# A population of values has a normal distribution with \mu=154.5 and \sigma=96.1. You intend to draw a random sample of size n=134. Find the probabilit

A population of values has a normal distribution with $\mu =154.5$ and $\sigma =96.1$. You intend to draw a random sample of size $n=134$.
Find the probability that a sample of size $n=134$ is randomly selected with a mean greater than 167.
$P\left(M>167\right)=$?
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Macsen Nixon
${X}_{1},{X}_{2},\dots {X}_{134}$ be a random sample from Normal distribution with $\mu =154.5$ and $\sigma =96.1$.
Then the sample mean
$M=\frac{1}{134}\sum _{i=1}^{134}{X}_{i}$ follows Normal distribution with mean 154.5 and standard deviation $\frac{96.1}{\sqrt{134}}$.
Hence, the probability that sample mean greater than 167.
$P\left(M>167\right)$
$=1-P\left(M\le 167\right)$
$=1-P\left(\frac{M-154.5}{\frac{96.1}{\sqrt{134}}}\le \frac{167-154.5}{\frac{96.1}{\sqrt{134}}}\right)$
$=1-P\left(Z\le 1.506\right),Z=\left(\frac{M-154.5}{\frac{96.1}{\sqrt{134}}}\right)$ follows Normal (0,1)
$=1-\varphi \left(1.506\right),\varphi \left(1.506\right)$ calculated from Normal distribution table.
$=1-0.934=0.066$
Therefore, the probability that a sample of size $n=134$ is randomly selected with a mean greater than 167 is 0.066.