Questions: Following is the probability distribution for age of a student at a certain public high school. x 13 14 15 16 17 18 P(x) 0.08 0.23 0.25 0.28 0.14 0.02 Part 1 of 2 (a) Find the variance of the ages. Round the answer to at least four decimal places. The variance of the ages is .

Solution Steps

To find the variance of the ages, we need to follow these steps:

1. Calculate the mean (expected value) of the ages.
2. Use the mean to calculate the variance by finding the expected value of the squared differences from the mean.

The mean (expected value) of the ages is calculated using the formula: \[ \mu = \sum_{i=1}^{n} x_i P(x_i) \] Given the ages \( x = [13, 14, 15, 16, 17, 18] \) and their corresponding probabilities \( P(x) = [0.08, 0.23, 0.25, 0.28, 0.14, 0.02] \), we have: \[ \mu = 13 \cdot 0.08 + 14 \cdot 0.23 + 15 \cdot 0.25 + 16 \cdot 0.28 + 17 \cdot 0.14 + 18 \cdot 0.02 = 15.23 \]

The variance is calculated using the formula: \[ \sigma^2 = \sum_{i=1}^{n} P(x_i) (x_i - \mu)^2 \] Substituting the values, we get: \[ \sigma^2 = 0.08 \cdot (13 - 15.23)^2 + 0.23 \cdot (14 - 15.23)^2 + 0.25 \cdot (15 - 15.23)^2 + 0.28 \cdot (16 - 15.23)^2 + 0.14 \cdot (17 - 15.23)^2 + 0.02 \cdot (18 - 15.23)^2 = 1.5171 \]

Final Answer