Question

Task #1 Construct the Models This term: This term, out of 85 students who took the beginning-of-term survey, 29 of them love pineapple on pizza. This gives a sample statistic of p-hat = 29/85 = 0.341. Last two terms: In the Fall and Spring, out of 60 students that took the beginning-of-term survey, 12 of them love pineapple on pizza. This gives a sample statistic of p-hat = 12/60 = 0.2. a) Discuss the assumptions and conditions required to use the Central Limit Theorem and whether they apply here. Regardless, we will use the Central Limit Theorem to do the rest of the analysis, but if any of the conditions are not met or are borderline, we are less convinced by our conclusions that we make. b) Construct the sampling distribution predicted by the Central Limit Theorem for each sample statistic. Task #2: Confidence Intervals c) Construct a 92% confidence interval for each sample statistic. Be sure to explain how you are constructing these. d) Describe what these confidence intervals mean. e) Compare the margins of error for this and the previous sample. Are they similar or drastically different? f) Combine these samples into one large sample. What do you predict will happen to the margin of error for a 92% confidence interval? Why? g) Construct a 92% confidence interval for the combined sample. Task #3: Comparing to a Claim The ::“public opinion and data company” Yougov.com claims:: that p=0.17 of people in the Western US say that they love pineapple on pizza. h) Assume that this claim is true, that p=0.17. Draw the normal distribution that the Central Limit Theorem gives us, and compute the standard deviation when n = 85. Then, find the area in the tail from the sample statistic from this term. Draw a picture of what this represents and explain how you can interpret your results. i) Repeat this for the sample from last two terms. Readjust your model, since n changed. j) Repeat this for the large combined sample. k) Do any of these results lead you to be suspicious ofYougov.com’s claim? Why or why not?

Answer 1

So, this term we have the following information:-

Proportion denotes that out of 85 students 29 of them love pineapple on pizza.

Over the last two terms we have the following information:-

Proportion denotes that out of 60 students 12 of them love pineapple on pizza.

1) Assumptions required to use the central limit theorem :-

a) Randomization:- As the students are randomly selected through a survey. Therefore, randomization is satisfied.

As they are randomized. So, condition of independence also satisfied.

b) Sample size should be sufficiently large. As sample size for both the cases is greater than 30. So, sample size is sufficiently large.

c) 10% condition:- The sample size is not more than 10% of population.

2) The 92% confidence interval for each sample statistic is given by

This term:

From the above confidence interval, we are 92% confidence that the population proportion is in the interval 0.251 and 0.431.

Last two terms :

From the above confidence interval, we are 92% confidence that the population proportion is in the interval 0.11 and 0.29.

3) The margins of error for this term is 0.09003193 .

The  margins of error for lasr two term is 0.09042125.

We can see that they are similar .

4) When we combine them then the proportion will be

The margin of error for the combine proportion is  0.06550206. So, the margin of error is decreased