# A population of values has a normal distribution with \mu = 181 and \sigma = 41.8. You intend to draw a

A population of values has a normal distribution with $\mu =181$ and $\sigma =41.8$. You intend to draw a random sample of size $n=144$.
Find the probability that a single randomly selected value is between 176.5 and 183.1.
$P\left(176.5?

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Tasneem Almond

Given that,
$\mu =181$,
$\sigma =41.8$
$n=144$
The probability that a sample size, $n=144$ is randomly selected with a mean between 176.5 and 183.1 is,
$P\left(176.5<\stackrel{―}{X}<183.1\right)=P\left(\frac{176.5-181}{\frac{41.8}{\sqrt{144}}}<\frac{\stackrel{―}{X}-\mu }{\frac{\sigma }{\sqrt{144}}}<\frac{183.1-181}{\frac{41.8}{\sqrt{144}}}\right)$
$=P\left(-1.292
$=P\left(z<0.603\right)-P\left(z<-1.292\right)$

$=0.7267-0.0982=0.6285$
Therefore, the required probability is, 0.6285.

Jeffrey Jordon