Question

Between January and June of 2011, The Gallup survey company conducted telephone interviews. The interviews were part of the Gallup-Healthways Well-Being Index survey. The researchers interviewed a random sample of adults, aged 18 and older. The adults lived in all 50 of the U.S. states and in the District of Columbia. Approximately 500 of the people who were interviewed were men and 500 were women. The people who were interviewed were asked about how much and how often they exercise. The researchers asked these people if they exercise for 30 minutes a day, three or more times each week. Among the women, 47.8% answered “Yes” whereas 54.8% of the men answered “Yes”. The researchers estimated that the 95% confidence interval for the true difference between the proportions of all U.S adult women and men who would answer “Yes” to the question was between -13.2% and -0.82%.

Part 1. Which of the following statements is the best interpretation of the 95% confidence interval?

A. We are 95% confident that, across the United States, men are between 0.82% and 13.2% more likely to exercise for 30 minutes at least 3 times a week than are women.

B. There is between a 0.82% and 13.2% chance that the true percentage of U.S. men who exercise for 30 minutes at least 3 times a week is higher than the percentage of women.

C. We are 95% confident that in the U.S, there are between 0.82% and 13.2% less women exercising for 30 minutes at least 3 times a week than men.

Part 2. Think about the results of the percentage of women and men in the United States who exercise for 30 minutes at least 3 times a week. Which of the following statements are supported by the poll results?

A. There is no difference between the percentage of women and men because both sample percents are close to 50%.

B. There is no difference between the percentage of women and men because the confidence interval includes the -5% level of significance.

C. There is a difference between the percentage of women and men because the sample percentage for women who said “Yes” is below 50%, but above 50% for the sample of men.

D. There is a difference between the percentage of women and men because the confidence interval does not include 0%.

Part 3. Which of the following statements is the best interpretation of the statistical term “95% confidence”?

A. 95% of the time that the same methods are used, the interval estimate will include the true difference between the percentages for women and men in the United States.

B. Adults from 95% of all the states in the United States were included in the poll that was conducted between January and June of 2011.

C. There is a 95% chance that the true percentage of women in the United States who exercise less often than men is between 0.82% and 13.2%.

D. Between 0.82% and 13.2% of the men in 95% of the states in the United States exercised more often than women.

Answer 1

The 95% confidence interval for the true difference between the proportions of all U.S adult women and men who would answer “Yes” to the question was between -13.2% and -0.82%.

with 95% confidence.

1.

A. is considered here. So this is not best interpretation.

B. The term '95% confidence interval' is ignored here. So this is not best interpretation.

C. This is the best interpretation.

Hence, the best interpretation is (C).

2.

A. This is not a proper statistical testing conclusion as there is no use of standard deviation term here.

B. Since level of significance denotes probability of type I error, it can not be negative. So, this is an invalid statement.

C. This is not a proper statistical testing conclusion as there is no use of standard deviation term here.

D. As , this is the statement that is supported by the poll results.

Hence, (D).

3.

The statement suggest that if we take more and more samples, on average in 95% of the time we shall obtain the difference in the interval .

Hence, the best interpretation is (A).