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 .

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 $\square$ .
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Solution

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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.
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: \[ \mu = 13 \cdot 0.08 + 14 \cdot 0.23 + 15 \cdot 0.25 + 16 \cdot 0.28 + 17 \cdot 0.14 + 18 \cdot 0.02 = 15.23 \]

Step 2: Calculate the Variance

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

The variance of the ages is: \[ \boxed{1.5171} \]

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