Question

A population has a mean is $25$ and a standard deviation of five. The sample mean is $24$, and the sample size is $108$. What distribution should you use to perform a hypothesis test?

Short answer

When the standard deviation is known and the population size is large, the Normal distribution approach is appropriate for performing the hypothesis test.

Step-by-step solution

Given Information

The population has a mean of $25$and a standard deviation of $5$. According to the data, the population has a mean of $25$and a standard deviation of $5$. The sample size is $108$people, and the sample mean is $24$. Now, for a hypothesis test.

Explanation

Typically, a hypothesis test is conducted on a sample of data drawn from a broader population. This test is used to generate results from sample data based on a hypothesis. Because we have the standard mean, standard deviation, sample mean, and population size, the hypothesis test may be performed using the normal distribution approach. The normal distribution is a continuous probability distribution with a bell shape in which all values cluster around the mean.

Calculation

We must use the following formula to calculate using the normal distribution:

The sample mean is $\overline{X}$, the population mean is $\mu$, the standard deviation is $\sigma$, and the sample size is $\eta$.

$Z=\frac{\overline{X}}{\sigma }\frac{-\mu }{/\sqrt{\eta }}$

$\frac{24-25}{5/\sqrt{108}}$

$$\begin{gathered} =\frac{-1}{5/10.3923} \\ =-2.078 \end{gathered}$$

As a result, the normal distribution approach is suitable for larger populations with established standard deviation values.