# A population of values has a normal distribution with \mu = 192.3 and \sigma = 66.5. You intend to draw a random sample of size n = 15. Find the proba

A population of values has a normal distribution with $\mu =192.3$ and $\sigma =66.5$. You intend to draw a random sample of size $n=15$.
Find the probability that a sample of size $n=15$ is randomly selected with a mean less than 185.4.
$P\left(M<185.4\right)=$?

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Cristiano Sears

The known values are,
$\mu =192.3$,
$\sigma =66.5$,
$n=15$
The probability that a sample size $n=15$ is randomly selected with a mean less than 185.4 is,
$P\left(M<185.4\right)=P\left(\frac{M-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{185.4-192.3}{\frac{66.5}{\sqrt{15}}}\right)$
$=P\left(z<-0.402\right)$
$=0.343842\approx 0.3438$
Therefore, the required probability is, 0.3438