Let’s break down each part step by step. --- ## **a. Probability that an EV costs less than $45,000** **Step 1: Find the z-score for $45,000.** Formula: \( z = \frac{X - \mu}{\sigma} \) Where: - \( X = \$45,000 \) - \( \mu = \$50,798 \) (mean) - \( \sigma = \$6,000 \) (standard deviation) \[ z = \frac{45,000 - 50,798}{6,000} = \frac{-5,798}{6,000} \approx -0.97 \] **Explanation:** The z-score tells us how many standard deviations $45,000 is below the mean. --- **Step 2: Use the z-table to find the probability for \( z = -0.97 \).** Looking up \( z = -0.97 \) in a standard normal table, we get: \[ P(Z < -0.97) \approx 0.1660 \] **Explanation:** This is the probability that a value is less than 0.97 standard deviations below the mean. --- **Step 3: State the final answer.** **Final Answer:** The probability that a randomly selected EV in the US in December 2023 costs less than $45,000 is **approximately 0.166 (or 16.6%)**. --- ## **b. Probability EV price in December 2023 is between $42,000 and $62,000** **Step 1: Find the z-scores for $42,000 and $62,000.** For $42,000: \[ z_1 = \frac{42,000 - 50,798}{6,000} = \frac{-8,798}{6,000} \approx -1.47 \] For $62,000: \[ z_2 = \frac{62,000 - 50,798}{6,000} = \frac{11,202}{6,000} \approx 1.87 \] **Explanation:** These z-scores tell us how many standard deviations each value is from the mean. --- **Step 2: Find the probabilities for each z-score using the z-table.** For \( z_1 = -1.47 \): \[ P(Z < -1.47) \approx 0.0708 \] For \( z_2 = 1.87 \): \[ P(Z < 1.87) \approx 0.9693 \] **Explanation:** These probabilities indicate the area under the normal curve to the left of each z-score. --- **Step 3: Subtract to find the probability between the two values.** \[ P(42,000 < X < 62,000) = P(Z < 1.87) - P(Z < -1.47) = 0.9693 - 0.0708 = 0.8985 \] **Final Answer:** The probability that a randomly selected EV in the US in December 2023 costs between $42,000 and $62,000 is **approximately 0.899 (or 89.9%)**. --- **Summary Table:** | Part | Probability | Explanation | |------|-------------|-------------| | (a) | 0.166 | P(X < $45,000) | | (b) | 0.899 | P($42,000 < X < $62,000) |