Question

The average height of students in a freshman class of a certain school has been 149.45 cm with a population standard deviation of 7.74 cm. Is there a reason to believe that there has been a change in the average height if a random sample of 43 students in the present freshman class has an average height of 164 cm? Use a 0.01 level of significance.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=149.45$

$H_1:\mu\not=149.45$

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 $\alpha = 0.01,$ and the critical value for a two-tailed test is $z_c = 2.5758.$

The rejection region for this two-tailed test is $R = \{z:|z|>2.5758\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{164-149.45}{7.74/\sqrt{43}}=12.3270$$

Since it is observed that $|z|=12.3270>2.5758=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=2P(z>12.3270)= 0,$ and since $p= 0<0.01=\alpha,$ it is concluded that the null hypothesis is rejected.

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

is different than 4.25, at the $\alpha = 0.01$ significance level.