Question

(1 point) Comfortable retirement. According to a recent Gallup poll on personal finances, 43% of U.S. workers say they feel they will have enough money to live comfortably when they retire. The superintendent of a large school district thinks that her employees might not be as optimistic. She takes a simple random sample of 50 district employees, and only 19 of them say they feel they will have enough money to live comfortably when they retire. Round all answers to four decimal places.

1. The superintendent frames her research question using the sentences below but can't remember which one is the null hypothesis and which one is the alternative hypothesis. Help her by labeling the statements.

? H_0 H_A  The percentage of employees in the district who feel they will have enough money to live comfortably on when they retire is less than 43% and the difference in the sample is not due to chance.

? H_0 H_A  The percentage of employees in the district who feel they will have enough money to live comfortably on when they retire is the same as the nationally reported value of 43% and the difference in the sample is due to chance.

2. The paragraph below describes the set up for a randomization test, if we were to conduct a hypothesis test without using software. Fill in the blanks with a number or select the appropriate choice out of those provided.

To setup a simulation for this situation, we let each person surveyed be represented with a card. We take 100 cards,  red cards representing people who feel they will have enough money and  black cards representing people who do not. Shuffle the cards and draw  cards with replacement representing the random sample of employees. Calculate the proportion of  ? black red  cards in the  ? sample deck  and call it p^simp^sim. Repeat 3,000 times and plot the resulting sample proportions. The p-value for the test will be the proportion of simulations where p^simp^sim is  ? less than greater than beyond   .

3. Report your p-value rounded to four decimal places:

p-value =

4. Based on the p-value, we have:
A. some evidence
B. extremely strong evidence
C. little evidence
D. strong evidence
E. very strong evidence
that the null model is not a good fit for our observed data.

Answer 1

1. Ho: The percentage of employees in the district who feel they will have enough money to live comfortably on when they retire is the same as the nationally reported value of 43% and the difference in the sample is due to chance.

Ha: The percentage of employees in the district who feel they will have enough money to live comfortably on when they retire is less than 43% and the difference in the sample is not due to chance.

2. We take 100 cards,  red cards representing people who feel they will have enough money and black cards representing people who do not. Shuffle the cards and draw cards with replacement representing the random sample of employees. Calculate the proportion of red cards in the sample and call it p^simp^sim. Repeat 3,000 times and plot the resulting sample proportions. The p-value for the test will be the proportion of simulations where p^simp^sim is less than.

3. The test statistic is given by:

, here the test statistic follows a standard normal distribution under Ho.

thus the p-value is: P(Z<-0.714) = 0.2376

4. Based on the p-value, we have:
C. little evidence that the null model is not a good fit for our observed data.