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**lllll** For a given exam the standard deviation for all seniors in a particular high school is 2.5. If we take a sample of 100 seniors, find the probability that the difference between the sample mean and the mean of all seniors does not exceed 0.5.

so far I have:

$\displaystyle P\left( \frac{\overline{Y}-X}{2.5 / \sqrt{100}} \leq \frac{0.5}{0.25} \right) \longrightarrow P\left( \frac{\overline{Y}-X}{0.25} \leq 2 \right)$

if I let $\displaystyle P(Z\leq 2) = 1- 0.228 = 0.9972$

but the answer in the back of the book is 0.9544 which would correspond to $\displaystyle P(Z\leq1.69)$. I'm guessing it has something to do with the population mean, but I can't figure it out.