# A population of values has a normal distribution with \mu = 116.5 and \sigma = 63.7. You intend to draw a random sample of size n = 244.

A population of values has a normal distribution with $\mu =116.5$ and $\sigma =63.7$. You intend to draw a random sample of size $n=244$.
Find the probability that a sample of size $n=244$ is randomly selected with a mean between 104.7 and 112.8.
$P\left(104.7?

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Aubree Mcintyre

A population of values has a normal distribution with $\mu =116.5$ and $\sigma =63.7$.
A random sample of size $\left(n\right)=244$ is drawn.
We need to find the probability that a sample of size $n=244$ is randomly selected with a mean(M) between 104.7 and 112.8.
$M\sim N\left(116.5,\frac{{63.7}^{2}}{n}\right)$ and $n=244$
So, $Z=\frac{\sqrt{n}\left(M-116.5\right)}{63.7}\sim N\left(0,1\right)$
Thus, $P\left[104.7
$=P\left[\frac{\sqrt{n}\left(104.7-116.5\right)}{63.7}<\frac{\sqrt{n}\left(M-116.5\right)}{63.7}<\frac{\sqrt{n}\left(112.8-116.5\right)}{63.7}\right]$
$=P\left[-2.893
$=P\left[Z\le -0.9073\right]-P\left[Z\le -2.8934\right]$
$=0.1821-0.0019=0.1802$
So, the probability that a sample of size $n=244$ is randomly selected with a mean between 104.7 and 112.8 is 0.1802.