Answer to Question #208586 in Statistics and Probability for Abel

Question

the expected performance of a group of machines is that they should operate for 80% of the available time, the remaining 20% being scheduled for maintenance and setting-up. it is believed that one machine of the group is not achieving this target. the machine was observed at 500 random instants of time and was found to be working on 370 occasions. do these results suggest that the machine was running significantly below the target efficiency?

Answer 1

The following null and alternative hypotheses for the population proportion needs to be tested:

$H_0: p=0.8$

$H_1: p<0.8$

This corresponds to a left-tailed test, for which a z-test for one population proportion 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:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$$

$$\hat{p}=\dfrac{370}{500}=0.74$$

$$z=\dfrac{0.74-0.8}{\sqrt{\dfrac{0.8(1-0.8)}{500}}}=-3.3541$$

Since it is observed that $z=-3.3541<-1.6449=z_c,$ it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $p$ is less than $0.8,$ at the $\alpha=0.05$ significance level.

Using the P-value approach: The p-value is $p=P(z<-3.3541)=0.0004,$ and since $p=0.0004<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $p$ is less than $0.8,$ at the $\alpha=0.05$ significance level.

Therefore there is enough evidence to claim that the machine was running significantly below the target efficiency, at the $\alpha=0.05$ significance level.