# A population of values has a normal distribution with \mu = 49 and \sigma = 79.5. You intend to draw a random sample of size n=84. Find the probabilit

A population of values has a normal distribution with $\mu =49$ and $\sigma =79.5$. You intend to draw a random sample of size $n=84$.
Find the probability that a a sample of size $n=84$ is randomly selected with a mean greater than 72.4.
$P\left(M>72.4\right)=$?
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Step 1
Given,
Mean = 49
Standard deviation = 79.5
Sample size = 84
Step 2
Consider,
$P\left(M>72.4\right)=P\left(\frac{M-\mu }{\frac{\sigma }{\sqrt{n}}}>\frac{72.4-\mu }{\frac{\sigma }{\sqrt{n}}}\right)$
$=P\left(Z>\frac{72.4-\mu }{\frac{\sigma }{\sqrt{n}}}\right)$
$=P\left(Z>\frac{72.4-49}{\frac{79.5}{\sqrt{84}}}\right)$
$=P\left(Z>2.70\right)$
$=1-P\left(Z\le 2.70\right)$
$=1-0.9965=0.0035$ (Using standard normal table)
The probability that a sample of size $n=84$ is randomly selected with a mean greater than 72.4 is, 0.0035.