Answer to Question #345181 in Statistics and Probability for Josh

Question

The leader of the Factory Worker claims that the average pay of factory workers in

Marikina City is less than 500 php. A random sample of 16 factory workers was

interviewed and resulted on the average of 350 php. Using a significance level of

0.05 and standard deviation of 92. Determine if there is sufficient evidence that the

claim is true.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\ge500$

$H_1:\mu<500$

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 $R = \{z<-1.6449\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{350-500}{92/\sqrt{16}}\approx-6.5217$$

Since it is observed that $z=-6.5217<-1.6449=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=P(z<-6.5217)=0,$ 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 less than 500, at the $\alpha = 0.05$ significance level.