Question

The official product store clerk at a university states that at least 40% of students have purchased one of the shirts with the university logo. However, the store owner believes that this figure is not true and that the design of the T-shirt should be changed to a more attractive one. To find out if the clerk is right, the owner asks 80 students if they had ever bought at least one of the university's T-shirts, 28 of them in them had. Assuming a significance level of 0.05, would we have enough evidence to claim that the clerk is wrong?

Type: proportion, 1 population

n = 80

p bar = 0.35

null hypothesis H_0: p = 0.4

alternative hypothesis H_1: p < 0.4

What is the p value?

The answer should be 0.213, I am not sure how they got this, I kept getting another answer. Please show any formulas and work used.

Answer 1

The following information is provided: The sample size is N = 80, the number of favorable cases is X = 28 and the sample proportion is pˉ​=X/N​=28/80​=0.35, and the significance level is α=0.05 (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p >=0.4 Ha: p <0.4 This corresponds to a Left-tailed test, for which a z-test for one population proportion needs to be used. (2a) Critical Value Based on the information provided, the significance level is α=0.05, therefore the critical value for this Left-tailed test is Zc​=-1.6449. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Left-tailed test is Z<-1.6449 (3) Test Statistics The z-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is p =P(Z<-0.9129)=0.1807 Using excel formula =NORM.S.DIST(-0.9129,TRUE) (The answer is not 0.213, it is not possible) (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that Z=-0.9129 > Zc​=-1.6449, it is then concluded that the null hypothesis is Not rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.1807, and since p=0.1807>0.05, it is concluded that the null hypothesis is Not rejected. (6) Conclusion It is concluded that the null hypothesis Ho is Not rejected. Therefore, there is Not enough evidence to claim that the population proportion p is less than 0.4, at the 0.05 significance level.