Answer to Question #341653 in Statistics and Probability for yumii

Question

According to a study conducted by grade 11 students, Php155 is the monthly expense for cellphone loads of high school students on their province. A statistics students claims that this amount has increased since January of this year. Do you think his claim is acceptable if a random sample of 50 students has an average monthly expense of Php165 on cellphone loads? Using 0.05 level of significance, assume that a population standard deviation of Php52.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\le155$

$H_1:\mu>155$

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:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{165-155}{52/\sqrt{50}}\approx1.35982$$

Since it is observed that $z=1.35982<1.6449=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=P(z>1.35982)=0.086943,$ and since $p=0.086943>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

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

is greater than 155, at the $\alpha = 0.05$ significance level.