slope \( b \) is calculated using the formula: \[ b = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum X^2) - (\sum X)^2} \] - **Intercept Calculation**: The y-intercept \( a \) can be calculated using: \[ a = \frac{\sum Y - b(\sum X)}{n} \] # Solution 1. **Calculate Required Sums**: - Number of data points \( n = 5 \) - \( \sum X = 40 + 50 + 60 + 70 + 80 = 300 \) - \( \sum Y = 42 + 55 + 63 + 68 + 75 = 303 \) - \( \sum XY = (40 \cdot 42) + (50 \cdot 55) + (60 \cdot 63) + (70 \cdot 68) + (80 \cdot 75) \) - \( = 1680 + 2750 + 3780 + 4760 + 6000 = 18970 \) - \( \sum X^2 = 40^2 + 50^2 + 60^2 + 70^2 + 80^2 \) - \( = 1600 + 2500 + 3600 + 4900 + 6400 = 21000 \) 2. **Calculate Slope \( b \)**: \[ b = \frac{5(18970) - (300)(303)}{5(21000) - (300)^2} \] Calculating the numerator: \[ 5 \cdot 18970 = 94850 \] \[ 300 \cdot 303 = 90900 \] \[ \text{Numerator} = 94850 - 90900 = 3950 \] Calculating the denominator: \[ 5 \cdot 21000 = 105000 \] \[ 300^2 = 90000 \] \[ \text{Denominator} = 105000 - 90000 = 15000 \] Now calculating \( b \): \[ b = \frac{3950}{15000} \approx 0.2633 \] 3. **Calculate Intercept \( a \)**: \[ a = \frac{\sum Y - b(\sum X)}{n} \] Substituting the values: \[ a = \frac{303 - 0.2633(300)}{5} \] Calculating \( b(\sum X) \): \[ 0.2633 \cdot 300 \approx 78.99 \] Now substituting: \[ a = \frac{303 - 78.99}{5} \approx \frac{224.01}{5} \approx 44.802 \] 4. **Final Regression Equation**: Putting values of \( a \) and \( b \) into the regression equation: \[ Y = 0.2633X + 44.802 \] ### Closest Option The closest option to our calculated regression equation is not explicitly listed, but if we round \( b \) and \( a \), we might find a match with one of the provided options. # Summary The regression equation of \( Y \) on \( X \) is approximately: \[ Y \approx 0.2633X + 44.802 \] None of the provided options exactly match this result. Further approximation or rounding may be necessary to find the closest choice.

# Given Information - Marks in Subject X: - \( X = [40, 50, 60, 70, 80] \) - Marks in Subject Y: - \( Y = [42, 55, 63, 68, 75] \) # What To Find - The regression equation of \( Y \) on \( X \). # Definitions or Concepts Used - **Linear Regression Equation**: The regression line can be expressed in the form: \[ Y = a + bX \] where \( b \) is the slope and \( a \) is the y-intercept. - **Slope Calculation**: The slope \( b \) is calculated using: \[ b = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2} \] - **Intercept Calculation**: The y-intercept \( a \) can be calculated as: \[ a = \bar{y} - b\bar{x} \] # Solution ### Step 1: Calculate Means Calculate the mean of \( X \) and \( Y \). For \( X \): \[ \bar{x} = \frac{40 + 50 + 60 + 70 + 80}{5} = \frac{300}{5} = 60 \] For \( Y \): \[ \bar{y} = \frac{42 + 55 + 63 + 68 + 75}{5} = \frac{303}{5} = 60.6 \] ### Step 2: Compute Values Compute the required values for the slope calculation: | \( X \) | \( Y \) | \( x - \bar{x} \) | \( y - \bar{y} \) | Product | Square | |---------|---------|--------------------|--------------------|-----------|----------| | 40 | 42 | -20 | -18.6 | 372 | 400 | | 50 | 55 | -10 | -5.6 | 56 | 100 | | 60 | 63 | 0 | 2.4 | 0 | 0 | | 70 | 68 | 10 | 7.4 | 74 | 100 | | 80 | 75 | 20 | 14.4 | 288 | 400 | Now calculate the sums: \[ \sum (x - \bar{x})(y - \bar{y}) = 372 + 56 + 0 + 74 + 288 = 790 \] \[ \sum (x - \bar{x})^2 = 400 + 100 + 0 + 100 + 400 = 1000 \] ### Step 3: Calculate Slope \( b \) Using the slope formula: \[ b = \frac{790}{1000} = 0.79 \] ### Step 4: Calculate Intercept \( a \) Using the intercept formula: \[ a = \bar{y} - b \bar{x} = 60.6 - (0.79 \times 60) = 60.6 - 47.4 = 13.2 \] ### Step 5: Write the Regression Equation Putting it all together, the regression equation is: \[ Y = 0.79X + 13.2 \] # Summary The regression equation of \( Y \) on \( X \) is: \[ Y = 0.79X + 13.2 \]

# Given Information The frequency distribution of defective items is as follows: | Defects | Frequency | |---------|-----------| | 0 | 25 | | 1 | 40 | | 2 | 20 | | 3 | 10 | | 4 | 5 | # What To Find - The approximate variance of the distribution. # Definitions or Concepts Used - **Mean (\(\mu\))**: The average value calculated as: \[ \mu = \frac{\sum (x \cdot f)}{N} \] where \( x \) is the number of defects, \( f \) is the frequency, and \( N \) is the total frequency. - **Variance (\( \sigma^2 \))**: This measures the dispersion of the data. It is calculated using: \[ \sigma^2 = \frac{\sum (f \cdot (x - \mu)^2)}{N} \] # Solution ### Step 1: Calculate Total Frequency \( N \) \[ N = 25 + 40 + 20 + 10 + 5 = 100 \] ### Step 2: Calculate the Mean (\(\mu\)) First, calculate \( \sum (x \cdot f) \): \[ \sum (x \cdot f) = (0 \cdot 25) + (1 \cdot 40) + (2 \cdot 20) + (3 \cdot 10) + (4 \cdot 5) \] \[ = 0 + 40 + 40 + 30 + 20 = 130 \] Now, calculate the mean: \[ \mu = \frac{130}{100} = 1.3 \] ### Step 3: Calculate Variance (\( \sigma^2 \)) Now calculate \( \sum (f \cdot (x - \mu)^2) \): - For \( x = 0 \): \[ (0 - 1.3)^2 = 1.69 \quad \Rightarrow \quad 25 \cdot 1.69 = 42.25 \] - For \( x = 1 \): \[ (1 - 1.3)^2 = 0.09 \quad \Rightarrow \quad 40 \cdot 0.09 = 3.6 \] - For \( x = 2 \): \[ (2 - 1.3)^2 = 0.49 \quad \Rightarrow \quad 20 \cdot 0.49 = 9.8 \] - For \( x = 3 \): \[ (3 - 1.3)^2 = 2.89 \quad \Rightarrow \quad 10 \cdot 2.89 = 28.9 \] - For \( x = 4 \): \[ (4 - 1.3)^2 = 7.29 \quad \Rightarrow \quad 5 \cdot 7.29 = 36.45 \] Now sum these values: \[ \sum (f \cdot (x - \mu)^2) = 42.25 + 3.6 + 9.8 + 28.9 + 36.45 = 120 \] Finally, calculate the variance: \[ \sigma^2 = \frac{120}{100} = 1.2 \] # Summary The approximate variance of the distribution is \( 1.2 \). The closest answer from the options provided is: **B) 1.24**.

# Given Information - Average of 5 numbers: \( \text{Average} = 18 \) - Number of values: \( n = 5 \) # What To Find - The total sum of the 5 numbers. # Definitions or Concepts Used - **Average**: The average (or mean) is calculated using the formula: \[ \text{Average} = \frac{\text{Total Sum}}{n} \] Where \( n \) is the number of values. # Solution To find the total sum, we can rearrange the average formula: \[ \text{Total Sum} = \text{Average} \times n \] Substituting the known values: \[ \text{Total Sum} = 18 \times 5 \] Calculating this gives: \[ \text{Total Sum} = 90 \] # Summary The total sum of the 5 numbers is \( 90 \). The correct answer is **B) 90**.

# Given Information - Probability of a component being defective: \( p = 0.06 \) (or \( 6\% \)) - Probability of a component being non-defective: \( q = 1 - p = 0.94 \) - Number of components selected: \( n = 12 \) - Number of defective components we want to find the probability for: \( k = 2 \) # What To Find - The probability that exactly two components are defective. # Definitions or Concepts Used - **Binomial Distribution**: The probability of obtaining exactly \( k \) successes in \( n \) trials is given by the formula: \[ P(X = k) = \binom{n}{k} p^k q^{n-k} \] where: - \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \) - \( p \) is the probability of success (defective component) - \( q \) is the probability of failure (non-defective component) - \( n \) is the total number of trials - \( k \) is the number of successes # Solution 1. **Calculate the Binomial Coefficient**: \[ \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \] 2. **Calculate the Probability**: Using the binomial formula: \[ P(X = 2) = \binom{12}{2} p^2 q^{12-2} \] Substituting the values: \[ P(X = 2) = 66 \cdot (0.06)^2 \cdot (0.94)^{10} \] 3. **Calculating \( p^2 \) and \( q^{10} \)**: \[ (0.06)^2 = 0.0036 \] \[ (0.94)^{10} \approx 0.5277 \quad (\text{Using a calculator}) \] 4. **Final Calculation**: \[ P(X = 2) = 66 \cdot 0.0036 \cdot 0.5277 \approx 66 \cdot 0.00190032 \approx 0.125 \] 5. **Refining**: Multiplying out: \[ 66 \cdot 0.0036 \cdot 0.5277 \approx 0.125 \text{ (as computed above)} \] # Summary The probability that exactly two components are defective is approximately \( 0.125 \). From the options provided, the closest match is **B) 0.212**.