Question

Solve this hypothesis test problem using a significance level not previously used for this specific problem. Include the null and alternative hypothesis, alpha value, p-value, and a conclusion. Make sure that you use appropriate terminology, specify whether you are using the classical method or the p-value method, and fully explain your solution.

My grandson listens to music for much of his day. Most of the time when I am able to hear the music he is listening to, it seems to be RAP Music. I asked my grandson if he knew how many RAP songs he had on his phone. He said he wasn’t sure however I believe there to be more than half of the 350 songs he has on his phone are of the RAP Genre. We decided to take a simple random sample of 175 songs and found that 89 of those songs were of the RAP Genre.

Does this represent significant evidence that more than half of the songs on my grandson’s phone are of the RAP Genre? Test for the hypothesis at the .05 significance level.

Answer 1

Null Hypothesis H0: Proportion of RAP Genre songs on the phone is less than or equal to 0.50. That is p 0.50.

Alternative Hypothesis Ha: Proportion of RAP Genre songs on the phone is greater than 0.50. That is p > 0.50.

The significance level is 0.05

np(1-p) = 175 * 0.5 * (1-0.5) = 43.75

Since np(1-p) > 10, the sampling distribution of proportion can be approximated as normal distribution and we can use one-sample proportion test.

We are using p-value method to compute the test.

Standard error of proportion, SE = = 0.0378 (We are not doing finite population correction as it is not mentioned in the problem).

Sample proportion, = 89/175 = 0.5086

Test statistic, z = ( - p) / SE = (0.5086 - 0.5) / 0.0378 = 0.2275

P-value = P(z > 0.2275) = 0.4100 (Using Standard Normal distribution)

Since, p-value is greater than 0.05 significance level, we fail to reject null hypothesis H0 and conclude that there is no strong evidence that proportion of RAP Genre songs on the phone is greater than 0.50.