# A population of values has a normal distribution with

A population of values has a normal distribution with $\mu =200.7$ and $\sigma =10$. You intend to draw a random sample of size $n=178$
Find the probability that a sample of size $n=178$ is randomly selected with a mean less than 198.6.
$P\left(M<198.6\right)=$?

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tabuordg

Step 1
From the provided information,
Mean $\left(\mu \right)=200.7$
Standard deviation $\left(\sigma \right)=10$
Let X be a random variable which represents the value.
$X\sim N\left(200.7,10\right)$
Sample size $\left(n\right)=178$
Step 2
The required probability that a sample of size $n=178$ is randomly selected with a mean less than 198.6 can be obtained as:
$P\left(M<198.6\right)=P\left(\frac{M-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{198.6-200.7}{\frac{10}{\sqrt{178}}}\right)$
$=P\left(Z<-2.802\right)=0.0025$ (Using standard normal table)
Thus, the required probability is 0.0025.