Question

Close Friends: The 2004 General Social Survey (GSS) interviewed a random sample of adult Americans. For one question the interviewer asked: “From time to time, most people discuss important matters with other people. Looking back over the last six months – who are the people with whom you discussed matters important to you? Just tell me their first names or initials.” The interviewer then recorded how many names or initials the respondent mentioned. Results are tallied in the following table: Number of Close Friends 0 1 2 3 4 5 6 Total Count (number of respondents) 397 281 263 232 128 96 70 1467 Answer the following questions in complete sentences. When showing work, calculator commands with correct parameters is sufficient. (You do not have to include keystrokes).

d. Which two of the following are reasonable interpretations of this confidence interval and its confidence level.

1. You can be 90% confident that then mean number of close friends in the population is between the endpoints of this interval.

2. Ninety percent of all people in this sample reported a number of close friends within this interval.

3. If you took another sample of 1467 people, there is a 90% chance that its sample mean would fall within this interval.

4. If you repeatedly took random samples of 1467 people, this interval would contain 90% of your sample means in the long run.

5. If you repeatedly took random samples of 1467 people and constructed tintervals in this same manner, 90% of the intervals in the long run would include the population mean number of close friends.

6. This interval captures the number of close friends for 90% of the people in the population.

e. Choose one of the four incorrect interpretations in part d and explain why it is incorrect.

f. Describe how the interval would change if all else remained the same except the following change was made: 1. The sample size was larger 2. The sample mean was larger 3. The sample values were less spread out 4. Every person in the sample reported one more close friend Now, from this same data g. Produce and interpret a 90% confidence interval for the population proportion of people who would report having 0 close friends. h. What information is gained in reporting a t-interval for the population mean rather than for these intervals?

Answer 1

d) At 90% confidence interval and df = 1467-1=1466, critical value, t* = 1.646

90% confidence interval for mean

Answer 1. You can be 90% confident that then mean number of close friends in the population is between the 1.911 and 2.063 of this interval.

4. If you repeatedly took random samples of 1467 people, this interval would contain 90% of your sample means in the long run.

e) 2. Ninety percent of all people in this sample reported a number of close friends within this interval.

Its is incorrect because it interval does not make interpratation about the sample.

f) 1. The sample size was larger : the confidence interval will decrease.

2. The sample mean was larger : the confidence interval will shift. but will will remain same.

3. The sample values were less spread out : the confidence interval will decrease.

4. Every person in the sample reported one more close friend :  the confidence interval will remain same.