Answer to Question #302361 in Statistics and Probability for princess

Question

A Average Senior high school annual cost of tuition fee for all private schools last year was php 43.700. a random sample of cost this year for 45 private schools indicated that the sample mean was 45,800 and a sample standard deviation was php5,600.at level of significance, is there sufficient evidence to conclude that the cost increased?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0: \mu\le43700$

$H_1:\mu>43700$

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 $\alpha = 0.05,df=n-1=44$ degrees of freedom, and the critical value for a right-tailed test is $t_c = 1.68023.$

The rejection region for this right-tailed test is $R = \{t: t > 1.68023\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{X}-\mu}{s/\sqrt{n}}=\dfrac{45800-43700}{5600/\sqrt{45}}=2.51558$$

Since it is observed that $t = 2.51558 > t_c = 1.68023,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for right-tailed $t=2.51558,$ $df=44$ degrees of freedom is $p=0.007804,$ and since $p=0.007804<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

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

is greater than $43700,$ at the $\alpha = 0.05$ significance level.