Question

a simple random sample of size ( n = 36 ) is obtained from a population with ( mu = 104 ) and ( sigma = 23 ).
enter your answer as an area under the curve with 4 decimal places.
( p(overline{x}<100)= )________

Explanation:

Step1: Calculate the standard error

The formula for the standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$.
Given $\sigma = 23$ and $n = 36$, we have $\sigma_{\bar{x}}=\frac{23}{\sqrt{36}}=\frac{23}{6}\approx3.8333$.

Step2: Calculate the z - score

The formula for the z - score is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$.
Here, $\bar{x} = 100$, $\mu = 104$, and $\sigma_{\bar{x}}\approx3.8333$.
So, $z=\frac{100 - 104}{3.8333}=\frac{-4}{3.8333}\approx - 1.04$.

Step3: Find the probability using the z - table

We want to find $P(\bar{X}<100)$, which is equivalent to $P(Z < - 1.04)$.
Looking up the value of $P(Z < - 1.04)$ in the standard normal table, we get $P(Z < - 1.04)=0.1492$.

Answer:

$0.1492$