Answer to Question #175105 in Statistics and Probability for earl

Question

According to a study done last year, the average monthly expenses for mobile phone loads of college students in San Mateo, Rizal was P 400.00. A statistics student believes that this amount has decreased since January of this year. Is there a reason to believe that this amount has decreased if a random sample of 50 students has an average monthly expense for mobile phone loads of P 380.00? Use .05 level of significance. Assume the population standard deviation is P 75.00.  

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\geq400$

$H_1:\mu<400$

This corresponds to a left-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 left-tailed test is $z_c=-1.6449.$

The rejection region for this left-tailed test is $R:\{z:z<-1.6449\}.$

The z-statistic is computed as follows:

$$z=\dfrac{x-\mu}{\sigma/\sqrt{n}}=\dfrac{380-400}{75.00/\sqrt{50}}=-1.8856$$

Since it is observed that $z=-1.8856<-1.6449=z_c,$  it is then concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 400, at the 0.05 significance level.

Using the P-value approach: The p-value is $p=P(Z<-1.8856)=0.0297,$ and since $p=0.0297<0.05=\alpha,$ it is concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 400, at the 0.05 significance level.