Question

2. Based on past data, it is believed that on Thanksgiving, 70% of people eat pumpkin pie. Suppose we take a sample of 100 people and find that 75 of the people in that sample eat pumpkin pie on Thanksgiving. We want to see if there is evidence that the percentage of people who eat pumpkin pie on Thanksgiving is increasing. a. If I wanted to control my margin of error and set it to 3% with 99% confidence, what sample size would I need to take instead of the 100? b. Using my original sample size of 100, what would be the 99% confidence interval for the population proportion? c. What are the null and alternative hypotheses? d. What is the critical value at 99% confidence? e. Calculate the test statistic (using the sample of 100). f. Find the p-value. g. What conclusion would be made here at the 99% confidence level?

Answer 1

a) From standard normal tables, we have:
P( -2.576 < Z < 2.576) = 0.99

Also as we have a prior proportion estimate here as 0.7, the margin of error is computed here as:

Therefore 1549 is the required sample size here.

b) The sample proportion here is computed as:

p = x/n = 75/100 = 0.75

The 99% confidence interval here is computed as:

This is the required 99% confidence interval here.

c) As we are testing here whether the percentage of people who eat pumpkin pie on Thanksgiving is increasing that is more than 0.7, therefore the null and the alternate hypothesis here are given as:

d) From standard normal tables, we have here:
P( Z < 2.326) = 0.99

Therefore 2.326 is the required critical value here.

e) The test statistic here is computed as:

Therefore 1.0911 is the test statistic value here.

f) As this is an upper tailed test, the p-value here is computed from the standard normal tables as:

p = P(Z > 1.0911) = 0.1376

Therefore 0.1376 is the required p-value here.

g) As the p-value here is 0.1376 > 0.01 which is the level of significance, therefore the test is not significant and we cannot reject the null hypothesis here. Therefore we dont have sufficient evidence here that the mean is increasing here.