Section 1-1: Statistical and Critical Thinking 1
Chapter 1: Introduction to Statistics
Section 1-1: Statistical and Critical Thinking
1. The respondents are a voluntary response sample or a self-selected sample. Because those with strong interests
in the topic are more likely to respond, it is very possible that their responses do not reflect the opinions or
behavior of the general population.
2. a. The sample consists of the 1046 adults who were surveyed. The population consists of all adults.
b. When asked, respondents might be inclined to avoid the shame of the unhealthy habit of not washing their
hands, so the reported rate of 70% might well be much higher than it is in reality. It is generally better to
observe or measure human behavior than to ask subjects about it.
3. Statistical significance is indicated when methods of statistics are used to reach a conclusion that a treatment is
effective, but common sense might suggest that the treatment does not make enough of a difference to justify its
use or to be practical. It is possible for a study to have statistical significance, but not practical significance.
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4. No. Correlation does not imply causation. The example illustrates a correlation that is clearly not the result of
any interaction or cause effect relationship between per capita consumption of margarine and the divorce rate in
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Maine.
5. Yes, there does appear to be a potential to create a bias.
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6. No, there does not appear to be a potential to create a bias.
7. No, there does not appear to be a potential to create a bias.
8. Yes, there does appear to be a potential to create a bias.
9. The sample is a voluntary response sample and has strong potential to be flawed.
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10. The samples are voluntary response samples and have potential for being flawed, but this approach might be
necessary due to ethical considerations involved in randomly selecting subjects and somehow imposing
treatments on them.
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11. The sampling method appears to be sound. 12. The sampling method appears to be sound.
13. The Ornish weight loss program has statistical significance, because the results are so unlikely (3 chances in
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1000) to occur by chance. It does not have practical significance because the amount of lost weight (3.3 lb) is so
small.
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14. Because there is only one chance in a thousand of getting such success rates by chance, the difference does
appear to have statistical significance. The 92% success rate for surgery appears to be substantially better than
the 72% success rate for splints, so the difference does appear to have practical significance.
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15. The difference between Mendel’s 25% rate and the result of 26% is not statistically significant. According to
Mendel’s theory, 145 of the 580 peas would have yellow pods, but the results consisted of 152 peas with yellow
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pods. The difference of 7 peas with yellow pods among the 580 offspring does not appear to be statistically
significant. The difference does not appear to have practical significance.
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16. Because there is a 25% chance of getting such results with a program that has no effect, the program does not
appear to have statistical significance. Because the average increase is only 3 IQ points, the program does not
appear to have practical significance.
17. The sample percentage of males is 49.5%, and it appears to be very close to the percentage expected under
normal circumstances. It does not appear to have statistical significance, nor does it appear to have practical
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significance.
18. Because there is a 15% chance of getting such results with a medication that has no effect, the medication does
not appear to have statistical significance. Because the average decrease is only 2 mmHg, the medication does
not appear to have practical significance.
19. Because there is an 8% chance of getting such nausea rates by chance, the results do not appear to have
statistical significance. Also, they do not appear to have practical significance.
20. Because there is less than a 1% chance of getting the results obtained in this study, the results have statistical
significance. They also appear to have practical significance.
21. Yes. Each column of 8 AM and 12 AM temperatures is recorded from the same subject, so each pair is
matched.
22. No. The source is from university researchers who do not appear to gain from distorting the data.
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,2 Chapter 1: Introduction to Statistics
23. The data can be used to address the issue of whether there is a correlation between body temperatures at
8 AM and at 12 AM. Also, the data can be used to determine whether there are differences between body
temperatures at 8 AM and at 12 AM.
24. Because the differences could easily occur by chance (with a 64% chance), the differences do not appear to
have statistical significance.
25. No. The white blood cell counts measure a different quantity than the red blood cell counts, so their differences
are meaningless.
26. The issue that can be addressed is whether there is a correlation, or association, between white blood cell counts
and red blood cell counts.
27. No. The National Center for Health Statistics has no reason to collect or present the data in a way that is biased.
28. No. Correlation does not imply causation, so a statistical correlation between white blood cell counts and red
blood cell counts should not be used to conclude that higher white blood cell counts are the cause of higher red
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blood cell counts.
29. It is questionable that the sponsor is the Idaho Potato Commission and the favorite vegetable is potatoes.
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30. The sample is a voluntary response sample, so there is a good chance that the results do not reflect the larger
population of people who have a water preference.
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31. The correlation, or association, between two variables does not mean that one of the variables is the cause of the
other. Correlation does not imply causation. Clearly, sour cream consumption is not directly related in any way
to motorcycle fatalities.
32. The sponsor of the poll is an electronic cigarette maker, so the sponsor does have an interest in the poll results.
The source is questionable.
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33. The correlation, or association, between two variables does not mean that one of the variables is the cause of the
other. Correlation does not imply causation. Common sense suggests that cheese consumption is not directly
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related in any way to fatalities from bedsheet entanglements.
34. The correlation, or association, between two variables does not mean that one of the variables is the cause of the
other. Correlation does not imply causation.
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35. The survey results are from subjects who chose to respond, so the results constitute a voluntary response
sample. Consequently, the results are questionable.
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36. Because the nutritionists are paid such large amounts of money, they might be more inclined to find favorable
results. It is very possible that the results represent desired outcomes instead of actual outcomes.
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37. a. 0.45(3014) 1356.3
b. No. The actual count of survey subjects who have at least one chronic condition must be a whole number.
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c. 1356 adults (Any value from 1342 to 1371 would yield a percentage that rounds to 45%.)
d. 1206 / (1206 1808) 0.40, or 40%
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38. a. 0.828(198) 163.944 patients
b. No. Because the result is a count of patients, the result must be a whole number.
c. 164 patients
d. 198 / (199 198) 0.499, or 49.9%, or 50% rounded
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39. The wording of the question is biased and tends to encourage negative responses. The sample size of 20 is too
small. Survey respondents are self-selected instead of being randomly selected by the newspaper. If 20 readers
respond, the percentages should be multiples of 5, so 87% and 13% are not possible results.
40. All percentages of success should be multiples of 5. The given percentages cannot be correct.
Section 1-2: Types of Data
1. The population consists of all drug tests of adults in the United States, and the sample is the 10 million drug
tests that were analyzed. The value of 4.2% is a statistic because it is obtained from the sample.
2. a. quantitative d. quantitative
b. categorical e. quantitative
c. categorical
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, Section 1-3: Collecting Sample Data 3
3. Only part (a) describes discrete data.
4. a. The sample is the 36,000 adults who were surveyed. The population is all adults in the United States.
b. statistic
c. ratio
d. discrete
5. statistic 17. discrete
6. statistic 18. continuous
7. parameter 19. discrete
8. parameter 20. continuous
9. statistic 21. nominal
10. parameter 22. ordinal
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11. parameter 23. ordinal
12. parameter 24. ratio
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13. continuous 25. interval
14. discrete 26. nominal
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15. continuous 27. ratio
16. discrete 28. interval
29. The numbers are not counts or measures of anything. They are at the nominal level of measurement, and it
makes no sense to compute the average (mean) of them.
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30. The ranks are at the ordinal level of measurement. Differences between medical schools cannot be interpreted,
so there is no way to know whether the difference between Harvard and New York University is the same as the
difference between New York University and Duke.
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31. The numbers on the pain scale are at the ordinal level of measurement. Differences cannot be determined, so
there is no way to know whether the first patient has twice as much pain as the second patient. Ratios such as
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“twice” make no sense with ordinal data.
32. The temperatures are at the interval level of measurement. Because there is no natural starting point with 0ºF
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representing no heat, it is wrong to state that the second patient is “5% cooler than the first patient.”
33. a. Continuous, because the number of possible values is infinite and not countable.
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b. Discrete, because the number of possible values is finite.
c. Discrete, because the number of possible values is finite.
d. Discrete, because the number of possible values is infinite and countable.
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34. Interval level of measurement. The direction of north represented by 0 is arbitrary, and 0 does not represent
“no direction.” Differences between degrees are meaningful; the difference between 30 and 60 is the same
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as the difference between 150 and 180. But ratios are not meaningful; the ratio of 60 to 30 does not result
in twice some direction. (These degree measurements are directions, not amounts of rotation.)
Section 1-3: Collecting Sample Data
1. The study is an experiment because subjects were given treatments.
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2. The subjects in the study did not know whether they were given the magnet treatment or the sham treatment,
and those who administered the treatments also did not know.
3. The group sample sizes are large enough so that the researchers could see the effects of the two treatments, but
it would have been better to have larger samples.
4. The sample appears to be a convenience sample. Given that the subjects were all patients at a Veterans Affairs
hospital, it is not likely that the sample is representative of the population, so it is questionable whether the
results can be generalized for the population of subjects with chronic low back pain.
5. The sample appears to be a convenience sample. By e-mailing the survey to a readily available group of Internet
users, it was easy to obtain results. Although there is a real potential for getting a sample group that is not
representative of the population, indications of which ear is used for cell phone calls and which hand is
dominant do not appear to be factors that would be distorted much by a sample bias.
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6. The study is an observational study because the subjects were not given any treatment.
7. With 717 responses, the response rate is 14%, which does appear to be quite low. In general, a very low
response rate creates a serious potential for getting a biased sample that consists of those with a special interest
in the topic.
8. Answers vary, but the following are good possibilities.
a. Obtain a printed copy of the class roster, assign consecutive numbers (integers), then use a computer to
randomly generate six of those numbers.
b. Select every third student leaving class until six students are chosen.
c. Randomly select three males and three females.
d. Randomly select a row, and then select the students in that row. (Use only the first six to meet the
requirement of a sample of size six.)
e. Select the first six students who enter the class.
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9. systematic 15. stratified
10. convenience 16. systematic
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11. random 17. random
12. stratified 18. cluster
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13. cluster 19. convenience
14. random 20. systematic
21. Observational study. The sample is a convenience sample consisting of subjects who decided to respond. Such
voluntary response samples have a high chance of not being representative of the larger population, so the
sample may well be biased, as it was in this case.
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22. Experiment. The sample subjects consist of male physicians only. It would have been better to include females.
Also, it would be better to include male and females who are not physicians.
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23. Experiment. This experiment would create an extremely dangerous and illegal situation that has a real potential
to result in injury or death. It’s difficult enough to drive in New York City while being completely sober.
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24. Observational study. The sample of eight respondents is too small.
25. Experiment. The biased sample created by using a small sample of college students cannot be fixed by using a
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larger sample. The larger sample will still be a biased sample that is not representative of the population of all
adults.
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26. Experiment. Calling the subjects and asking them to report their weights has a high risk of getting results that do
not reflect the actual weights. It would have been much better to somehow measure the weights instead of
asking the subjects to report them.
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27. Observational study. Respondents are not likely to respond honestly because there is a “social desirability bias”
causing respondents to reply in ways that will be viewed favorably.
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28. Observational study. The number of responses is very small, and the response rate of only 1.52% is far too
small. With such a low response rate, there is a real possibility that the sample of respondents is biased and
consists only of those with special interests in the survey topic.
29. prospective study 33. matched pairs design
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30. retrospective study 34. randomized block design
31. cross-sectional study 35. completely randomized design
32. prospective study 36. matched pairs design
37. Prospective: The experiment was begun and results were followed forward in time.
Randomized: Subjects were assigned to the different groups through a process of random selection, whereby
they had the same chance of belonging to each group.
Double-blind: The subjects did not know which of the three groups they were in, and the people who evaluated
results did not know either.
Placebo-controlled: There was a group of subjects who were given a placebo; by comparing the placebo group
to the two treatment groups, the effects of the treatments might be better understood.
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