Assume the random variable \( x \) is normally distributed with mean \( \mu=80 \) and standard deviation \( \sigma=4 \). Find the indica probability. \( P(68

Ask by Ball Powers. in the United States

Feb 03,2025

To find the probability \( P(68 < x < 72) \) for a normally distributed random variable \( x \) with mean \( \mu = 80 \) and standard deviation \( \sigma = 4 \), follow these steps: 1. **Standardize the Values:** Convert the original values to standard \( z \)-scores using the formula: \[ z = \frac{x - \mu}{\sigma} \] - For \( x = 68 \): \[ z_1 = \frac{68 - 80}{4} = \frac{-12}{4} = -3 \] - For \( x = 72 \): \[ z_2 = \frac{72 - 80}{4} = \frac{-8}{4} = -2 \] 2. **Find the Probability Using Standard Normal Distribution:** The probability \( P(-3 < z < -2) \) corresponds to the area under the standard normal curve between \( z = -3 \) and \( z = -2 \). 3. **Use Standard Normal Tables or a Calculator:** - \( \Phi(-2) \) (the cumulative probability up to \( z = -2 \)) is approximately 0.0228. - \( \Phi(-3) \) is approximately 0.0013. - Therefore: \[ P(-3 < z < -2) = \Phi(-2) - \Phi(-3) = 0.0228 - 0.0013 = 0.0215 \] 4. **Final Probability:** \[ P(68 < x < 72) = 0.0215 \] **Answer:** \( P(68 < x < 72) = 0.0215 \)