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 .
Transcript text: Following is the probability distribution for age of a student at a certain public high school.
\begin{tabular}{c|ccccccc}
$x$ & 13 & 14 & 15 & 16 & 17 & 18 \\
\hline$P(x)$ & 0.08 & 0.23 & 0.25 & 0.28 & 0.14 & 0.02
\end{tabular}
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 $\square$ .
Solution
Solution Steps
To find the variance of the ages, we need to follow these steps:
Calculate the mean (expected value) of the ages.
Use the mean to calculate the variance by finding the expected value of the squared differences from the mean.
Step 1: Calculate the Mean (Expected Value)
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:
\[
\mu = 13 \cdot 0.08 + 14 \cdot 0.23 + 15 \cdot 0.25 + 16 \cdot 0.28 + 17 \cdot 0.14 + 18 \cdot 0.02 = 15.23
\]