Question

Part 2: More Review of Confidence Intervals

The following questions might be more challenging, but we want you to wrestle through them and ask for clarification along the way. Talking through these problems with a neighbor can help, and we hope that, ultimately, working through these problems will strengthen your understanding of the big ideas behind confidence intervals.

1. A 95% confidence interval is constructed in order to estimate the average number of minutes college students spend on Facebook per day. The interval ends up being from 30.1 minutes to 47.1 minutes. Based on this interval, we know the sample mean must be __________________ and the margin of error must be _____________ . (Note that your answers should be in the form of numbers, and it might help to review the general format of the confidence interval presented earlier in this lab activity).

2. A 95% confidence interval for the proportion of college students who have texted during a class was 0.75 to 0.95. Which of the following is the 90% confidence interval from the same sample?

1. 0.05 to 0.25
2. 0.731 to 0.969
3. 0.766 to 0.934
4. 0.777 to 0.9

3. A 90% confidence interval is constructed in order to estimate the average number of hours college students spend studying per week. The resulting interval has a margin of error of 3 hours. Which of the following could be the margin of error for a 95% confidence interval based on the same sample of data?

A. 2 hours

B. 3 hours

C. 4 hours

D. 8 hours

E. This cannot be answered without knowing the sample size.

4. Which of the following statements about confidence intervals is not correct?  

1. A confidence interval is an interval of values computed from sample data that is likely to include the population parameter.
2. The general format of a confidence interval is “sample statistic ± margin of error.”
3. Doubling the population size will result in a more narrow confidence interval.
4. If you construct a confidence interval for a population mean, the size of the sample mean has no effect on the size of the margin of error.

5. Suppose that a survey is planned to estimate the proportion of the population of OSU students who are left-handed. The sample data will be used to form a confidence interval. Which one of the following combinations of sample size and confidence level will give the widest confidence interval?

1. n = 400, confidence level = 90%
2. n = 400, confidence level = 95%
3. n = 1000, confidence level = 90%
4. n = 1000, confidence level = 95%

6. A 95% confidence interval is calculated for the percentage of OSU students who believe the parking options offered at OSU are satisfactory. The resulting confidence interval is 59.5% to 64.4%. Based on this information, which of the following is not true?

1. The confidence interval was produced by a process that will capture the true population percentage 95% of the time.
2. We are 95% confident that the interval 59.5% to 64.4% contains the true population percentage of OSU students who believe the parking options offered at OSU are satisfactory.
3. We are 95% confident that the interval 59.5% to 64.4% contains the sample percentage of OSU students who believe the parking options offered at OSU are satisfactory.
4. The sample percentage was about 62%.

7. Based on a random sample of data, an administrator at Sweet Valley High School estimates, at a 99% confidence level, that 22% ± 8% of Sweet Valley High School students plan to take summer classes. If the school has 1420 students, this means the possible number of students who plan to take summer classes is from

1. 199 to 426 students.
2. 195 to 430 students.
3. 114 to 312 students.
4. 47 to 178 students.

Answer 1

(1)

Question:

A 95% confidence interval is constructed in order to estimate the average number of minutes college students spend on Facebook per day. The interval ends up being from 30.1 minutes to 47.1 minutes. Based on this interval, we know the sample mean must be __________________ and the margin of error must be _____________ .

Margin of Error = 38.6 - 30.1 = 8.5

So,

Answer is:

A 95% confidence interval is constructed in order to estimate the average number of minutes college students spend on Facebook per day. The interval ends up being from 30.1 minutes to 47.1 minutes. Based on this interval, we know the sample mean must be 38.6 and the margin of error must be 8.5.

(2)

Question:

A 95% confidence interval for the proportion of college students who have texted during a class was 0.75 to 0.95. Which of the following is the 90% confidence interval from the same sample?

Correct option:

C     0.766 to 0.934

Explanation:

The width of the confidence interval decreases as confidence level decreases from 95% to 90%, keeping the sample mean the same.

(3)

Question:

A 90% confidence interval is constructed in order to estimate the average number of hours college students spend studying per week. The resulting interval has a margin of error of 3 hours. Which of the following could be the margin of error for a 95% confidence interval based on the same sample of data?

Correct option:

E.   This cannot be answered without knowing the sample size.

Explanation:

The width of the confidence interval increases as confidence level increases from 90% to 95%, keeping the sample mean the same. So, Option:C 4 hours and Option D: 8 hours can be correct. We require the sample size to decide betweenthese 2 optons.

(4)

Question:

Which of the following statements about confidence intervals is not correct?  

Correct option:

Doubling the population size will result in a more narrow confidence interval.

Explanation:
Corrected answer is: Doubling the sample size will result in a more narrow confidence interval.

(5)

Question:

Suppose that a survey is planned to estimate the proportion of the population of OSU students who are left-handed. The sample data will be used to form a confidence interval. Which one of the following combinations of sample size and confidence level will give the widest confidence interval?

Correct option:

B.    n = 400, confidence level = 95%

Explanation:

The width of the confidence interval increases as confidence level increases. The width of the confidence interval increases as sample size decreases.

(6)

Question:

A 95% confidence interval is calculated for the percentage of OSU students who believe the parking options offered at OSU are satisfactory. The resulting confidence interval is 59.5% to 64.4%. Based on this information, which of the following is not true?

Correct option:

We are 95% confident that the interval 59.5% to 64.4% contains the sample percentage of OSU students who believe the parking options offered at OSU are satisfactory.

Explanation:

The 95% Confidence Interval 59.5% to 64.4% is a range of values we are 95% confident will contain unknown true population parameter.

(7)

Question:

Based on a random sample of data, an administrator at Sweet Valley High School estimates, at a 99% confidence level, that 22% ± 8% of Sweet Valley High School students plan to take summer classes. If the school has 1420 students, this means the possible number of students who plan to take summer classes is from

Correct option:

A.   199 to 426 students

Explanation:

1420 X 22/100 = 312.4

1420 X 8/100 = 113.6

312.4 -113.6= 198.8 =199 (Round to integer)

312.4 +113.6= 198.8 = 426