Sampling distribution and standard error

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Problem

A normal population has mean $10$ and standard deviation $4$ . For an i.i.d. sample of size $25$ , give the exact distribution of $\bar X$ , its standard error, and $P(|\bar X-10|\le1.6)$ .

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Normality makes the sample mean exactly normal:

\bar X\sim\mathcal N\left(10,\frac{4^2}{25}\right) =\mathcal N(10,0.64)

Its standard error is $4/\sqrt{25}=0.8$ . Standardizing $1.6/0.8=2$ gives

P(|\bar X-10|\le1.6)=P(|Z|\le2)=2\Phi(2)-1\approx0.9545

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Sampling distribution and standard error

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