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Chapter 19
Chapter 19
The Binomial Distribution
Section A
1. The binomial distribution arises when dealing with:
*
a) two samples at a time
b) two events at a time
c) events that can be classified as one of two outcomes
d) events, each of which occur twice
2. The binomial distribution more closely resembles the normal distribution as:
*
a) the number of events increases
b) the number of outcomes for each event increases
c) the number of samples increases
d) the probability of one of the outcomes approaches 1.0
3. The correction for continuity is needed only when:
*
a) P is either near 0 or 1.0
b) N is very large
c) the number of outcomes is small
d) the normal distribution is used to approximate the binomial distribution
4. If the probability of one of the outcomes is near zero, which of the following occurs as the
number of events increases?
*
a) the binomial distribution becomes more symmetrical
b) the binomial distribution becomes more like the normal distribution
c) the mean of the distribution increases
d) all of the above
5. For a given number of events, which of the following binomial distributions (in terms of the
probability of one of the outcomes) most closely resembles the normal distribution?
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a) P = .05
b) P = .4
c) P = .7
d) P = 1.0
6. A series of dichotomous events will follow the binomial distribution only if:
a) the events are normally distributed
b) the two outcomes are equally likely
c) the number of events is large
*
d) the events are independent of each other
7. After playing the same "slot machine" many times without winning, a friend of yours claims
that he or she is "overdue" for a jackpot, and wants to keep playing the same machine, because
he or she expects to win soon. Your friend's belief is an example of:
a) a stationary process
b) the distribution-free assumption
*
c) the gambler's fallacy
d) null hypothesis testing
8. The sign test should be performed instead of the matched t-test, whenever:
*
a) the direction of the difference can be determined for each pair, but the differences
cannot be quantified
b) the pairs of scores are not independent of each other
c) the number of pairs is small
d) it is important to avoid Type I errors
9. Compared to the matched t-test, a sign test on the same data:
*
a) usually has less power
b) usually leads to more Type I errors
c) requires more stringent assumptions
d) is easier to calculate, but always leads to the same statistical decision
10. The effect that ties have on the sign test is:
a) reduced by the correction for continuity
b) to increase the Type I error rate
*
c) to reduce the sample size
d) to make the null hypothesis distribution more symmetrical
Section B
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11. Imagine a primitive language that has only two different letters in its alphabet. If all the
words in this language have 7 letters, how many different words can be spelled?
a) 7 b) 14 c) 49 d) 128
-12. Referring to the primitive language described in the previous question, how many possible
words could be spelled that have six of one letter and only one of the other letter?
a) 7 b) 14 c) 49 d) 128
-13. Consider a true-false quiz that consists of only 10 questions. If someone were to answer
every question by guessing, what is the probability of getting exactly four correct?
a) .02 b) .04 c) .21 d) .40
-14. Consider a jury that consists of 15 people. Assuming that men and women are equally likely
to be selected for jury duty, what is the probability that a 15-person jury will consist of at least 10
people of the same gender?
a) .092 b) .151 c) .183 d) .302
-15. Sixty-five percent of the residents in Salem are Republicans and 35 percent are Democrats.
A local radio station polls a random sample of 100 of its listeners and finds that 75 of them are
Republicans. If the audience of that radio station really has the same proportion of Republicans
as the rest of Salem, what is the probability of getting as many (or more) Republicans as were
found in that survey?
a) .018 b) .023 c) .036 d) .046
-16. Seventy subjects are exposed to subliminal messages persuading them to buy Brand X cola.
Then each subject is asked to try two brands of cola (actually identical), and decide whether he
or she prefers Brand X or Brand Y. If 42 subjects prefer Brand X, what is the value of the zscore you would use (with continuity correction) to test whether the subliminal messages had any
effect at all?
a) 1.55 b) 1.67 c) 2.20 d) 2.37
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17. Eleven pairs of identical twins are tested for shyness, and it is found that for nine of the pairs
it is the twin born a few minutes earlier that is more shy. What is the probability of getting
results this extreme if birth order for twins has no effect on shyness in the population (i.e., what
is the two-tailed p value)?
a) .0164 b) .0328 c) .0656 d) .131
-18. A new and controversial treatment for arthritis is being tested. At the end of a three-month
clinical trial, 78 out of 140 patients have shown some improvement. What is the two-tailed
probability associated with the z-score you would use (with continuity correction) to test whether
the new treatment had any effect at all?
a) .089 b) .177 c) .204 d) .354
-19. Thirty-two beginning piano students are matched into 16 pairs based on a musical aptitude
test. Then one member of each pair is randomly assigned to a new form of musical training,
while the other member receives conventional training. At the end of training, the musical
ability of each subject is rated on a ten-point scale, as shown below:
Pair #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
New Training
7
5
8
4
6
5
6
8
5
6
7
3
8
6
Traditional
6
4
7
7
6
3
5
7
5
9
5
3
9
2
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6
6
5
Assume that the ratings and their differences cannot be used quantitatively (except to determine
the direction of the results for each pair), and perform a sign test on these data. What is the onetailed p value for these results?
a) > .05 b) < .05 c) < .025 d) < .01
-20. Suppose that twenty-eight similar studies have been conducted on whether men or women
have the better sense of smell. In 18 studies women were significantly better, in 8 studies men
were significantly better, and in two studies the results were not statistically significant.
Assuming that all of these studies are equivalent, perform a sign test on the 28 studies. What is
the z-score (with continuity correction) that would be used to determine the significance of that
sign test?
a) 1.57 b) 1.77 c) 1.96 d) 2.33
-Section C
21. A friend of yours is at a carnival and betting on a wheel that stops in one of ten places,
numbered one to ten. Imagine that your friend will lose if the number comes up either odd or
higher than eight on the next spin. What is your friend's probability of winning?
a) .2 b) .4 c) .5 d) .6
-22. Suppose that you are conducting an experiment that involves seven questionnaires. In how
many orders can these questionnaires be presented to the subjects?
a) 7 b) 49 c) 720 d) 5040
-23. Imagine that you are conducting a taste test between two popular cola drinks. Each subject
receives six taste trials, three of each cola. In how many distinct orders can these six trials be
presented?
a) 12 b) 20 c) 120 d) 720
-24. A drawer contains three white socks and three black socks. If you pick two socks from the
drawer without looking (and, of course, without replacement), what is the probability that you
will select a pair of matching socks?
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a) .2 b) .4 c) .5 d) .6
-25. Suppose that in a certain town only one out of every five people uses the internet. If you are
selecting people from the town at random for a survey about the internet, what is the probability
that the next three random selections will all be internet users (assume that the town is large
enough that sampling without replacement will not affect your answer)?
a) .04 b) .06 c) .008 d) .001
-Chapter 20
Chi-Square Tests
Section A
1. Suppose that you have obtained frequencies for two categories that are equally likely in the
population, and that you have calculated both the z-score for the normal approximation to the
binomial test, and the chi-square statistic (χ2). How will these two statistics compare?
*
a) they will be equal
b) χ2 will equal the square of z
c) χ2 will equal the square-root of z
d) χ2 will be larger than z, but the difference decreases as N increases
2. The critical value of the chi-square statistic for a one-way test increases as:
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a) the number of categories increases
b) N increases
c) alpha increases
d) all of the above
3. In the chi-square test, the sum of the expected frequencies is always equal to:
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a) 1.0
b) N divided by the number of categories
c) the sum of the observed frequencies
d) none of the above
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4. Suppose you have calculated the chi-square statistic for a one-way test with five categories,
and obtained the value of -17. Which of the following can be concluded?
*
a) a calculation error has been made
b) the sum of the observed frequencies is much less than the sum of the expected
frequencies
c) the null hypothesis can be rejected at the .05 level
d) it depends on the size of N
5. Suppose you have read about a one-way chi-square test with four categories, in which the chisquare statistic turned out to be .03. Without further information, which of the following could
you conclude?
*
a) a calculation error has been made
b) N must have been small
c) the null hypothesis could be rejected at the .05 level
d) the observed frequencies are similar to the expected frequencies
6. In a two-way chi-square test of association involving R rows and C columns, the expected
frequencies are found by:
*
a) dividing N equally by the number of cells (i.e., R * C)
b) dividing each of the observed frequencies by N
c) dividing each of the observed frequencies by R * C
d) none of the above
7. In a 2 X 2 chi-square test of association, the ratio of the expected frequencies in each column
of the contingency table is equal to:
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a) the ratio of the row sums
b) the ratio of the column sums
c) 1.0
d) .25 or 4.0, depending on how you look at it
8. Yates' correction for continuity:
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a) increases the power of the chi-square test, especially when N is small
b) applies only to the one-way chi-square test
*
c) is appropriate only when there is one degree of freedom
d) is needed when some of the expected frequencies are too small to meet the assumptions
of the chi-square test
9. Large discrepancies between expected and obtained frequencies in a chi-square test are likely
to lead to a chi-square statistic in:
*
a) the positive tail of the null hypothesis distribution
b) the negative tail of the null hypothesis distribution
c) either tail of the null hypothesis distribution
d) the middle area of the null hypothesis distribution
10. The chi-square distribution will be a reasonable approximation of the distribution of the chisquare statistic if:
*
a) the number of degrees of freedom is not too small
b) none of the expected frequencies is too small
c) the expected frequencies do not add up to more than N
d) all of the above
Section B
11. A college is studying the religious affiliations of its students. Each respondent in a random
sample of 40 students is classified into one of six categories. If a one-way chi-square will be
performed on the results, how many degrees of freedom will be associated with the test?
a) 5 b) 6 c) 36 d) 39
-12. In a two-way chi-square test of independence involving ninety-six subjects classified into six
religious categories and four political preferences, how many degrees of freedom will be
associated with the test?
a) 4 b) 15 c) 18 d) 24
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13. In a one-way chi-square test, 60 students are shown four photographs of potential professors,
differing in attractiveness, and asked to choose the one they would most like to have as an
instructor. If you want to test the null hypothesis that attractiveness is unrelated to teacher
preference, what would be the expected frequency for each photograph?
*
a) 4
b) 15
c) 60
d) it depends on the observed frequencies
14. A company is testing three new logos for its company name. If 20 subjects in a sample
prefer the first logo, 11 prefer the second, and 14 subjects prefer the third, what is the value of
the chi-square statistic for testing the null hypothesis that the three logos are equally preferred?
a) 2.78 b) 2.8 c) 3.12 d) 3.2
-15. Suppose that census data in one small town reveal that of the men over 18, 30% are single,
40% are married, 20% are divorced, and 10% are widowers. If a random sample of male readers
of a town newsletter consists of 22 singles, 10 married, 10 divorced and 8 widowers, what is the
value of the chi-square statistic you would use to test whether the readers of the newsletter have
the same distribution of marital status as the population of the town?
a) 10.1 b) 13.4 c) 15.5 d) 29.9
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16. Suppose that the data for a chi-square test are as shown in the table below:
fo
14
8
11
5
6
fe
10
8
6
8
12
What is the p value that corresponds to the chi-square statistic for these data?
*
a) > .05
b) between .05 and .025
c) between .025 and .01
d) < .01
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17. A political analyst is trying to determine whether there is any link between being a
vegetarian and being against the death penalty. The data collected for a random sample of 100
people led to the following cross-classification table:
Vegetarian
Not vegetarian
Against
16
34
For
8
42
What is the expected frequency for the upper right cell (i.e., not being a vegetarian, but being
against the death penalty)?
a) 25 b) 28 c) 34 d) 38
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18. A researcher believes she has found a gene that is associated with having panic attacks. Ten
out of her 15 panic patients have the gene, whereas only five out of 25 control subjects have the
gene. What is the value of the chi-square statistic she should use to test the association between
the suspected gene and having panic attacks?
a) 4.36 b) 5.23 c) 6.83 d) 8.71
-19. A developmental psychologist is studying cognitive functions in children. Three-, four-, and
five-year olds are asked to choose between one large cookie or three smaller cookies that add up
to only about half the mass of the large cookie. The number of children choosing each
alternative is shown for each age group below:
Three years
Four years
Five years
One cookie
5
8
14
Three cookies
15
12
6
What is the p value that corresponds to the chi-square statistic for these data?
a) > .05
b) between .05 and .025
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Test Items
c) between .025 and .01
d) < .01
20. A researcher is studying the influence of criminal history on the likelihood of a defendant
being convicted of a crime. All subjects hear the same evidence in a simulated jury trial, but
subjects are randomly assigned to one of three conditions. Subjects are told either that the
defendant had been previously: tried for a similar crime but acquitted; tried for a similar crime
and convicted; or, never accused of a similar crime. The verdicts of subjects in the three groups
are shown below:
Never Accused
Previously
Acquitted
Previously
Convicted
Guilty
6
10
20
Not Guilty
16
8
6
Unable to Decide
8
12
4
What is the value of the chi-square statistic you would use to test the independence of the two
factors?
a) 16.8 b) 18.3 c) 22.9 d) 24.4
-Section C
21. If some of the expected frequencies are too small to meet the assumptions of a 2 X 2 chisquare test, which of the following procedures can provide a valid test:
*
a) log-linear analysis
b) Fisher's exact test
c) Pearson's test of association
d) Yates' correction for continuity
22. The strength of association in a 2 X 2 contingency table can be measured directly by
calculating:
*
a) Pearson's r
b) the chi-square statistic
c) the binomial coefficient
d) Fisher's exact test
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23. A very large value for the chi-square statistic in a test of association in a 2 X 3 contingency
table implies that:
*
a) the two variables are strongly associated
b) Cramér's phi will be large
c) the contingency coefficient will be large
d) the p value will be small
24. If you increase the sample size (N) without changing the relative proportions in a 2 X 2
contingency table, which of the following will increase:
a) the phi coefficient
*
b) the chi-square statistic
c) the cross-product (odds) ratio
d) the Type II error rate
25. Cramér's phi is often preferred to the contingency coefficient:
a) when dealing with a 2 X 2 contingency table
b) when the sample size is small
c) because the size of the contingency coefficient depends on the sample size, whereas
the size of Cramér's phi does not
*
d) because Cramér's phi always ranges from zero to 1.0, whereas the contingency
coefficient may not
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Chapter 21
Statistical Tests for Ordinal Data
Section A
1. Statistical procedures that deal with ordinal statistics usually assume that the dependent
variable:
*
a) has been measured on a ratio scale
b) varies with changes in the independent variable
c) varies continuously
d) varies discretely
2. Statistical procedures that deal with ordinal statistics are generally not valid when:
*
a) the number of cases is very small
b) the dependent variable does not follow a normal distribution
c) the dependent variable has been measured on a ratio scale
d) a large proportion of the cases are tied
3. The Mann-Whitney rank-sum test is an alternative to:
*
a) the independent t-test
b) the matched t-test
c) the sign test
d) the Spearman test
4. To perform the Mann-Whitney test:
*
a) the scores are ranked separately for each group
b) the scores are combined into one large group before ranking
c) the two groups must be the same size
d) the ranks for one group are added to the ranks for the other group
5. The Mann-Whitney test is most likely to attain statistical significance when the sum of ranks
for one group:
a) is zero
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Test Items
b) is the same as the sum of ranks for the other group
c) is as different as possible from the sum of ranks of the other group
d) equals N
6. The sum of ranks when there are ties is:
*
a) less than the sum of ranks without ties
b) greater than the sum of ranks without ties
c) the same as the sum of ranks without ties
d) less than N
7. If the ranks of two groups do not overlap at all (this is the best case for the Mann-Whitney
test), the value for the U statistic will be:
*
a) zero
b) equal to the sum of the two sample sizes
c) equal to the product of the two sample sizes
d) equal to the sum of ranks for the smaller group
8. To perform the Wilcoxon signed-rank test you must first:
*
a) rank order the difference scores separately for positive and negative differences
b) rank order the difference scores ignoring their signs
c) rank order the scores before finding the differences
d) delete any nonzero difference scores that are tied
9. If your Wilcoxon signed-rank test leads to too many tied nonzero difference scores, you can
always resort to a less powerful alternative known as:
*
a) the matched t-test
b) the normal approximation
c) Cohen's Kappa
d) the sign test
10. When dealing with data measured on an ordinal scale, which of the following cannot be
validly computed:
a) the Spearman correlation coefficient
b) the Mann-Whitney rank-sum test
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Test Items
c) the Wilcoxon matched-pairs signed-rank test
d) the matched t-test
Section B
11. What is the sum of ranks from 21 to 60?
a) 780 b) 1599 c) 1620 d) 1830
-12. Suppose that you are comparing six obsessive-compulsive patients with 12 control subjects
in a Mann-Whitney test, and the patients tend to have the lower ranks on the variable being
measured. The sum of the ranks for the patients must be at or below which of the following
values to be considered statistically significant at the .01 level in a two-tailed test?
a) 30 b) 32 c) 35 d) 82
-13. Imagine that eight extroverts (E) and ten introverts (I) are rank-ordered together in terms of
their creativity in writing a short story. If you apply the Mann-Whitney test (two-tailed) to the
data below, which of the following p values corresponds to your test?
I, I, I, E, I, I, I, E, I, E, I, I, E, I, E, E, E, E
*
a) > .05
b) between .05 and .02
c) between .02 and .01
d) < .01
14. A psychologist is studying the impact of divorce on the social adjustment of teenagers.
Social adjustment ratings are shown below for teenagers whose parents are divorced, and for
those whose parents are still married.
Divorced Parents: 4, 5, 4, 6, 3, 2, 5, 1
Married Parents: 7, 4, 6, 6, 8, 3, 7, 5, 7, 8, 7, 5
If you assign low ranks to low scores, what is the sum of ranks you would use to test the
significance of the Mann-Whitney test?
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a) 30 b) 40 c) 50.5 d) 52.5
-15. Twenty children of professional athletes are being compared on motor coordination with 25
children whose parents are not athletes. The sum of the ranks for the children of the athletes is
600. What is the z-score for the normal approximation to the Mann-Whitney test for this study?
a) .57 b) 1.73 c) 3.2 d) 3.46
-16. Eight subjects have been tested for creativity before and after learning a new relaxation
method. If the four smallest differences are negative (i.e., creativity has been reduced), and the
rest are positive, what is the value for Wilcoxon's T?
a) 0
b) 4
--
c) 10
d) 18
17. Suppose that 30 pairs of subjects are measured before and after some treatment, and
Wilcoxon's T turns out to be 200. What z-score corresponds to this T score, if you assume a
normal distribution for T?
a) -.67
b) +2.67
c) +4.11
d) -4.78
--
18. Thirty-two beginning piano students are matched into 16 pairs based on a musical aptitude
test. Then one member of each pair is randomly assigned to a new form of musical training,
while the other member receives conventional training. At the end of training, the musical
ability of each subject is rated on a ten-point scale, as shown below:
Pair #
1
2
3
4
5
6
7
8
9
New Training
7
5
8
4
6
5
6
8
5
Traditional
6
4
7
7
6
3
5
7
5
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10
11
12
13
14
15
16
6
7
3
8
6
7
6
9
5
3
9
2
6
5
Despite the large number of tied difference scores, perform the Wilcoxon signed-rank test on
these data. What is the value for Wilcoxon's T?
a) 7
b) 19
--
c) 27.5
d) 35
19. In order to calculate the Spearman correlation coefficient:
*
a) the scores should first be ranked separately for each variable
b) the scores from the two variables are combined into one large group before ranking
c) a special short-cut formula must be used
d) both variables must have been initially measured on an ordinal scale
20. In order for the Spearman correlation coefficient to be either +1 or -1, the relationship
between the two variables must be:
*
a) proportional
b) continuous
c) linear
d) monotonic
Section C
21. The Kruskal-Wallis test can be used as an alternative to:
*
a) the independent-groups t-test
b) the one-way independent-groups ANOVA
c) the Mann-Whitney (rank-sum) test
d) all of the above
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22. The size of the Kruskal-Wallis test statistic, H, increases as the sums of ranks for each
group:
a) increase
b) decrease
c) become more alike
d) become more different
*
23. When there are only two repeated conditions, the Friedman test is the same as:
*
a) the sign test
b) the matched t-test
c) the Mann-Whitney test
d) the Kruskal-Wallis test
24. Suppose that ten subjects are each measured under four conditions. Before applying the
formula for the Friedman test:
a) all 40 scores are ranked together
b) the ten scores in each condition are ranked separately
c) the four scores for each subject are ranked separately
d) the ranks are summed separately for each subject
*
25. If five different raters rank order the same ten items, and their coefficient of concordance
(i.e., Kendall's W) is .6, what is the average Spearman correlation for all possible pairs of raters?
a) .50
b) .56
c) .60
d) .66
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