QQ. 2 Find probabillties using the normal distribution I QA9 \( X \) is a normally distributed random variable with mean 5 and standard deviation 3 . What is the probability that \( X \) is less than 2 ? Use the \( 0.68-0.95-0.997 \) rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

To find the probability that \( X \) is less than 2 for a normally distributed random variable with mean \( \mu = 5 \) and standard deviation \( \sigma = 3 \), we can use the **68-95-99.7 rule** (also known as the empirical rule). ### Step-by-Step Solution: 1. **Calculate the Z-Score:** The Z-score helps us determine how many standard deviations away a value is from the mean. \[ Z = \frac{X - \mu}{\sigma} = \frac{2 - 5}{3} = \frac{-3}{3} = -1 \] 2. **Apply the 68-95-99.7 Rule:** - **68%** of the data falls within **±1** standard deviation from the mean. - **95%** falls within **±2** standard deviations. - **99.7%** falls within **±3** standard deviations. Since \( Z = -1 \) corresponds to one standard deviation below the mean: - **68%** of the data is between \( Z = -1 \) and \( Z = 1 \). - This leaves **32%** of the data in the two tails, with **16%** in each tail. Therefore: \[ P(X < 2) = P(Z < -1) = 16\% = 0.16 \] ### Final Answer: The probability that \( X \) is less than 2 is **0.160**.