Question

According to the school librarian, the average number of pages of books in the reference section is 240. To

test her claim, she collected a sample of 15 books and after noting the number of pages of each book, she

determined that the mean number of pages is 224.6 with a standard deviation of 4.1. At α = 0.01, will the

librarian be able to prove her claim?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=240$

$H_1:\mu\not=240$

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha = 0.01,$ $df=n-1=14$ and the critical value for a two-tailed test is $t_c =2.976842.$

The rejection region for this two-tailed test is $R = \{t:|t|>2.976842\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{224.6-240}{4.1/\sqrt{15}}=-14.5473$$

Since it is observed that $|t|=14.5473>2.976842=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for two-tailed, $df=14$ degrees of freedom, $t=-14.5473$ is $p=0,$ 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 240, at the $\alpha = 0.01$ significance level.