 # A population of values has a normal distribution with \mu=192.6 and \sigma=34.4. defazajx 2021-02-25 Answered

A population of values has a normal distribution with $\mu =192.6$ and $\sigma =34.4$. You intend to draw a random sample of size $n=173$.
Find the probability that a sample of size $n=173$ is randomly selected with a mean less than 186.1.
$P\left(M<186.1\right)=$?
Write your answers as numbers accurate to 4 decimal places.

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Step 1
Given:
$\mu =192.6$,
$\sigma =34.4$,
$n=173$.
The Z-score follows standard normal distribution.
Step 2
$P\left[X<186.1\right]=P\left[\frac{X-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{186.1-192.6}{\frac{34.4}{\sqrt{173}}}\right]$
$=P\left[Z<-2.49\right]=0.0066$ (Use standard normal table)
The probability that a sample of size $n=173$ is randomly selected with a mean less than 186.1 is 0.0066.

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