Answer to Question #337689 in Statistics and Probability for mikay

Question

Average Senior High School annual cost of tuition fee for all private schools last year was P46,300. A random sample of costs this year for 45 private schools indicated that the sample mean was P47,800 and a sample standard deviation was P5,600. At the 0.10 level of significance, is there sufficient evidence to conclude that the cost has increased?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\le 46300$

$H_a:\mu>46300$

This corresponds to a right-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 $\alpha = 0.10,$ $df=n-1=44$ degrees of fredom, and the critical value for a right-tailed test is $t_c =1.30109.$

The rejection region for this right-tailed test is $R = \{t: t >1.30109\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{47800-46300}{5600/\sqrt{45}}\approx1.79684$$

Since it is observed that $t = 1.79684> 1.30109=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for right-tailed, $df=44$ degrees of freedom, $t=1.79684$ is $p = 0.039614,$ and since $p= 0.039614<0.10=\alpha,$ it is concluded that the null hypothesis is rejected.

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

is greater than 46300, at the $\alpha = 0.10$ significance level.