Question

Company A makes a large shipment to Company B. Company B can reject the shipment if they can conclude that the proportion of defective items in the shipment is larger than 0.1. In a sample of 400 items from the shipment, Company B finds that 53 are defective. Conduct the appropriate hypothesis test for Company B using a 0.1 level of significance.
a) What are the appropriate hypotheses?

H0: p = 0.1 versus Ha: p ≠ 0.1

H0: p = 0.1 versus Ha: p > 0.1 (I got this as the correct answer)

H0: p = 0.1 versus Ha: p < 0.1

H0: μ = 0.1 versus Ha: μ > 0.1

b) What is the test statistic? Give your answer to four decimal places. -> the corret answer I got is 2.16666

c) What is the critical point for the test? Give your answer to four decimal places. -> the incorrect answer I got is 1.64868

d) What is the appropriate conclusion?

- Fail to reject the claim that the defective proportion in the shipment is 0.1 because the test statistic is smaller than the critical point.

- Conclude that the defective proportion in the shipment is greater than 0.1 because the test statistic is smaller than the critical point.

- Fail to reject the claim that the defective proportion in the shipment is 0.1 because the test statistic is larger than the critical point. (This is the answer I picked and it is incorrect)

- Conclude that the defective proportion in the shipment is greater than 0.1 because the test statistic is larger than the critical point.

Answer 1

Answer: a) $H_0: p=0.1\\H_a:p>0.1$

b) 2.166666

c) 1.2816

d) Conclude that the defective proportion in the shipment is greater than 0.1 because the test statistic is larger than the critical point.

Step-by-step explanation:

Let p be the population proportion.

As per given , Company B can reject the shipment if they can conclude that the proportion of defective items in the shipment is larger than 0.1.

Thus, the appropriate hypotheses :

a) $H_0: p=0.1\\H_a:p>0.1$, since alternative hypothesis is right-tailed , so the test is a right-tailed test.

Also , n=400

sample proportion of defective : $\hat{p}=\dfrac{53}{400}=0.1325$

b) Test statistic : $z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}$

$z=\dfrac{0.1325-0.1}{\sqrt{\dfrac{0.1(1-0.1)}{400}}}\approx2.1666666$

c) Using , z-value table ,

Critical value for 0.1 level of significance.:$z_{0.1}=1.2816$

Since, the test statistic value(2.166666) is greater than the critical value (1.2816) so we reject the null hypothesis.

d) Conclusion: We conclude that the defective proportion in the shipment is greater than 0.1 because the test statistic is larger than the critical point.