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CHAPTER 19
THE DIVERSITY OF SAMPLES FROM THE SAME
POPULATION
Narrative: Bananas
Suppose a researcher asks the question: What is the average weight of bananas selected for purchase
by customers in grocery stores? Assume the distribution of weights is the same across all stores.
1.
{Bananas narrative} How can the researcher go about answering this question? Give a general
description of the process.
ANSWER: TAKE A SAMPLE, RECORD THE AVERAGE WEIGHT OF THE SAMPLE,
AND THEN USE THAT TO ANSWER THE QUESTION ABOUT THE POPULATION
(ALONG WITH SOME MEASURE OF ERROR).
2.
{Bananas narrative} Suppose the researcher takes a sample of 100 shoppers and finds the average
weight of their banana purchases to be 2.1 pounds. Can he just go ahead and report that the
average banana purchase for all grocery store customers is 2.1 pounds? Why or why not?
ANSWER: NO; SAMPLE RESULTS WILL VARY FROM SAMPLE TO SAMPLE, SO
THE SAMPLE AVERAGE WILL PROBABLY NOT BE EXACTLY EQUAL TO THE
POPULATION AVERAGE.
3.
Which of the following statements is false?
a. Sample results will always be very close to their respective population values.
b. Sample results vary from one sample to the next.
c. The key to interpreting statistical results 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
4.
Suppose you knew that most samples were likely to provide an answer that is within 10% of the
population value. What would also be true in that case?
a. The population value should be within 10% of whatever our specific sample gave us.
b. 10% of the population values should be close to whatever our specific sample gave us.
c. The chance that a specific sample answer is correct (equal to the population value) is
90%.
d. All of the above statements are true.
ANSWER: A
Narrative: Politics
Suppose a population contains 60% Republicans and 40% Democrats.
5.
{Politics narrative} Suppose you take a random sample of 10 people from this population. Are you
certain that you would get 6 Republicans and 4 Democrats in your sample? Explain your answer.
ANSWER: NO; SAMPLE RESULTS VARY FROM SAMPLE TO SAMPLE; OR, THE
LONG-TERM AND SHORT-TERM PROBABILITIES ARE NOT THE SAME.
6.
{Politics narrative} Is it possible to get a random sample that does not represent the population
well, in terms of Democrats and Republicans? Explain your answer.
ANSWER: YES; THIS CAN HAPPEN JUST BY CHANCE, BECAUSE SAMPLE
RESULTS VARY, ESPECIALLY WITH SMALL SAMPLES.
7.
{Politics narrative} Suppose you take a random sample of 10 people from this population. Does
the rule for sample proportions apply in this situation? Explain your answer.
ANSWER: NO; THE SAMPLE SIZE IS NOT LARGE ENOUGH. THE EXPECTED
NUMBER OF DEMOCRATS IS ONLY 4, WHICH IS LESS THAN THE 5 NEEDED FOR
THE RULE TO HOLD.
8.
{Politics narrative} Suppose numerous random samples of size 1,000 are taken from this
population. How will the shape, mean, and standard deviation of the frequency curve for the
proportions of Democrats in the samples differ from the shape, mean, and standard deviation of
the frequency curve for the proportions of Republicans in the samples?
ANSWER: BOTH WILL BE BELL-SHAPED; BOTH WILL HAVE THE SAME
STANDARD DEVIATION (.015); THE MEANS WILL DIFFER (.60 FOR THE
REPUBLICANS, AND .40 FOR THE DEMOCRATS).
9.
In practice you don’t know the population value, and you take a sample in order to estimate what
the population value is. Once you take a specific sample, is it possible to determine whether or not
that sample is an accurate reflection of the population? Explain your answer.
ANSWER: NO; YOU WOULD HAVE TO KNOW THE POPULATION VALUE, AND
YOU DON’T, WHICH WAS THE WHOLE POINT OF TAKING THE SAMPLE. EVEN
RANDOM SAMPLES CAN BE MISREPRESENTATIVE, JUST BY CHANCE.
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell
phone owners.
10. {Cell phone owners narrative} The frequency curve made from proportions of cell phone owners
from the various samples of size 2,500 from this population will have what approximate shape?
ANSWER: BELL-SHAPED CURVE
11. {Cell phone owners narrative} The frequency curve made from proportions of cell phone owners
from the various samples of size 2,500 from this population will have what approximate mean and
standard deviation?
ANSWER: MEAN: .20 OR 20%; STANDARD DEVIATION: .008 OR .8%.
12. {Cell phone owners narrative} What is the chance that a sample of size 2,500 from this population
will contain at least 20% cell phone owners?
ANSWER: 50% OR .50.
13. {Cell phone owners narrative} Suppose you took a random sample of size 2,500 from this
population and found that 17.6% of them owned a cell phone. Is this considered to be a reasonable
value given the size of this sample? Use the standardized score in your answer.
ANSWER: THE STANDARDIZED SCORE IS -3. THIS IS NOT WITHIN THE
REASONABLE EXPECTATIONS FOR A SAMPLE OF THIS SIZE.
14. {Cell phone owners narrative} Suppose you took a random sample of size 2,500 from this
population and found that 21.6% of the people in this sample owned a cell phone. Is this
considered to be a reasonable value given the size of this sample? Use the standardized score in
your answer.
ANSWER: THE STANDARDIZED SCORE IS +2. THIS IS WITHIN THE REASONABLE
EXPECTATIONS FOR A SAMPLE OF THIS SIZE, BUT IS ON THE MARGIN.
15. {Cell phone owners narrative} What is the chance that less than 20.8% of the people in a sample
of size 2,500 from this population will own a cell phone?
ANSWER: 84% OR .84. (THE STANDARDIZED SCORE IS 1.)
16. {Cell phone owners narrative} What is the chance that the proportion of cell phone owners in a
sample of size 2,500 from this population will be more than two standard deviations from the
expected mean?
ANSWER: 5% OR .05.
17. {Cell phone owners narrative} What range of proportions of cell phone owners is reasonable to
expect from this population (assuming your sample size is 2,500)? Justify your answer
ANSWER: 95% OF THE SAMPLE PROPORTIONS SHOULD LIE BETWEEN .184 AND
.216.
18. {Cell phone owners narrative} Suppose your sample size was only 250. What range of proportions
of cell phone owners is reasonable to expect from this population? Justify your answer
ANSWER: 95% OF THE SAMPLE PROPORTIONS WILL LIE BETWEEN .149 AND
.250.
19. {Cell phone owners narrative} How would the frequency curve made from proportions of cell
phone owners from the various samples of size 2,500 compare to the frequency curve made from
proportions of cell phone owners from the various samples of size 250?
ANSWER: BOTH WILL BE BELL-SHAPED; BOTH WILL HAVE A MEAN OF .20 OR
20%; THE STANDARD DEVIATION FOR SAMPLES OF SIZE 2,500 IS .008 OR 0.8%,
WHILE THE STANDARD DEVIATION FOR SAMPLES OF SIZE 250 IS .025 OR 2.5%.
20. In which of the following situations does the rule for sample proportions apply?
a. A pollster takes a random sample of 1,000 Americans and asks their opinion on the
President (approve/disapprove/neutral). He is interested in the percentage who approve of
the President.
b. You want to know whether or not people like the new CD by your favorite artist. You ask
5 people and record the percentage who say they like it.
c. A researcher weighs the same newborn baby each week for one year, and records
whether or not the child is within the normal weight range. At the end of the year, he
records the percentage of times that the child was within the normal weight range.
d. All of the above.
ANSWER: A
21. If numerous large random samples or repetitions of the same size are taken from a population, the
frequency curve made from proportions from the various samples will have what approximate
shape?
a. A bar graph with two bars, one for the proportion having the trait of interest, and the
other for the proportion not having the trait of interest.
b. A bell-shape.
c. A flat shape; each outcome should be equally likely.
d. Unknown; it can change every time.
ANSWER: B
22. If numerous large random samples or repetitions of the same size are taken from a population, the
proportions from the various samples will have what approximate mean?
a. The true population proportion.
b. The true population average.
c. 95% because most of them will be within 2 standard deviations of the true population
value.
d. None of the above.
ANSWER: A
23. {Politics narrative} Suppose numerous random samples of size 1,000 are taken from this
population. The proportions of Democrats from the various samples of size 1,000 will have what
approximate standard deviation?
a. .24 or 24%
b. .00024 or .024%
c. .015 or 1.5%
d. None of the above.
ANSWER: C
24. {Politics narrative} Suppose numerous random samples of size 1,000 are taken from this
population. The proportions of Republicans from the various samples of size 1,000 will have what
approximate standard deviation?
a. .24 or 24%
b. .00024 or .024%
c. .015 or 1.5%
d. None of the above.
ANSWER: C
25. If numerous large random samples or repetitions of the same size are taken from a population, the
frequency curve made from proportions from the various samples will have an approximate
__________ shape.
ANSWER: BELL
26. If numerous large random samples or repetitions of the same size are taken from a population, the
frequency curve made from proportions from the various samples will have a mean that is
__________ the true population proportion.
ANSWER: EQUAL TO
27. The standard deviation of the proportions from numerous random samples of size 1,000 from a
population will be __________ the standard deviation of the proportions from numerous random
samples of size 10,000 from the same population.
ANSWER: GREATER THAN
Narrative: Test scores
Suppose that test scores on a particular exam have a mean of 77 and standard deviation of 5, and
that they have a bell-shaped curve.
28. {Test scores narrative} Suppose you take numerous random samples of size 100 from this
population. Describe the shape and give the mean and standard deviation of the resulting
frequency curve.
ANSWER: BELL-SHAPED; MEAN=77; STANDARD DEVIATION=0.5.
29. {Test scores narrative} Suppose you take a single random sample of size 100 people from this
population. What is the chance that their average test score will be above 77?
ANSWER: 50% OR .50.
30. {Test scores narrative} Suppose you take a single random sample of size 100 from this population,
and you get a mean test score of 76. Is this something that you would have expected? Use a
probability to justify your answer.
ANSWER: 2.5% OR .025 OF THE MEAN TEST SCORES FOR A SAMPLE OF THIS
SIZE WILL LIE AT OR BELOW 76 (OR, THE STANDARDIZED SCORE IS -2). THIS IS
ON THE BORDERLINE OF WHAT WE WOULD EXPECT FOR THIS POPULATION.
31. {Test scores narrative} Suppose you take a single random sample of size 100 from this population,
and you get a mean test score of 79. Is this something that you would have expected? Use a
probability to justify your answer.
ANSWER: NO. THE STANDARD SCORE IS +4. SAMPLE MEANS OUT THIS FAR
HAVE VIRTUALLY NO CHANCE OF OCCURRING FOR THIS POPULATION.
32. {Test scores narrative} Suppose you randomly select a single individual from this population.
Where would you expect his/her test score to fall?
ANSWER: 95% OF THE TIME IT WOULD FALL BETWEEN 67 AND 87.
33. {Test scores narrative} Suppose you randomly select a sample of size 100 from this population.
Where would you expect their average test score to fall? Compare your answer to what you would
expect from a single individual selected at random from this population.
ANSWER: WE EXPECT THE AVERAGE OF 100 TEST SCORES TO BE BETWEEN 76
AND 78. WE EXPECT AN INDIVIDUAL TO SCORE BETWEEN 67 AND 87.
34. {Test scores narrative} Find and compare the answers to the following two questions; explain why
your answers are the same or different. 1) One individual is selected at random from the
population. What range of test scores is reasonable to expect for this person? 2) A sample of 100
individuals is selected at random from the population. What range of average test scores is
reasonable to expect for this group? ANSWER: WE EXPECT AN INDIVIDUAL SCORE TO
BE BETWEEN 67 AND 87. WE EXPECT THE AVERAGE OF 100 TEST SCORES TO BE
BETWEEN 76 AND 78. THE MEAN FOR A SAMPLE OF SIZE 100 HAS A SMALLER
STANDARD DEVIATION AND GIVES A TIGHTER RANGE OF POSSIBLE VALUES.
35. {Test scores narrative} Suppose that the test scores are not bell-shaped, but are skewed to the
right. You want to take a random sample and estimate the average test scored for this population.
You want to be able to use the rule of sample means to interpret your results in this case. Under
what (if any) conditions is this possible?
ANSWER: YOU CAN USE IT ONLY IF THE SAMPLE IS LARGE ENOUGH (AT LEAST
30).
36. Explain, in words that a non-statistics student would understand, why the standard deviation of the
various sample means taken from a population is smaller than the standard deviation of the
individuals in the population? (In the first situation, assume all samples are of the same size, and
that size is large.)
ANSWER: SAMPLE MEANS VARY LESS, BECAUSE THEY ARE BASED ON MORE
INFORMATION.
.
37. Which of the following are examples where you would be interested in estimating the population
mean?
a. About how long do left-handed people live?
b. Do most people support or oppose the President’s foreign policy?
c. What size was the viewing population who tuned in to the ABC News special last night?
d. All of the above.
ANSWER: A
38. In which of the following situations does the rule for sample means not apply?
a. A pollster takes a random sample of 1,000 Americans and asks them to give their opinion
of the President on a scale from 1 (completely disapprove) to 100 (completely approve).
He is interested in the average rating.
b. You take a random sample of 20 students’ scores from the ACT exam and record the
average score. Assume ACT scores are bell-shaped.
c. A sports fan takes a random sample of 20 NBA players and records their salaries. He
wants to estimate the average salary for the entire NBA.
d. The rule for sample means does not apply in any of these situations.
ANSWER: C
39. If numerous large random samples or repetitions of the same size are taken from a population, the
frequency curve made from means from the various samples will have what approximate shape?
a. A flat shape; each outcome should be equally likely.
b. A bell-shape.
c. A histogram.
d. Unknown; it can change every time.
ANSWER: B
40. Larger samples tend to result in __________ accurate estimates of population values than do
smaller samples.
ANSWER: MORE
41. Samples of size 2,500 will produce estimates of the population value that are __________ times
more accurate than samples of size 25. (Assume the population is bell-shaped. Use standard
deviation as a measure of accuracy.)
ANSWER: 10