\( X \) is a normally distributed random variable with mean 14 and standard deviation 8 What is the probability that \( X \) is between 29 and \( 87 ? \) Write your answer as a decimal rounded to the nearest thousandth.

To find the probability that a normally distributed random variable \( X \) is between two values, we can use the cumulative distribution function (CDF) of the normal distribution. Given: - Mean (\( \mu \)) = 14 - Standard deviation (\( \sigma \)) = 8 - We want to find \( P(29 < X < 87) \). We can calculate this probability using the following steps: 1. **Standardize the values**: Convert the values 29 and 87 to their corresponding z-scores using the formula: \[ z = \frac{x - \mu}{\sigma} \] 2. **Calculate the z-scores**: - For \( x = 29 \): \[ z_1 = \frac{29 - 14}{8} \] - For \( x = 87 \): \[ z_2 = \frac{87 - 14}{8} \] 3. **Find the probabilities**: Use the z-scores to find the probabilities from the standard normal distribution table or a calculator. 4. **Calculate the probability**: The probability that \( X \) is between 29 and 87 is given by: \[ P(29 < X < 87) = P(Z < z_2) - P(Z < z_1) \] Now, let's perform the calculations step by step. ### Step 1: Calculate the z-scores Calculating \( z_1 \) and \( z_2 \): \[ z_1 = \frac{29 - 14}{8} \] \[ z_2 = \frac{87 - 14}{8} \] ### Step 2: Calculate the probabilities Now, I will calculate the z-scores and then find the corresponding probabilities. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(87-14\right)}{8}\) - step1: Remove the parentheses: \(\frac{87-14}{8}\) - step2: Subtract the numbers: \(\frac{73}{8}\) Calculate or simplify the expression \( (29 - 14) / 8 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(29-14\right)}{8}\) - step1: Remove the parentheses: \(\frac{29-14}{8}\) - step2: Subtract the numbers: \(\frac{15}{8}\) It seems that I am unable to directly calculate the probabilities from the standard normal distribution. However, I can provide the values based on standard normal distribution tables or common statistical software. ### Step 1: Z-scores Recap - \( z_1 = 1.875 \) - \( z_2 = 9.125 \) ### Step 2: Probability Values Using standard normal distribution tables or calculators: - \( P(Z < 1.875) \approx 0.9693 \) - \( P(Z < 9.125) \approx 1 \) (since this is far into the tail of the distribution) ### Step 3: Calculate the Probability Now we can calculate the probability that \( X \) is between 29 and 87: \[ P(29 < X < 87) = P(Z < 9.125) - P(Z < 1.875) \approx 1 - 0.9693 = 0.0307 \] ### Final Answer The probability that \( X \) is between 29 and 87 is approximately \( 0.031 \) when rounded to the nearest thousandth.