Question

the population mean and standard deviation are given below. find the indicated probability and determine whether the given sample mean would be considered unusual.
for a sample of ( n = 40 ), find the probability of a sample mean being less than 12,751 or greater than 12,754 when ( mu = 12,751 ) and ( sigma = 1.8 ).
for the given sample, the probability of a sample mean being less than 12,751 or greater than 12,754 is
(round to four decimal places as needed.)
would the given sample mean be considered unusual?
a. the sample mean would not be considered unusual because there is a probability less than 0.05 of the sample mean being within this range.
b. the sample mean would be considered unusual because there is a probability less than 0.05 of the sample mean being within this range.
c. the sample mean would be considered unusual because there is a probability greater than 0.05 of the sample mean being within this range.
d. the sample mean would not be considered unusual because there is a probability greater than 0.05 of the sample mean being within this range.

Explanation:

Step1: Calculate the standard error

The standard error $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 1.8$ and $n = 40$, we have $\sigma_{\bar{x}}=\frac{1.8}{\sqrt{40}}\approx0.2846$.

Step2: Calculate the z - scores

For $\bar{x}_1 = 12751$, $z_1=\frac{\bar{x}_1-\mu}{\sigma_{\bar{x}}}=\frac{12751 - 12751}{0.2846}=0$.
For $\bar{x}_2 = 12754$, $z_2=\frac{\bar{x}_2-\mu}{\sigma_{\bar{x}}}=\frac{12754 - 12751}{0.2846}=\frac{3}{0.2846}\approx10.54$.

Step3: Calculate the probability

$P(\bar{X}<12751\ or\ \bar{X}>12754)=P(Z < 0)+(1 - P(Z<10.54))$.
Since $P(Z < 0)=0.5$ and $P(Z<10.54)\approx1$ (from the standard normal table), then $P(\bar{X}<12751\ or\ \bar{X}>12754)=0.5+(1 - 1)=0.5$.

Answer:

The probability is $0.5000$.
For the multiple - choice part:
A sample mean is considered unusual if the probability of obtaining it (or a more extreme value) is less than $0.05$. Since $P = 0.5>0.05$, the sample mean would not be considered unusual.
So the answer for the multiple - choice is D. The sample mean would not be considered unusual because there is a probability greater than $0.05$ of the sample mean being within this range.