# A population of values has a normal distribution with \mu=197 and \sigma=68.

A population of values has a normal distribution with $\mu =197$ and $\sigma =68$. You intend to draw a random sample of size $n=181$
Find the probability that a sample of size $n=181$ is randomly selected with a mean between 198.5 and 205.6.
$P\left(198.5?

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Step 1
From the provided information,
Mean $\left(\mu \right)=197$
Standard deviation $\left(\sigma \right)=68$
Let X be a random variable which represents the value.
$X\sim N\left(197,68\right)$
Step 2
Sample size $\left(n\right)=181$
The required probability that a sample of size $n=181$ is randomly selected with a mean between 198.5 and 205.6 $P\left(198.5 can be obtained as:
$P\left(198.5
$=P\left(0.297
$=P\left(Z<1.701\right)-P\left(Z<0.297\right)$
$=0.9555-0.6168=0.3387$ (Using standard normal table)
Thus, the required probability is 0.3387.