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CHAPTER 21
1.
What three items are needed in order to construct a confidence interval to estimate the mean of a
population?
ANSWER: 1) SAMPLE MEAN; 2) SAMPLE STANDARD DEVIATION; 3) SAMPLE SIZE.
2.
The formula for a confidence interval for the population mean is based on the rule for sample means,
which has assumptions that need to be met. What are those assumptions? (Hint: there are different
assumptions depending on whether or not the population is bell-shaped.)
ANSWER: 1) IF THE POPULATION IS BELL-SHAPED, ANY SIZE RANDOM SAMPLE
CAN BE TAKEN; 2) IF THE POPULATION IS NOT BELL-SHAPED, A LARGE ENOUGH
SAMPLE MUST BE TAKEN (AT LEAST 30 MEASUREMENTS).
Narrative: Quiz times
The introductory biology class at a large university is taught to hundreds of students each semester. For
planning purposes, the instructor wants to find out the average amount of time that students would use to
take the first quiz, if they could have as long as necessary to take it. She takes a random sample of 100
students from this population and finds that their average time for taking the quiz is 24 minutes, and the
standard deviation is 16 minutes.
3.
{Quiz times narrative} Find a 95% confidence interval for the average time to take this quiz for the
whole population of students who take the class.
ANSWER: 24 +/- 2(1.6) MINUTES; OR BETWEEN 20.8 AND 27.2 MINUTES.
4.
{Quiz times narrative} Suppose the professor expects the average time to take the exam is 18
minutes. Do you have enough evidence to say that the professor is wrong in her estimation of the
average time to take this quiz? Use statistics to justify your answer.
ANSWER: YES. YOU ESTIMATE WITH HIGH CONFIDENCE THAT THE AVERAGE
TIME NEEDED TO TAKE THIS QUIZ, OVER THE ENTIRE POPULATION OF
STUDENTS TAKING IT, IS BETWEEN 20.8 AND 27.2 MINUTES. THIS ENTIRE
INTERVAL LIES ABOVE 18 MINUTES, SO THE PROFESSOR IS WRONG IN HER
ASSUMPTION.
5. Which of the following statements is not true regarding a 95% confidence interval for the mean of a
population?
a. In 95% of all samples, the sample mean will fall within 2 standard errors of the true
population mean.
b. 95% of the population values will lie within 2 standard errors of the sample mean.
c. In 95% of all samples, the true population mean will be within 2 standard errors of the
sample mean.
d. If you add and subtract two standard errors to/from the sample mean, in 95% of all cases
you will have captured the true population mean.
ANSWER: B
Narrative: Male weight loss
Suppose a 95% confidence interval for the average amount of weight loss on a diet program for males is
between 13.4 and 18.3 pounds. These results were based on a sample of 42 male participants who were
deemed to be overweight at the start of the 4-month study.
6.
{Male weight loss narrative} What is the sample mean?
a. 2.45
b. 13.4
c. 15.85
d. 18.3
ANSWER:
C
7.
{Male weight loss narrative} What is the margin of error?
a. 4.9
b. 2.45
c. 1.225
d. None of the above
ANSWER:
B
8.
{Male weight loss narrative} What is the standard error of the sample mean?
a. 4.9
b. 2.45
c. 1.225
d. None of the above
ANSWER:
C
9. {Male weight loss narrative} Suppose an individual from this population wants to join this diet
program, and wants to know how much weight he can expect to lose. What can you say, given the
results above?
a. He should expect to lose about ‘up to’ about 18.3 pounds.
b. He should expect to lose ‘up to’ about 31.7 pounds.
c. He should expect to lose about 15.85 pounds.
d. He should expect to lose between 13.4 and 18.3 pounds.
ANSWER:
B
10. You are taking a class with a large number of students. The professor notes that the midterm exam
scores were bell-shaped but doesn’t tell you the mean. You take a random sample of 10 students and
construct a 95% confidence interval for the mean exam score. Now the professor wants to give
everyone 5 extra points on the exam. How will your confidence interval change?
a. The sample mean will increase by 5 points and the margin of error will also.
b. The sample mean will increase by 5 points but the margin of error will stay the same.
c. The sample mean will stay the same but the margin of error will increase by 5 points.
d. Both the sample mean and the margin of error will remain unchanged.
ANSWER: B
11. The standard deviation of all possible sample means (from same sized samples) is called the
_______________ of the mean.
ANSWER: STANDARD ERROR
12. In practice the population standard deviation, which appears in the formula for the standard error of
the mean, is not known. Statisticians replace it with the _______________, computed from the data.
ANSWER: SAMPLE STANDARD DEVIATION
CONFIDENCE INTERVALS FOR THE DIFFERENCE BETWEEN
TWO MEANS
13. Describe a general scenario where you would use a confidence interval for the difference between
two means.
ANSWER: WHEN YOU ARE INTERESTED IN COMPARING THE POPULATION
MEANS UNDER TWO CONDITIONS OR FOR TWO GROUPS.
14. How should you collect your data in order to use it in a confidence interval for the difference of two
means?
ANSWER: COLLECT A LARGE SAMPLE OF AT LEAST 30 OBSERVATIONS,
INDEPENDENTLY, UNDER EACH CONDITION COMPARED, OR FROM EACH
GROUP STUDIED.
Narrative: Cell phone use
How do men and women compare when it comes to talking on the cell phone? Suppose you take a
random sample of 100 male cell phone owners and a random sample of 100 female cell phone owners.
The average number of minutes for the women per month was 280 with a standard deviation of 10; the
average number of minutes for the men per month was 190 with a standard deviation of 30.
15. {Cell phone use narrative} Based on these sample results, what is the difference in average time
spent on the cell phone for females versus males?
ANSWER: WE ARE 95% CONFIDENT THAT THE DIFFERENCE IN AVERAGE TIMES
(FEMALES – MALES) PER MONTH WAS 90 MINUTES, PLUS OR MINUS 6.32
MINUTES, OR BETWEEN 83.68 AND 96.32 MINUTES.
16. {Cell phone use narrative} Suppose the women had only talked for 195 minutes per month, on
average, and all the other statistics remained the same. Find the lower and upper limits for the
resulting confidence interval (females – males), and interpret the results. Who’s talking now?
ANSWER: -1.32 TO 11.32 MINUTES FOR THE DIFFERENCE (FEMALES – MALES).
SINCE ZERO IS IN THIS INTERVAL, YOU CAN’T CONCLUDE THAT THERE IS A
REAL DIFFERENCE DETECTABLE IN THE POPULATION WITH THESE SAMPLES,
ALTHOUGH THE EVIDENCE SUGGESTS FEMALES STILL TALK MORE BECAUSE
MANY MORE OF THE NUMBERS IN THE INTERVAL ARE POSITIVE.
17. Which of the following is the correct way to calculate the standard error of the difference between
two (independent) means? Let SEM1 represent the standard error of the first mean, and let SEM2
represent the standard error of the second mean.
a. SEM1 + SEM2
b. SEM1 − SEM2.
c. The square root of (SEM1)2 + (SEM2)2.
d. The square root of (SEM1)2 − (SEM2)2.
ANSWER: C
Narrative: Classes 1 and 2
Suppose a 95% confidence interval for the difference in test scores between Class 1 and Class 2 (in that
order) is the following: 9 +/− 2. These results were based on independent samples of size 100 from each
class.
18. {Classes 1 and 2 narrative} What can you conclude?
a. You are confident that the averages for Class 1 and Class 2 are significantly different.
b. You are confident that the average for Class 1 was 7 to 11 points higher than for Class 2.
c. You are confident that the observed difference found in the two samples (plus or minus the
margin of error) will carry over to their respective populations.
d. All of the above.
ANSWER: D
19. What does it mean for a confidence interval for the difference of two means to contain zero?
a. You are unable to say there is a difference in the population means.
b. Different samples could give results in either direction; completely above zero, completely
below zero, or containing zero.
c. The confidence interval will contain some negative numbers and some positive numbers.
d. All of the above.
ANSWER: D
20. You can determine confidence intervals for individual means or for the difference in two means (for
large, independent samples) as long as you are given these three things: the means, the standard
deviations, and the __________.
ANSWER: SAMPLE SIZES
21. You can determine confidence intervals for individual means or for the difference in two means (for
large, independent samples) as long as you are given these two things: the means and the
__________.
ANSWER: STANDARD ERRORS OF THE MEANS
22. Are all confidence intervals (whether they be for population means, proportions, or anything else)
symmetric around their respective sample values? Explain.
ANSWER: NO. THE RULE FOR SAMPLE PROPORTIONS AND THE RULE FOR
SAMPLE MEANS SAY THAT THIS IS TRUE FOR MEANS AND PROPORTIONS, IF THE
SAMPLE SIZE IS LARGE ENOUGH. BUT FOR OTHER POPULATION VALUES, THEIR
SAMPLE ESTIMATES MAY NOT HAVE A SYMMETRIC SHAPE.
23. Suppose you want to calculate a confidence interval to estimate the population mean. What happens
if you want to be more than 95% confident in your results but you can’t take another sample?
(Assume data quality is not an issue.)
a. You need a larger confidence level than 95%.
b. You need to use a multiplier that is larger than 2 to create your margin of error.
c. You will have to live with a wider confidence interval.
d. All of the above.
ANSWER:
D
24. For a 95% confidence interval, the value of 95% is called the _______________.
ANSWER: CONFIDENCE LEVEL
25. A confidence level is a measure of how much confidence we have that the __________ we used to
generate the interval worked.
ANSWER: PROCEDURE (OR SAMPLE)