Question

In a certain population, it is claimed that the mean number of years of education is 13.2, while the standard deviation is 2.95 years. A random sample of 60 people is drawn from this population, and the sample mean is 13.87 years. What are the hypotheses and test statistic to be used in this case?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=13.2$

$H_1:\mu\not=13.2$

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:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{13.87-13.2}{2.95/\sqrt{60}}=1.76$$

Since it is observed that $|z|=1.76<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=2P(z>1.76)= 0.078408,$ and since $p=0.078408>0.05=\alpha,$ 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 13.2, at the $\alpha = 0.05$ significance level.