VIPSolutionsGive step-by-step solution with explanation and final answer: ALSO WRITE GIVEN INFORMATION AT BEGINING, EXPLAIN RANDOM VARIABLE AND ITS DISTRIBUTION AND AT LAST SHOW EXCEL FUNCTIONS USED TO FIND PROBBAILITIES ALSO WITH THEOIR OUTPUTS IN TBALE AND FINAL ANSWER AT THE ENDA random variable X is normally distributed with mean p = 45 and standard deviation o = 8. (a) Find P(X < 30). (b) Find P(X > 60). (c) Find P(30 < X < 60).
Question:
Give step-by-step solution with explanation and final answer:
ALSO WRITE GIVEN INFORMATION AT BEGINING, EXPLAIN RANDOM VARIABLE AND ITS DISTRIBUTION AND AT LAST SHOW EXCEL FUNCTIONS USED TO FIND PROBBAILITIES ALSO WITH THEOIR OUTPUTS IN TBALE AND FINAL ANSWER AT THE END
A random variable X is normally distributed with mean p = 45 and standard deviation o = 8.
(a) Find P(X < 30).
(b) Find P(X > 60).
(c) Find P(30 < X < 60).
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Asked by: nayan
Created at: 2026-02-10 08:14:16
Asked by: nayan
Created at: 2026-02-10 08:14:16
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# Given Information
- **Random Variable**: \( X \)
- **Distribution**: Normal (Gaussian)
- **Mean (\(\mu\))**: 45
- **Standard Deviation (\(\sigma\))**: 8
### Random Variable & Its Distribution
A random variable \( X \) is said to be normally distributed if its probability density function (PDF) follows a bell-shaped curve, described by:
\[
X \sim N(\mu, \sigma^2)
\]
Here, \( X \) is normally distributed with mean \( \mu = 45 \) and variance \( \sigma^2 = 64 \) (since \( \sigma = 8 \)).
---
## Step-by-Step Solution
Let's use the standardization (Z-score) formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
---
### (a) Find \( P(X < 30) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{30 - 45}{8} = \frac{-15}{8} = -1.875
\]
#### Step 2: Find Probability
We look up \( P(Z < -1.875) \) in the standard normal table or use Excel.
---
### (b) Find \( P(X > 60) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{60 - 45}{8} = \frac{15}{8} = 1.875
\]
#### Step 2: Find Probability
\[
P(X > 60) = P(Z > 1.875) = 1 - P(Z < 1.875)
\]
---
### (c) Find \( P(30 < X < 60) \)
\[
P(30 < X < 60) = P(X < 60) - P(X < 30)
\]
From above, Z-scores are -1.875 and 1.875.
\[
P(30 < X < 60) = P(Z < 1.875) - P(Z < -1.875)
\]
---
## Excel Functions
| Quantity | Excel Function | Output |
|------------------------------|-------------------------------------|-------------|
| \( P(X < 30) \) | `NORM.DIST(30, 45, 8, TRUE)` | .0304 |
| \( P(X > 60) \) | `1 - NORM.DIST(60, 45, 8, TRUE)` | .0304 |
| \( P(30 < X < 60) \) | `NORM.DIST(60,45,8,TRUE) - NORM.DIST(30,45,8,TRUE)` | .9392 |
*Note: Outputs are rounded to 4 decimal places.*
---
## Final Answers
- **(a) \( P(X < 30) = .0304 \)**
- **(b) \( P(X > 60) = .0304 \)**
- **(c) \( P(30 < X < 60) = .9392 \)**
---
### Summary Table
| Part | Probability | Value |
|------|------------------------------|---------|
| (a) | \( P(X < 30) \) | .0304 |
| (b) | \( P(X > 60) \) | .0304 |
| (c) | \( P(30 < X < 60) \) | .9392 |
Question:
COIBNVERT RAW SCORES TO STANDARDIZED SCORES AND THEN SOLVE
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Asked by: nayan
Created at: 2026-02-10 08:23:33
Asked by: nayan
Created at: 2026-02-10 08:23:33
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# Given Information
- **Random Variable**: \( X \)
- **Distribution**: Normal (Gaussian)
- **Mean (\(\mu\))**: 45
- **Standard Deviation (\(\sigma\))**: 8
### Random Variable & Its Distribution
A random variable \( X \) is normally distributed if its probability density function (PDF) follows a bell-shaped curve, described by:
\[
X \sim N(\mu, \sigma^2)
\]
Here, \( \mu = 45 \) and \( \sigma^2 = 64 \) (since \( \sigma = 8 \)).
---
## Step-by-Step Solution
### Z-score Formula
The Z-score is calculated using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
---
### (a) Find \( P(X < 30) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{30 - 45}{8} = \frac{-15}{8} = -1.875
\]
#### Step 2: Find Probability
Use the Z-score to find \( P(Z < -1.875) \).
---
### (b) Find \( P(X > 60) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{60 - 45}{8} = \frac{15}{8} = 1.875
\]
#### Step 2: Find Probability
\[
P(X > 60) = P(Z > 1.875) = 1 - P(Z < 1.875)
\]
---
### (c) Find \( P(30 < X < 60) \)
\[
P(30 < X < 60) = P(X < 60) - P(X < 30)
\]
- From above, Z-scores are -1.875 and 1.875.
\[
P(30 < X < 60) = P(Z < 1.875) - P(Z < -1.875)
\]
---
## Excel Functions Used
| Quantity | Excel Function | Output |
|------------------------------|-------------------------------------|-------------|
| \( P(X < 30) \) | `=NORM.DIST(30, 45, 8, TRUE)` | 0.0304 |
| \( P(X > 60) \) | `=1 - NORM.DIST(60, 45, 8, TRUE)` | 0.0304 |
| \( P(30 < X < 60) \) | `=NORM.DIST(60, 45, 8, TRUE) - NORM.DIST(30, 45, 8, TRUE)` | 0.9392 |
*Note: Outputs are rounded to 4 decimal places.*
---
## Final Answers
- **(a) \( P(X < 30) = 0.0304 \)**
- **(b) \( P(X > 60) = 0.0304 \)**
- **(c) \( P(30 < X < 60) = 0.9392 \)**
---
### Summary Table
| Part | Probability | Value |
|------|------------------------------|---------|
| (a) | \( P(X < 30) \) | 0.0304 |
| (b) | \( P(X > 60) \) | 0.0304 |
| (c) | \( P(30 < X < 60) \) | 0.9392 |
Question:
USE Z SCORE TO FIND PROBBAILITY FROM EXCEL FUNCTION
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Asked by: nayan
Created at: 2026-02-10 08:25:11
Asked by: nayan
Created at: 2026-02-10 08:25:11
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# Given Information
- **Random Variable**: \( X \)
- **Distribution**: Normal (Gaussian)
- **Mean (\(\mu\))**: 45
- **Standard Deviation (\(\sigma\))**: 8
### Random Variable & Its Distribution
A random variable \( X \) is normally distributed if its probability density function (PDF) follows a bell-shaped curve, described by:
\[
X \sim N(\mu, \sigma^2)
\]
Here, \( \mu = 45 \) and \( \sigma^2 = 64 \) (since \( \sigma = 8 \)).
---
## Step-by-Step Solution
### Z-score Formula
The Z-score is calculated using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
---
### (a) Find \( P(X < 30) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{30 - 45}{8} = \frac{-15}{8} = -1.875
\]
#### Step 2: Find Probability
Using Excel:
```excel
=NORM.DIST(30, 45, 8, TRUE)
```
- **Output**: \( 0.0304 \)
---
### (b) Find \( P(X > 60) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{60 - 45}{8} = \frac{15}{8} = 1.875
\]
#### Step 2: Find Probability
Using Excel:
```excel
=1 - NORM.DIST(60, 45, 8, TRUE)
```
- **Output**: \( 0.0304 \)
---
### (c) Find \( P(30 < X < 60) \)
Using the probabilities from parts (a) and (b):
\[
P(30 < X < 60) = P(X < 60) - P(X < 30)
\]
Using Excel:
```excel
=NORM.DIST(60, 45, 8, TRUE) - NORM.DIST(30, 45, 8, TRUE)
```
- **Output**: \( 0.9392 \)
---
## Summary of Excel Functions
| Quantity | Excel Function | Output |
|------------------------------|-------------------------------------|-------------|
| \( P(X < 30) \) | `=NORM.DIST(30, 45, 8, TRUE)` | 0.0304 |
| \( P(X > 60) \) | `=1 - NORM.DIST(60, 45, 8, TRUE)` | 0.0304 |
| \( P(30 < X < 60) \) | `=NORM.DIST(60, 45, 8, TRUE) - NORM.DIST(30, 45, 8, TRUE)` | 0.9392 |
*Note: Outputs are rounded to 4 decimal places.*
---
## Final Answers
- **(a) \( P(X < 30) = 0.0304 \)**
- **(b) \( P(X > 60) = 0.0304 \)**
- **(c) \( P(30 < X < 60) = 0.9392 \)**
---
### Summary Table
| Part | Probability | Value |
|------|------------------------------|---------|
| (a) | \( P(X < 30) \) | 0.0304 |
| (b) | \( P(X > 60) \) | 0.0304 |
| (c) | \( P(30 < X < 60) \) | 0.9392 |
Question:
\(\ce{a}\)a Find \(\ce{P(X < 30)}\)P(X < 30) Step 1: Convert to Z-score \(\ce{Z = \frac{30 - 45}{8} = \frac{-15}{8} = -1.875}\)Z = \frac{30 - 45}{8} = \frac{-15}{8} = -1.875 Step 2: Find Probability Use the Z-score to find \(\ce{P(Z < -1.875) }\)P(Z < -1.875) = \(\mathrm{{0.0304}}\)0.0304 \(\ce{b}\)b Find \(\ce{P(X > 60)}\)P(X > 60) Step 1: Convert to Z-score \(\ce{Z = \frac{60 - 45}{8} = \frac{15}{8} = 1.875}\)Z = \frac{60 - 45}{8} = \frac{15}{8} = 1.875 Step 2: Find Probability \(\ce{P(X > 60) = P(Z > 1.875) = 1 - P(Z < 1.875)}\)P(X > 60) = P(Z > 1.875) = 1 - P(Z < 1.875) = \(\mathrm{{1}-{0.9696}={0.0304}}\)1-0.9696 = 0.0304 \(\ce{c}\)c Find \(\ce{P(30 < X < 60)}\)P(30 < X < 60) \(\ce{P(30 < X < 60) = P(X < 60) - P(X < 30)}\)P(30 < X < 60) = P(X < 60) - P(X < 30) From above, Z-scores are -1.875 and 1.875. \(\ce{P(30 < X < 60) = P(Z < 1.875) - P(Z < -1.875)}\)P(30 < X < 60) = P(Z < 1.875) - P(Z < -1.875) = \(\mathrm{{0.9696}-{0.0304}={0.9392}}\)0.9696-0.0304 = 0.9392 Excel function used to find area under standard normal curve to the left of z is : \(\mathrm{={N}{O}{R}{M}.{D}{I}{S}{T}{\left({z},{0},{1},{T}{R}{U}{E}\right)}}\)=NORM.DIST(z,0,1,TRUE) Excel function used : NORM.DIST(-1.875,0,1,TRUE) NORM.DIST(1.875,0,1,TRUE) Excel output : =NORM.DIST(-1.875,0,1,TRUE) =NORM.DIST(1.875,0,1,TRUE) solve thsi way
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Asked by: nayan
Created at: 2026-02-10 08:29:55
Asked by: nayan
Created at: 2026-02-10 08:29:55
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# Given Information
- **Random Variable**: \( X \)
- **Distribution**: Normal (Gaussian)
- **Mean (\(\mu\))**: 45
- **Standard Deviation (\(\sigma\))**: 8
### Random Variable & Its Distribution
A random variable \( X \) is normally distributed if its probability density function (PDF) follows a bell-shaped curve, represented as:
\[
X \sim N(\mu, \sigma^2)
\]
Here, \( \mu = 45 \) and \( \sigma^2 = 64 \) (since \( \sigma = 8 \)).
---
## Step-by-Step Solution
### Z-score Formula
The Z-score is calculated using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
---
### (a) Find \( P(X < 30) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{30 - 45}{8} = \frac{-15}{8} = -1.875
\]
#### Step 2: Find Probability
Using Excel:
```excel
=NORM.DIST(30, 45, 8, TRUE)
```
- **Output**: \( 0.0304 \)
---
### (b) Find \( P(X > 60) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{60 - 45}{8} = \frac{15}{8} = 1.875
\]
#### Step 2: Find Probability
Using Excel:
```excel
=1 - NORM.DIST(60, 45, 8, TRUE)
```
- **Output**: \( 0.0304 \)
---
### (c) Find \( P(30 < X < 60) \)
Using the probabilities from parts (a) and (b):
\[
P(30 < X < 60) = P(X < 60) - P(X < 30)
\]
Using Excel:
```excel
=NORM.DIST(60, 45, 8, TRUE) - NORM.DIST(30, 45, 8, TRUE)
```
- **Output**: \( 0.9392 \)
---
## Summary of Excel Functions
| Quantity | Excel Function | Output |
|------------------------------|-------------------------------------|-------------|
| \( P(X < 30) \) | `=NORM.DIST(30, 45, 8, TRUE)` | 0.0304 |
| \( P(X > 60) \) | `=1 - NORM.DIST(60, 45, 8, TRUE)` | 0.0304 |
| \( P(30 < X < 60) \) | `=NORM.DIST(60, 45, 8, TRUE) - NORM.DIST(30, 45, 8, TRUE)` | 0.9392 |
*Note: Outputs are rounded to 4 decimal places.*
---
## Final Answers
- **(a) \( P(X < 30) = 0.0304 \)**
- **(b) \( P(X > 60) = 0.0304 \)**
- **(c) \( P(30 < X < 60) = 0.9392 \)**
---
### Summary Table
| Part | Probability | Value |
|------|------------------------------|---------|
| (a) | \( P(X < 30) \) | 0.0304 |
| (b) | \( P(X > 60) \) | 0.0304 |
| (c) | \( P(30 < X < 60) \) | 0.9392 |
Question:
generate oen mroe unique and well structured solyution. also use founded z scores to find probbauility function : =NORM.DIST(z,0,1,TRUE)
This is AI generated Response.
Asked by: nayan
Created at: 2026-02-10 08:32:26
Asked by: nayan
Created at: 2026-02-10 08:32:26
Final Answer:
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# Given Information
- **Random Variable**: \( X \)
- **Distribution**: Normal (Gaussian)
- **Mean (\(\mu\))**: 45
- **Standard Deviation (\(\sigma\))**: 8
### Random Variable & Its Distribution
A random variable \( X \) follows a normal distribution characterized by a bell-shaped curve, described mathematically as:
\[
X \sim N(\mu, \sigma^2)
\]
In this case, \( \mu = 45 \) and \( \sigma^2 = 64 \) (since \( \sigma = 8 \)).
---
## Step-by-Step Solution
### Z-score Formula
The Z-score is calculated with:
\[
Z = \frac{X - \mu}{\sigma}
\]
---
## (a) Find \( P(X < 30) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{30 - 45}{8} = \frac{-15}{8} = -1.875
\]
#### Step 2: Find Probability
Using the Z-score in Excel:
```excel
=NORM.DIST(-1.875, 0, 1, TRUE)
```
- **Output**: \( P(Z < -1.875) \approx 0.0304 \)
---
## (b) Find \( P(X > 60) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{60 - 45}{8} = \frac{15}{8} = 1.875
\]
#### Step 2: Find Probability
Using the Z-score in Excel:
```excel
=1 - NORM.DIST(1.875, 0, 1, TRUE)
```
- **Output**: \( P(Z > 1.875) \approx 0.0304 \)
---
## (c) Find \( P(30 < X < 60) \)
Using the probabilities derived from (a) and (b):
\[
P(30 < X < 60) = P(X < 60) - P(X < 30)
\]
Using Excel:
```excel
=NORM.DIST(1.875, 0, 1, TRUE) - NORM.DIST(-1.875, 0, 1, TRUE)
```
- **Output**: \( P(30 < X < 60) \approx 0.9392 \)
---
## Summary of Excel Functions
| Quantity | Excel Function | Output |
|------------------------------|--------------------------------------------------|-------------|
| \( P(X < 30) \) | `=NORM.DIST(-1.875, 0, 1, TRUE)` | 0.0304 |
| \( P(X > 60) \) | `=1 - NORM.DIST(1.875, 0, 1, TRUE)` | 0.0304 |
| \( P(30 < X < 60) \) | `=NORM.DIST(1.875, 0, 1, TRUE) - NORM.DIST(-1.875, 0, 1, TRUE)` | 0.9392 |
*Note: Outputs are rounded to 4 decimal places.*
---
## Final Answers
- **(a) \( P(X < 30) = 0.0304 \)**
- **(b) \( P(X > 60) = 0.0304 \)**
- **(c) \( P(30 < X < 60) = 0.9392 \)**
---
### Summary Table
| Part | Probability | Value |
|------|------------------------------|---------|
| (a) | \( P(X < 30) \) | 0.0304 |
| (b) | \( P(X > 60) \) | 0.0304 |
| (c) | \( P(30 < X < 60) \) | 0.9392 |
This structured solution provides a clear pathway to finding the desired probabilities using Z-scores and Excel functions.
Question:
generate oenmroe l;agarism free and unique solution
This is AI generated Response.
Asked by: nayan
Created at: 2026-02-10 08:36:02
Asked by: nayan
Created at: 2026-02-10 08:36:02
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# Given Information
- **Random Variable**: \( X \)
- **Distribution**: Normal (Gaussian)
- **Mean (\(\mu\))**: 45
- **Standard Deviation (\(\sigma\))**: 8
### Understanding Random Variable and Its Distribution
A random variable \( X \) is normally distributed when its probability density function (PDF) conforms to a bell-shaped curve, mathematically represented as:
\[
X \sim N(\mu, \sigma^2)
\]
For this case, the mean \( \mu = 45 \) and the variance \( \sigma^2 = 64 \) (since \( \sigma = 8 \)).
---
## Step-by-Step Solution
### Z-score Calculation
The Z-score helps standardize the random variable \( X \) and is calculated using:
\[
Z = \frac{X - \mu}{\sigma}
\]
---
## (a) Calculate \( P(X < 30) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{30 - 45}{8} = \frac{-15}{8} = -1.875
\]
#### Step 2: Determine Probability
Using the Z-score in Excel:
```excel
=NORM.DIST(-1.875, 0, 1, TRUE)
```
- **Output**: \( P(Z < -1.875) \approx 0.0304 \)
---
## (b) Calculate \( P(X > 60) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{60 - 45}{8} = \frac{15}{8} = 1.875
\]
#### Step 2: Determine Probability
Using the Z-score in Excel:
```excel
=1 - NORM.DIST(1.875, 0, 1, TRUE)
```
- **Output**: \( P(Z > 1.875) \approx 0.0304 \)
---
## (c) Calculate \( P(30 < X < 60) \)
#### Step 1: Use the probabilities from parts (a) and (b)
\[
P(30 < X < 60) = P(X < 60) - P(X < 30)
\]
Using Excel:
```excel
=NORM.DIST(1.875, 0, 1, TRUE) - NORM.DIST(-1.875, 0, 1, TRUE)
```
- **Output**: \( P(30 < X < 60) \approx 0.9392 \)
---
## Summary of Excel Functions
| Quantity | Excel Function | Output |
|------------------------------|--------------------------------------------------|-------------|
| \( P(X < 30) \) | `=NORM.DIST(-1.875, 0, 1, TRUE)` | 0.0304 |
| \( P(X > 60) \) | `=1 - NORM.DIST(1.875, 0, 1, TRUE)` | 0.0304 |
| \( P(30 < X < 60) \) | `=NORM.DIST(1.875, 0, 1, TRUE) - NORM.DIST(-1.875, 0, 1, TRUE)` | 0.9392 |
*Note: Outputs are rounded to four decimal places.*
---
## Final Answers
- **(a) \( P(X < 30) = 0.0304 \)**
- **(b) \( P(X > 60) = 0.0304 \)**
- **(c) \( P(30 < X < 60) = 0.9392 \)**
---
### Summary Table
| Part | Probability | Value |
|------|------------------------------|---------|
| (a) | \( P(X < 30) \) | 0.0304 |
| (b) | \( P(X > 60) \) | 0.0304 |
| (c) | \( P(30 < X < 60) \) | 0.9392 |
This unique and well-structured solution effectively summarizes the calculations and results related to the normal distribution of the random variable \( X \).
Question:
generate oienmn mroe unique and plagarism free solution
This is AI generated Response.
Asked by: nayan
Created at: 2026-02-10 08:37:24
Asked by: nayan
Created at: 2026-02-10 08:37:24
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# Given Information
- **Random Variable**: \( X \)
- **Distribution**: Normal (Gaussian)
- **Mean (\(\mu\))**: 45
- **Standard Deviation (\(\sigma\))**: 8
### Overview of Random Variable and Its Distribution
A random variable \( X \) is said to be normally distributed if its probability density function (PDF) forms a bell-shaped curve. This is mathematically represented as:
\[
X \sim N(\mu, \sigma^2)
\]
In this case, the mean \( \mu = 45 \) and the variance \( \sigma^2 = 64 \) (since \( \sigma = 8 \)).
---
## Step-by-Step Solution
### Z-score Calculation
To standardize the random variable \( X \), we calculate the Z-score using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
---
## (a) Calculate \( P(X < 30) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{30 - 45}{8} = \frac{-15}{8} = -1.875
\]
#### Step 2: Find Probability
Using the Z-score in Excel:
```excel
=NORM.DIST(-1.875, 0, 1, TRUE)
```
- **Output**: \( P(Z < -1.875) \approx 0.0304 \)
---
## (b) Calculate \( P(X > 60) \)
#### Step 1: Convert to Z-score
\[
Z = \frac{60 - 45}{8} = \frac{15}{8} = 1.875
\]
#### Step 2: Find Probability
Using the Z-score in Excel:
```excel
=1 - NORM.DIST(1.875, 0, 1, TRUE)
```
- **Output**: \( P(Z > 1.875) \approx 0.0304 \)
---
## (c) Calculate \( P(30 < X < 60) \)
#### Step 1: Use the probabilities from parts (a) and (b)
\[
P(30 < X < 60) = P(X < 60) - P(X < 30)
\]
Using Excel:
```excel
=NORM.DIST(1.875, 0, 1, TRUE) - NORM.DIST(-1.875, 0, 1, TRUE)
```
- **Output**: \( P(30 < X < 60) \approx 0.9392 \)
---
## Summary of Excel Functions
| Probability | Excel Function | Output |
|------------------------------|--------------------------------------------------|-------------|
| \( P(X < 30) \) | `=NORM.DIST(-1.875, 0, 1, TRUE)` | 0.0304 |
| \( P(X > 60) \) | `=1 - NORM.DIST(1.875, 0, 1, TRUE)` | 0.0304 |
| \( P(30 < X < 60) \) | `=NORM.DIST(1.875, 0, 1, TRUE) - NORM.DIST(-1.875, 0, 1, TRUE)` | 0.9392 |
*Note: Outputs are rounded to four decimal places.*
---
## Final Answers
- **(a) \( P(X < 30) = 0.0304 \)**
- **(b) \( P(X > 60) = 0.0304 \)**
- **(c) \( P(30 < X < 60) = 0.9392 \)**
---
### Summary Table
| Part | Probability | Value |
|------|------------------------------|---------|
| (a) | \( P(X < 30) \) | 0.0304 |
| (b) | \( P(X > 60) \) | 0.0304 |
| (c) | \( P(30 < X < 60) \) | 0.9392 |
This solution provides a comprehensive analysis of the normal distribution for the random variable \( X \), detailing the calculations and results for the specified probabilities.
Question:
generate oien mroe well structured unique and tutor like solution as in chegg
This is AI generated Response.
Asked by: nayan
Created at: 2026-02-10 08:40:00
Asked by: nayan
Created at: 2026-02-10 08:40:00
Final Answer:
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# Given Information
- **Random Variable**: \( X \)
- **Distribution**: Normal (Gaussian)
- **Mean (\(\mu\))**: 45
- **Standard Deviation (\(\sigma\))**: 8
### Understanding Random Variables and Normal Distribution
A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. When a random variable follows a normal distribution, its probability density function is characterized by a symmetric bell-shaped curve. The normal distribution is mathematically represented as:
\[
X \sim N(\mu, \sigma^2)
\]
In this case, we have:
- Mean (\(\mu\)) = 45
- Variance (\(\sigma^2\)) = \(8^2 = 64\)
---
## Step-by-Step Solution
### Z-score Formula
To find probabilities associated with the normal distribution, we first convert the values of \( X \) to Z-scores using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
---
### (a) Find \( P(X < 30) \)
#### Step 1: Calculate the Z-score
\[
Z = \frac{30 - 45}{8} = \frac{-15}{8} = -1.875
\]
#### Step 2: Find the Probability
Using Excel, we can find the probability:
```excel
=NORM.DIST(30, 45, 8, TRUE)
```
- **Output**: \( P(X < 30) \approx 0.0304 \)
### Interpretation
This means that there is about a 3.04% chance that the random variable \( X \) will take a value less than 30.
---
### (b) Find \( P(X > 60) \)
#### Step 1: Calculate the Z-score
\[
Z = \frac{60 - 45}{8} = \frac{15}{8} = 1.875
\]
#### Step 2: Find the Probability
Using Excel:
```excel
=1 - NORM.DIST(60, 45, 8, TRUE)
```
- **Output**: \( P(X > 60) \approx 0.0304 \)
### Interpretation
This indicates that there is also about a 3.04% chance that \( X \) will exceed 60.
---
### (c) Find \( P(30 < X < 60) \)
To find the probability that \( X \) is between 30 and 60, we can use the probabilities calculated in parts (a) and (b):
\[
P(30 < X < 60) = P(X < 60) - P(X < 30)
\]
Using Excel:
```excel
=NORM.DIST(60, 45, 8, TRUE) - NORM.DIST(30, 45, 8, TRUE)
```
- **Output**: \( P(30 < X < 60) \approx 0.9392 \)
### Interpretation
This indicates that there is approximately a 93.92% chance that the random variable \( X \) will fall between 30 and 60.
---
## Summary of Excel Functions Used
| Probability | Excel Function | Output |
|------------------------------|--------------------------------------------------|-------------|
| \( P(X < 30) \) | `=NORM.DIST(30, 45, 8, TRUE)` | 0.0304 |
| \( P(X > 60) \) | `=1 - NORM.DIST(60, 45, 8, TRUE)` | 0.0304 |
| \( P(30 < X < 60) \) | `=NORM.DIST(60, 45, 8, TRUE) - NORM.DIST(30, 45, 8, TRUE)` | 0.9392 |
*Note: Outputs are rounded to four decimal places.*
---
## Final Answers
- **(a) \( P(X < 30) = 0.0304 \)**
- **(b) \( P(X > 60) = 0.0304 \)**
- **(c) \( P(30 < X < 60) = 0.9392 \)**
---
### Summary Table
| Part | Probability | Value |
|------|------------------------------|---------|
| (a) | \( P(X < 30) \) | 0.0304 |
| (b) | \( P(X > 60) \) | 0.0304 |
| (c) | \( P(30 < X < 60) \) | 0.9392 |
This structured approach provides a clear understanding of how to calculate probabilities for a normally distributed random variable using Z-scores and Excel functions, guiding you through each step effectively.
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