Answer to Question #344563 in Statistics and Probability for vickyteodosio

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.