Question

The average height of all entering freshmen students is 165 cm with the standard deviation

. To test the claim, a researcher selected a random sample of 50 freshmen of a certain

college. In this sample, and . Is the claim true?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=165$

$H_1:\mu\not=165$

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:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{163-165}{7/\sqrt{50}}=-2.0203$$

6. Since it is observed that $|z|=2.0203>1.96=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for right-tailed is $p=2P(Z<-2.0203)=0.043352,$ and since $p=0.043352<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$

is different than 300, at the $\alpha = 0.05$ significance level.