CHAPTER 20
ESTIMATING PROPORTIONS WITH
CONFIDENCE
1.
Explain (in words that a non-statistics student would understand) what is meant by a ‘95%
confidence interval.’
ANSWER: AN INTERVAL OF VALUES THAT COVERS THE TRUE POPULATION
VALUE FOR 95% OF THE SAMPLES SELECTED.
2.
Statisticians have a phrase, “Being a statistician means never having to say you’re certain.” Do
you know whether the confidence interval constructed by your sample actually contains the true
population value? Why or why not?
ANSWER: NO, BECAUSE SAMPLE RESULTS VARY. 95% OF THE SAMPLES WILL
RESULT IN CONFIDENCE INTERVALS THAT ARE CORRECT, AND 5% WON’T.
YOU NEVER KNOW WHETHER A PARTICULAR CONFIDENCE INTERVAL IS
CORRECT UNLESS THE TRUTH IS REVEALED AT SOME FUTURE DATE.
3.
A 95% confidence interval means that 95% of all possible random samples will result in an
interval that contains the true population value, and 5% of them won’t. How could a confidence
interval that is based on a random sample not contain the true population value?
ANSWER: 5% OF THE SAMPLE RESULTS WILL BE UNUSUALLY HIGH OR
UNUSUALLY LOW COMPARED TO THE TRUE POPULATION VALUE, JUST BY
CHANCE.
4.
What is the most common level of confidence used to construct confidence intervals?
a. 5%
b. 90%
c. 95%
d. 100%
ANSWER: C
5.
Which of the following is a correct interpretation of a 90% confidence interval?
a. 90% of the random samples you could select would result in intervals that contain the
true population value.
b. 90% of the population values should be close to our sample results.
c. Once a specific sample has been selected, the probability that its resulting confidence
interval contains the true population value is 90%.
d. All of the above statements are true.
ANSWER: A
6.
Which of the following statements is false?
a. Confidence intervals are always close to their true population values.
b. Confidence intervals vary from one sample to the next.
c. The key to constructing confidence intervals is to understand what kind of dissimilarity
we should expect to see in various samples from the same population.
d. None of the above statements are false.
ANSWER: A
7.
How does a 90% confidence interval compare to a 95% confidence interval?
a. Fewer of the samples will result in intervals that contain the true population value in the
90% case.
b. Fewer of the samples will result in incorrect intervals in the 90% case.
c. With the 90% confidence interval you are less willing to take a chance on missing the
true value.
d. All of the above.
ANSWER: A
8.
A(n) _______________ is an interval of values computed from the sample data that is almost sure
to cover the true population number.
ANSWER: CONFIDENCE INTERVAL
9.
Give an example of a confidence interval for a proportion with a margin of error of 5% that falls
below zero for some of its values.
ANSWER: ANY ANSWER THAT CONTAINS NEGATIVE NUMBERS AS PART OF
THE INTERVAL AND HAS WIDTH OF 10%, SUCH AS 2% PLUS OR MINUS 5%,
WHICH WOULD BE -3% TO 7%.
10. Name two different situations that can result in a confidence interval for a proportion falling into
negative numbers for some of its values (which does not make sense). Give an example of each.
ANSWER: 1) A LARGE MARGIN OF ERROR CAUSED BY A SMALL SAMPLE SIZE
(FOR EXAMPLE, 10% PLUS OR MINUS 20%); OR 2) A SMALL SAMPLE
PROPORTION AND A MARGIN OF ERROR THAT’S LARGER THAN THE SAMPLE
PROPORTION (FOR EXAMPLE, 2% PLUS OR MINUS 5%).
11. Which would be wider, a 90% confidence interval or a 95% confidence interval? (Assume both of
them were calculated using the same sample data.) Explain your answer.
ANSWER: THE 95% CONFIDENCE INTERVAL IS WIDER. SINCE THE DATA
SAMPLE SIZE IS THE SAME AND YOU WANT MORE CONFIDENCE OF COVERING
THE TRUE POPULATION VALUE, YOU NEED A LARGER MARGIN OF ERROR,
WHICH RESULTS IN A WIDER CONFIDENCE INTERVAL.
12. Suppose a survey was conducted to find out what proportion of Americans intend to vote in the
next Presidential election. For which of the following confidence intervals would it be fair to
conclude, with high confidence, that a majority of Americans will vote in the next Presidential
election?
a. 52% plus or minus 3%
b. 52% plus or minus 2%
c. 52% plus or minus 1%
d. All of the above are “too close to call.”
ANSWER: C
Narrative: Bread machines
Suppose a survey was done this year to find out what percentage of all Americans own a bread
machine. Out of their random sample of 1,000 Americans, 317 own a bread machine. The margin of
error for this survey was plus or minus 3%.
13. {Bread machine narrative} What proportion of the entire American population owns a bread
machine, based on these results?
a. Definitely 317/1,000 = .317 or 31.7%
b. Probably between 28.7% and 34.7%
c. 31.7% times the population of the U.S.
d. None of the above.
ANSWER: B
14. {Bread machine narrative} Suppose three years ago, 29% of Americans owned a bread machine.
Based on the results of this current survey, what would you conclude (with high confidence) about
the population of all Americans now compared with three years ago?
a. A larger proportion of Americans own a bread machine now compared to three years ago.
b. A smaller proportion of Americans own a bread machine now compared to three years
ago.
c. The proportion of Americans who own a bread machine has not significantly changed
from three years ago.
d. None of the above.
ANSWER: C
15. Suppose the U.S. government reports that the total number of people in the U.S. who are currently
infected with HIV is likely to be between 300,000 and 1,000,000. What is the margin of error for
these findings? (Assume a symmetric 95% confidence interval.)
a. +/ 350,000
b. +/ 700,000
c. +/ 5%
d. Not enough information to tell
ANSWER: A
Narrative: Oranges
Suppose a shipment of oranges is advertised to weigh 5 pounds per bag. We know that not every bag
can contain exactly 5 pounds of oranges. We decide to take a random sample of 100 bags of oranges
and find out what they tell us about the population of all bags in this shipment. We are only interested
in whether or not the bags are underweight, so each bag is weighed and counted as underweight if it
weighs less than 5 pounds. Five bags in our sample of 100 were found to be underweight.
16. {Oranges narrative} What percentage of all the bags in the entire shipment do you think are
underweight? Give the most complete answer you can.
ACCEPTABLE ANSWERS: .050 PLUS OR MINUS .044; .006 TO .094; 5% PLUS OR
MINUS 4.4%; 0.6% TO 9.4%.
17. {Oranges narrative} What is the margin of error for a 95% confidence interval for the proportion
of bags in the shipment that are underweight?
ANSWER: PLUS OR MINUS .044 (OR 4.4%).
18. {Oranges narrative} Suppose the grocery store who ordered the oranges will reject the shipment if
they believe, based on these sample results, that more than 10% of the bags in the entire truckload
are underweight. Based on our sample, will they have to return this shipment? Explain your
answer.
ANSWER: NO; A 95% CONFIDENCE INTERVAL FOR THE PROPORTION OF BAGS
IN THE SHIPMENT THAT ARE UNDERWEIGHT IS .006 TO .094 (5% PLUS OR
MINUS 4.4%). SINCE .10 (10%) IS NOT IN THIS INTERVAL, THE SHIPMENT IS NOT
REJECTED.
19. Using one divided by the square root of the sample size is known as a ‘conservative’ formula for
the margin of error for a sample proportion. Explain what that means.
ANSWER: THE TRUE MARGIN OF ERROR FOUND BY THE FORMULA 2  (S.D.) IS
ALWAYS LESS THAN OR EQUAL TO THIS VALUE. IT IS EQUAL WHEN THE
PROPORTION USED IN THE FORMULA FOR S.D. IS .50.
20. The formula for calculating a confidence interval for a population proportion is based on the rule
of sample proportions, which has assumptions that need to be met. What is the most important
assumption that you need to check before applying the confidence interval formula to a sample
proportion?
ANSWER: THE SAMPLE SIZE MUST BE LARGE ENOUGH SO THAT YOU ARE
LIKELY TO SEE AT LEAST FIVE OF EACH OF THE TWO POSSIBLE RESPONSES
OR OUTCOMES.
21. Which of the following statements is true regarding a 95% confidence interval? Assume numerous
large samples are taken from the population.
a. In 95% of all samples, the sample proportion will fall within 2 standard deviations of the
mean, which is the true proportion for the population.
b. In 95% of all samples, the true proportion will fall within 2 standard deviations of the
sample proportion.
c. If we add and subtract 2 standard deviations to/from the sample proportion, in 95% of all
cases we will have captured the true population proportion.
d. All of the above.
ANSWER: D
22. Sampling methods and confidence intervals are routinely used for financial audits of large
companies. Which of the following is an advantage of doing it this way, versus having a complete
audit of all records?
a. It is much cheaper.
b. A sample can be done more carefully than a complete audit.
c. A well-designed sampling audit may yield a more accurate estimate than a less carefully
carried out complete audit or census.
d. All of the above.
ANSWER: D
23. In which of the following situations can you construct a confidence interval for the population
proportion with only what is given?
a. The sample proportion and the margin of error.
b. The sample proportion and the sample size.
c. The number of individuals in the sample with the trait of interest, and the total sample
size.
d. All of the above.
ANSWER: D
24. In which of the following situations can a confidence interval provide useful information for
making a decision?
a. To assess whether or not at least 90% of a company’s financial entries are correct.
b. To estimate the percentage of married couples in which the wife is taller than the
husband.
c. To help determine whether or not someone has ESP (versus just being a lucky guesser).
d. All of the above.
ANSWER:
D
25. To obtain a 99.7% (virtually all-encompassing) confidence interval for the true population
proportion, you would add and subtract about __________ standard deviations to/from the sample
proportion.
ANSWER: THREE
26. Using the same sample data, the margin of error for an 80% confidence interval is __________
than the margin of error for a 90% confidence interval.
ANSWER: LARGER