CHAPTER 13
STATISTICAL SIGNIFICANCE FOR 2 x 2 TABLES
SECTION 13.1
MEASURING THE STRENGTH OF THE RELATIONSHIP
FREE RESPONSE QUESTIONS
1.
Whether or not we can rule out chance as an explanation for the relationship observed in a sample
depends on what two things?
ANSWER: 1) THE STRENGTH OF THE RELATIONSHIP OBSERVED IN THE
SAMPLE; AND 2) HOW MANY PEOPLE WERE INVOLVED IN THE STUDY.
2.
Suppose you observe a relationship between two categorical variables in a sample based on 22,000
people. If this observed relationship were based on a sample of 600 people rather than 22,000,
how would this affect the believability of the results (if at all)? Explain your answer.
ANSWER: AN OBSERVED RELATIONSHIP IS MUCH MORE BELIEVABLE IF IT IS
BASED ON A MUCH LARGER SAMPLE SIZE. SAMPLE RESULTS FROM LARGER
SAMPLES ARE LESS LIKELY TO VARY FROM SAMPLE TO SAMPLE AND ARE
MORE LIKELY TO REPRESENT THE TRUTH ABOUT THE POPULATION.
MULTIPLE CHOICE QUESTIONS
3.
Which of these does not apply to a ‘statistically significant’ relationship between two categorical
variables?
a. The relationship between the two variables is very important from a practical standpoint.
b. The relationship observed in the sample was unlikely to have occurred unless there really
is a relationship in the population.
c. The notion that this relationship could have happened by chance is deemed to be
implausible.
d. All of the above apply to a ‘statistically significant’ relationship.
ANSWER: A
4.
Which of the following differences in percentages for the two categories of an explanatory
variable would be considered to be statistically significant?
a. 50% - 49 % = 1%
b. 50% - 45% = 5%
c. 50% - 40% = 10%
d. Not enough information to tell.
ANSWER: D
FILL-IN-THE-BLANK QUESTIONS
5.
When researchers look for a relationship between two categorical variables for individuals in the
__________, they measure those categorical variables on individuals in the __________.
ANSWERS (RESPECTIVELY): POPULATION; SAMPLE
6.
The question researchers are really asking when they are looking for a relationship between two
variables is whether or not that relationship is present in the __________.
ANSWER: POPULATION
SECTION 13.2
STEPS FOR ASSESSING STATISTICAL SIGNIFICANCE
FREE RESPONSE QUESTIONS
7.
The first two basic steps for conducting a hypothesis test are to: 1) determine the null and
alternative hypotheses; and 2) collect the data and summarize it with a ‘test statistic.’ What is the
third step in the process that allows you to then make a decision?
ANSWER: DETERMINE HOW UNLIKELY THE TEST STATISTIC WOULD BE IF
THE NULL HYPOTHESIS WERE TRUE.
8.
Suppose you want to investigate whether there is a relationship between the gender of college
students and whether or not they wear hats in school. What would be your null hypothesis and
your alternative hypothesis (in words)? Be sure to label clearly which hypothesis is which.
ANSWER: NULL HYPOTHESIS: NO RELATIONSHIP BETWEEN GENDER AND HAT
WEARING IN COLLEGE STUDENTS; ALTERNATIVE HYPOTHESIS: THERE IS A
RELATIONSHIP.
MULTIPLE CHOICE QUESTIONS
.
9.
Suppose you find a statistically significant relationship between two categorical variables (with no
other supporting evidence available). When can such results correctly lead you conclude a cause
and effect relationship?
a. Never.
b. Only when the data were from a randomized experiment.
c. Only when the data were from a random sample.
d. Always; a statistically significant relationship wouldn’t be significant unless there is a
cause and effect relationship.
ANSWER: B
10. Your mother-in-law says that rubbing a carpet stain with toothpaste before applying stain remover
helps take the stain out. You are skeptical. You asked an impartial neighbor to set up a randomized
experiment to find out whether there is a relationship between the stain removing method and the
outcome. She went home to collect and analyze the data, then called you and said “In a chi-square
test, I would choose the alternative hypothesis,” and then hung up. What do you conclude?
a. The toothpaste didn’t make a difference.
b. The toothpaste did help after all.
c. The toothpaste had a significant effect but you don’t know in which way yet.
d. The toothpaste made things worse.
ANSWER: C
FILL-IN-THE-BLANK QUESTIONS
11. Another name for the alternative hypothesis is the __________ hypothesis.
ANSWER: RESEARCH
12. The __________ hypothesis is usually written to express the fact that ‘nothing is happening.’
ANSWER: NULL
SECTION 13.3
THE CHI-SQUARE TEST
FREE RESPONSE QUESTIONS
For Questions 13-15, use the following narrative
Narrative: Aspirin and polyps
A well-designed experiment by the Ohio State University Medical Center studied whether or not
taking an aspirin a day would affect colon cancer patients’ chances of getting subsequent colon polyps.
In the study, 635 patients with colon cancer participated; 317 of them were randomly assigned to the
aspirin group, and the other 318 patients were assigned to a placebo (non-aspirin) group. The results
showed that 17% of the patients in the aspirin group developed subsequent polyps, compared to 27%
of the patients in the non-aspirin group. The results are summarized in the table below.
ASPIRIN
PLACEBO
TOTAL
POLYPS
54
86
140
NO POLYPS
263
232
495
TOTAL
317
318
635
13. {Aspiring and polyps narrative} Write the expected counts in each of the four cells of the table in
parentheses next to each of the observed counts. ANSWER: SEE ANSWERS IN
PARENTHESES IN THE TABLE BELOW.
ASPIRIN
PLACEBO
TOTAL
POLYPS
54 (69.89)
86 (70.11)
140
NO POLYPS
263 (247.11)
232 (247.89)
495
TOTAL
317
318
635
14. {Aspirin and polyps narrative} Compute the test statistic for the data.
ANSWER: 9.25
15. {Aspirin and polyps narrative} Is there a statistically significant relationship between taking
aspirin and developing subsequent polyps in colon cancer patients? If so, what is the relationship?
Justify your answer.
ANSWER: THE CHI-SQUARE STATISTIC IS 9.25 > 3.84 (P-VALUE < .05), THE
RELATIONSHIP IS STATISTICALLY SIGNIFICANT. SINCE THE ASPIRIN GROUP
EXPERIENCED FEWER POLYPS (17% COMPARED TO 27%), IT APPEARS THAT
ASPIRIN REDUCES THE CHANCES OF DEVELOPING SUBSEQUENT POLYPS IN
COLON CANCER PATIENTS.
16. Suppose a researcher wanted to find out which of two chemicals works best to kill poison ivy. He
conducted a controlled experiment involving 200 poison ivy plants, randomly assigned half of
them to each treatment (chemical #1 or chemical #2). Five days after applying the chemicals, he
recorded the data below. Is there a statistically significant relationship between type of chemical
used and the results when applied to poison ivy? Conduct a hypothesis test to find out.
DIED
LIVED
TOTAL
CHEMICAL #1
CHEMICAL #2
TOTAL
78
89
167
22
11
33
100
100
200
ANSWER: YES; THE EXPECTED CELL COUNTS ARE IN PARENTHESES. THE CHISQUARE TEST STATISTIC IS 4.39 > 3.84, SO THE RELATIONSHIP IS
STATISTICALLY SIGNIFICANT. NOTE THAT CHEMICAL #2 SEEMS TO DO
BETTER SINCE IT KILLED A HIGHER PERCENTAGE OF PLANTS IN THE SAMPLE
(89% COMPARED TO 78%).
CHEMICAL #1
CHEMICAL #2
TOTAL
DIED
78 (83.5)
89 (83.5)
167
LIVED
22 (16.5)
11 (16.5)
33
TOTAL
100
100
200
MULTIPLE CHOICE QUESTIONS
17. Using the criterion of .05, which of the following results allows a researcher to conclude that a
relationship between two categorical variables is statistically significant?
a. p-value = .04
b. p-value = .50
c. p-value = .95
d. None of the above.
ANSWER: A
18. Which of the following statements is true about chi-square tests?
a. A large chi-square test statistic results in a large p-value.
b. A large p-value means that there is a good chance that the relationship is statistically
significant.
c. If the two variables are not related in the population, then less than 5% of the samples
you could ever take would give you a test statistic of 3.84 or larger.
d. All of the above.
ANSWER: C
19. Which of the following statements is not true about the chi-square statistic for a 2 x 2 contingency
table?
a. If it is greater than 3.84, reject the null hypothesis and accept the alternative hypothesis.
b. If it is greater than 3.84, the relationship in the table is considered to be statistically
significant.
c. 95% of the tables for sample data from populations in which there is no relationship will
have a chi-square statistic of 3.84 or greater.
d. All of the above are true.
ANSWER: C
20. Suppose you wanted to test to see if there is a relationship between gender and driving an SUV.
You take a random sample of 200 vehicles and their owners and use Minitab to analyze your data.
Your computer output is below, but unfortunately the last line of the output (which would have
contained the p-value) didn’t print. What can you conclude? (Assume the study was conducted
properly otherwise.)
Expected counts are printed below observed counts
SUV
Non-SUV
Male
45
67
49.84
62.16
TOTAL
112
Female
44
39.16
44
48.84
Total
89
111
88
200
Chi-Sq = .47 + .38 + .60 + .48 = 1.92
a.
b.
The relationship between gender and driving an SUV is not statistically significant.
There is a statistically significant relationship. Women are more likely to drive SUVs
than men.
c. There is a statistically significant relationship. More men are likely to drive SUVs than
women.
d. Not enough information to tell.
ANSWER: A
FILL-IN-THE-BLANK QUESTIONS
21. A measure of how unlikely the test statistic would be if the null hypothesis were true is the
__________.
ANSWER: P-VALUE
22. A __________ test is a statistical procedure that is used to determine whether or not there is a
relationship between two categorical variables.
ANSWER: CHI-SQUARE
SECTION 13.4
PRACTICAL VERSUS STATISTICAL SIGNIFICANCE
FREE RESPONSE QUESTIONS
23. If two variables have a statistically significant relationship that does not necessarily mean the two
variables have a relationship of practical importance. Describe how this could happen.
ANSWER: IT DEPENDS ON THE SAMPLE SIZE; A MINOR RELATIONSHIP
OBSERVED IN A SAMPLE BASED ON A LARGE ENOUGH NUMBER OF
OBSERVATIONS CAN EASILY ACHIEVE STATISTICAL SIGNIFICANCE.
24. Is it possible for an important relationship in a population to fail to achieve statistical significance
in the sample and therefore go undetected, even if the sample is selected properly and the data are
collected correctly? Explain your answer.
ANSWER: YES; WHY: 1) THERE MAY HAVE ONLY BEEN A FEW OBSERVATIONS;
OR 2) THE SAMPLE IS NOT REPRESENTATIVE OF THE POPULATION, JUST BY
CHANCE.
For Questions 25-26, use the following narrative
Narrative: Teenage voting
Suppose you take a random sample of teenagers aged 18-19, and you find that 53% of the males in
your sample are Democrats, and 54% of the females in your sample are Democrats, respectively.
25. {Teenage voting narrative} Should these results be considered significant from a practical
standpoint? Why or why not?
ANSWER: NO; THEY ARE NOT SIGNIFICANTLY DIFFERENT FROM A PRACTICAL
STANDPOINT. THE DIFFERENCE IS TOO SMALL.
26. {Teenage voting narrative} Could these results ever be statistically significant? Why or why not?
ANSWER: YES, IF THE SAMPLE SIZE IS LARGE ENOUGH, ANY NON-ZERO
DIFFERENCE CAN BECOME STATISTICALLY SIGNIFICANT.
MULTIPLE CHOICE QUESTIONS
27. Suppose a chi-square test statistic from a sample of size 1,000 is 38.2 with a p-value of .0001, so
the relationship is statistically significant. If the sample size for this study had only been based on
a sample size of 100 (but the percentages remained the same), what would the chi-square test
statistic have been and what conclusion would have been drawn?
a. Chi-square = 38.2; p-value = .0001; the relationship is statistically significant.
b. Chi-square = 3.82; p-value = .001; the relationship is statistically significant.
c. Chi-square = 3.82; p-value = .052; the relationship is not statistically significant.
d. None of the above.
ANSWER: C
28. Suppose researchers say they ‘failed to find a relationship’ between two variables that they
thought might have been related. What does this say to you, as an educated consumer of statistical
information?
a. You should check to make sure the study was not based on a small number of
individuals.
b. The researchers must not have observed any relationship in their sample.
c. There must be no relationship between these two variables in the population.
d. All of the above.
ANSWER: A
29. Suppose there is not enough evidence to conclude that the relationship in the population is real.
Which of the following is not an equivalent way of saying this?
a. The relationship is not statistically significant.
b. We cannot reject the null hypothesis.
c. We accept the null hypothesis.
d. All of the above.
ANSWER: C
30. Suppose that a medical study found a statistically significant relationship between wearing gold
jewelry and developing skin cancer. Suppose the study was based on a sample of 100,000 people
and had a p-value of .049. How would you react to these results?
a. You feel these results are very convincing since the sample size was so large.
b. You are impressed by the sample size but the p-value is too close to .05 to feel very
convinced; the result is marginal.
c. You are skeptical because the high sample size would lead to a small p-value even with a
miniscule relationship, and thus gives misleading results.
d. None of the above
ANSWER: C
FILL-IN-THE-BLANK QUESTIONS
31. A minor relationship that is found to be __________ significant is not necessarily __________
significant.
ANSWER: STATISTICALLY; PRACTICALLY
32. An interesting relationship in the population may fail to achieve __________ significance if there
are too _____ observations.
ANSWER: STATISTICAL; FEW
33. A minor relationship in the population may achieve __________ significance but fail to achieve
__________ significance if there are an extremely large number of observations.
ANSWERS (RESPECTIVELY): STATISTICAL; PRACTICAL