Answer to Question #346878 in Statistics and Probability for Yana

Question

The manager of the machine-made kretek kretek cigarette company RAENAK found that 12% of its production was below production standards (poor). One day, the manager received a report that on that day 17% of substandard cigarette products were obtained from a random sample of 1000 cigarettes. Test whether the manufacturing defects (17%) are random or permanent (machines need to be repaired)! Use a= 0.05!

Answer 1

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

$H_0:p\le0.12$

$H_a:p>0.12$

This corresponds to a right-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 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:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{0.17-0.12}{\sqrt{\dfrac{0.12(1-0.12)}{1000}}}=4.8656$$

Since it is observed that $z = 4.8656>1.6449= z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=P(Z>4.8656)=0.000001,$ and since $p=0.000001<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $p$ is greater than 0.12, at the $\alpha = 0.05$ significance level.