Answer to Question #333850 in Statistics and Probability for Bella

Question

2. A researcher claims that the average salary of a private school teacher is greater than P20,000 with a standard deviation of P5,000. A sample of 20 teachers has a mean salary of P25,000. Test the claim of the researcher. At 0.05 level of significance, test the claim of the researcher.

Answer the following:

Parameter:

Claim:

Claim ( in symbol ):

Ho: Ho:

Ha: Ha:

What is the significance level or a?

Is it two-tailed or one-tailed test?

Answer 1

Parameter: mean

Claim: the average salary of a private school teacher is greater than P20,000 

The following null and alternative hypotheses need to be tested:

$H_0:\mu\le20000$

$H_a:\mu>20000$

One-tailed test.

Significance level is $\alpha = 0.05.$

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:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{25000-20000}{5000/\sqrt{20}}\approx4.4721$$

Since it is observed that $z = 4.4721>1.6449= z_c ,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is $p=P(Z>4.4721)=0,$ and since $p=0<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

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

is greater than 20000, at the $\alpha = 0.05$ significance level.