Answer to Question #349087 in Statistics and Probability for Johnmark

Question

A sample of 160 people has a mean age of 27 with a population standard deviation of 5. Test the hypothesis that the population mean is 26.7 at =0.05

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=26.7$

$H_1:\mu\not=26.7$

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ and the critical value for a two-tailed test is $z_c =1.96.$

The rejection region for this two-tailed test is $R = \{z:|z|> 1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{27-26.7}{5/\sqrt{160}}=0.759$$

6. Since it is observed that $|z|=0.759<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value for two-tailed is $p=2P(Z>0.759)=0.447853,$ and since $p=0.447853>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$

is different than 26.7, at the $\alpha = 0.05$ significance level.