Question

Incoming freshmen are given entrance examinations in a number of fields, including English. Over a period of years, it has been found that the average score in English examination is 80 with a standard deviation of 7.8. An English instructor examines the scores for his class of 30 and finds that their average is 85. Can the instructor claim that the average score has increased?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\leq80$

$H_1:\mu>80$

This corresponds to a right-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 right-tailed test is $z_c = 1.6449.$

The rejection region for this right-tailed test is $R = \{z: z > 1.6449\}$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{85-80}{7.8/\sqrt{30}}=3.511$$

Since it is observed that $z = 3.511 > 1.6449=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=P(Z>3.511)=0.000223,$ and since $p=0.0002<0.05,$ it is concluded that the null hypothesis is rejected.

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

is greater than 80, at the $\alpha = 0.05$ significance level.

Therefore, there is enough evidence to claim that the average score has increased, at the $\alpha = 0.05$ significance level.