Question

Coffee: A popular chain of cafes has been receiving online complaints about one store location. Regular customers complained that the staff at this location consistently underfill their cups of coffee. The owner of the chain will personally visit the location to meet with the staff if more than 10% of the 16 ounce cups are underfilled by 1 fluid ounce. She arranges to have "mystery shoppers" visit the store. Mystery shoppers are people who are paid to pose as a regular customer and provide feedback to the owner about customer service. After 100 mystery shop visits, she finds that 18 of the 16 ounce cups of coffee were underfilled. She plans to test the hypotheses LaTeX: H_0H 0: LaTeX: p=0.10p = 0.10 versus LaTeX: H_aH a: LaTeX: p>0.10p > 0.10. What is the test statistic?

Answer 1

Answer:

The test statistic is z=2.5.

The null hypothesis is rejected.

There is enough evidence to support the claim that more than 10% of the 16 ounce cups are underfilled.

Step-by-step explanation:

This is a hypothesis test for a proportion.

The claim is that more than 10% of the 16 ounce cups are underfilled.

Then, the null and alternative hypothesis are:

$H_0: \pi=0.1\\\\H_a:\pi>0.1$

The significance level is assumed to be 0.05.

The sample has a size n=100.

The sample proportion is p=0.18.

$p=X/n=18/100=0.18$

The standard error of the proportion is:

$\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.1*0.9}{100}}\\\\\\ \sigma_p=\sqrt{0.001}=0.03$

Then, we can calculate the z-statistic as:

$z=\dfrac{p-\pi-0.5/n}{\sigma_p}=\dfrac{0.18-0.1-0.5/100}{0.03}=\dfrac{0.075}{0.03}=2.5$

This test is a right-tailed test, so the P-value for this test is calculated as:

$P-value=P(z>2.5)=0.006$

As the P-value (0.006) is smaller than the significance level (0.05), the effect is  significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that more than 10% of the 16 ounce cups are underfilled.