Deck 15: Nonparametric Tests

Question 1

A researcher decided to conduct an experiment to study the effect of caffeine by randomly dividing nine people into two groups. Group 1, consisting of five study subjects had a cup of coffee to start the experiment. Group 2, consisting of four people, had decaffeinated coffee. The study subjects did not know which group they belonged to. All study subjects were then asked to tap their finger. The measurement taken was the number of finger taps per minute. This type of data is often skewed and the researchers decided that they should not use a two-sample t test. They chose a Wilcoxon rank sum test and recorded the ranks of the subjects in the caffeine group. The standard deviation of this test statistic is given by:

A)16.67.
B)4.083.
C)22.5.
D)None of the answer options is correct.

Answer 1

4.083.

Question 2

The alternative hypothesis for the Kruskal-Wallis test is typically:

A)that not all median values from the populations are the same.
B)that the response variables are systematically different in some populations more than others.
C)Both options (a)and (b); they express the same thing in different ways.
D)Neither option (a)nor (b).

Answer 2

Both options (a)and (b); they express the same thing in different ways.

Question 3

The use of carpeting in hospitals, while having aesthetic value, raises an obvious question: Are carpeted floors sanitary? An experiment comparing airborne bacteria levels in carpeted rooms with that in uncarpeted rooms was performed in an effort to answer this question. Specifically, for each of eight carpeted rooms and eight uncarpeted rooms in a hospital, the number of bacteria per cubic foot of air was measured. The data are given below. Assume that room sizes are equal, and that the 16 rooms in this experiment were assigned randomly to the carpet and no-carpet groups. If W, the Wilcoxon rank sum test statistic, is the sum of the ranks assigned to the carpeted group, and if the two populations have the same continuous distribution, then the standard deviation of W is:

A)2.32.
B)8.51.
C)9.52.
D)90.67.

Answer 3

9.52.

Question 4

A tool manufacturer tests three types of cutters used in a lathe operation. One type is a laminated steel cutter made of very hard high-carbon steel sandwiched between two softer steels. Another is a special high-speed steel cutter developed using powder metallurgy. The final one is made from cryogenically treated steel. Cutters of each type are tested to see how long they will last (in hours) in continuous operation until they need to be sharpened. The times are recorded for each cutter used in the study, and the results are given below. The value of the Kruskal-Wallis statistic H for these data is:

A)2.75.
B)3.20.
C)4.76.
D)5.42.

Question 5

The use of carpeting in hospitals, while having aesthetic value, raises an obvious question: Are carpeted floors sanitary? An experiment comparing airborne bacteria levels in carpeted rooms with that in uncarpeted rooms was performed in an effort to answer this question. Specifically, for each of eight carpeted rooms and eight uncarpeted rooms in a hospital, the number of bacteria per cubic foot of air was measured. The data are given below. Assume that room sizes are equal, and that the 16 rooms in this experiment were assigned randomly to the carpet and no-carpet groups. If W, the Wilcoxon rank sum test statistic, is the sum of the ranks assigned to the carpeted group, then the value of W is:

A)18.5.
B)58.5.
C)62.5.
D)74.5.

Question 6

The Wilcoxon signed rank test is best described as:

A)a nonparametric alternative to one-way ANOVA that relaxes the Normality condition.
B)a nonparametric alternative to the two-sample t test that relaxes the Normality condition.
C)a nonparametric alternative to the paired t test that relaxes the Normality condition.
D)None of the answer options is correct.

Question 7

In a study to investigate the effect of regular exercise on raising HDL (good cholesterol) levels, a random sample of five male subjects known to have low HDL levels had their HDL measured. After six months on a regular exercise schedule, it was measured again. The data are given in the following table. The researcher is interested in testing H₀ : initial and final HDL measurements have the same distribution against the one-sided alternative H_a: final HDL levels are systematically lower. The P-value for the Wilcoxon signed rank test W⁺ is:

A)0.0281.
B)0.0329.
C)0.0495.
D)0.0694.

Question 8

The Wilcoxon rank sum test for comparing two independent samples tests which of the following hypotheses?

A)H₀: median₁ = median₂ for two distributions having the same shape vs. H_a: median₁  median₂.
B)H₀: the two distributions are the same vs, H_a: one distribution has values systematically larger than the other.
C)Both options (a)and (b).
D)None of the answer options is correct.

Question 9

A tool manufacturer tests three types of cutters used in a lathe operation. One type is a laminated steel cutter made of very hard high-carbon steel sandwiched between two softer steels. Another is a special high-speed steel cutter developed using powder metallurgy. The final one is made from cryogenically treated steel. Cutters of each type are tested to see how long they will last (in hours) in continuous operation until they need to be sharpened. The times are recorded for each cutter used in the study, and the results are given below. Under the null hypothesis that the three populations have the same continuous distribution:

A)H has approximately a chi-square distribution, with 3 degrees of freedom.
B)H has approximately an F(2, 6)distribution.
C)H has approximately a chi-square distribution, with 2 degrees of freedom.
D)H has a distribution that cannot be evaluated because the populations may not be Normal.

Question 10

The Kruskal-Wallis test is best described as:

A)a nonparametric alternative to one-way ANOVA that relaxes the Normality condition.
B)a nonparametric alternative to one-way ANOVA that relaxes the requirement of independent samples.
C)a nonparametric alternative to the two-sample t test that relaxes the Normality condition.
D)a nonparametric alternative to the two-sample t test that relaxes the requirement of independent samples.

Question 11

A tool manufacturer tests three types of cutters used in a lathe operation. One type is a laminated steel cutter made of very hard high-carbon steel sandwiched between two softer steels. Another is a special high-speed steel cutter developed using powder metallurgy. The final one is made from cryogenically treated steel. Cutters of each type are tested to see how long they will last (in hours) in continuous operation until they need to be sharpened. The times are recorded for each cutter used in the study, and the results are given below. The P-value of the Kruskal-Wallis statistic H for these data is:

A)less than 0.025.
B)between 0.025 and 0.05.
C)between 0.05 and 0.1.
D)greater than 0.1.

Question 12

The use of carpeting in hospitals, while having aesthetic value, raises an obvious question: Are carpeted floors sanitary? An experiment comparing airborne bacteria levels in carpeted rooms with that in uncarpeted rooms was performed in an effort to answer this question. Specifically, for each of eight carpeted rooms and eight uncarpeted rooms in a hospital, the number of bacteria per cubic foot of air was measured. The data are given below. Assume that room sizes are equal, and that the 16 rooms in this experiment were assigned randomly to the carpet and no-carpet groups. If W, the Wilcoxon rank sum test statistic, is the sum of the ranks assigned to the carpeted group, then z, the standardized value of W, is:

A)0.37.
B)0.68.
C)1.68.
D)2.32.

Question 13

The use of carpeting in hospitals, while having aesthetic value, raises an obvious question: Are carpeted floors sanitary? An experiment comparing airborne bacteria levels in carpeted rooms with that in uncarpeted rooms was performed in an effort to answer this question. Specifically, for each of eight carpeted rooms and eight uncarpeted rooms in a hospital, the number of bacteria per cubic foot of air was measured. The data are given below. Assume that room sizes are equal, and that the 16 rooms in this experiment were assigned randomly to the carpet and no-carpet groups. When ranking all the observations together, the bacteria count ranked 12th is:

A)11.8.
B)12.0.
C)12.1.
D)13.0.

Question 14

In a study to investigate the effect of regular exercise on raising HDL (good cholesterol) levels, a random sample of five male subjects known to have low HDL levels had their HDL measured. after six months on a regular exercise schedule, it was measured again. The data are given in the following table. The standardized value z of the Wilcoxon signed rank test W⁺ is:

A)0.4.
B)0.95.
C)1.48.
D)3.51.

Question 15

The Wilcoxon signed rank test is used for small samples when Normality cannot be established or is known not to hold for:

A)more than two groups.
B)two independent samples with unequal variances.
C)paired data.
D)All of the answer options are correct.

Question 16

The primary difference between the null hypothesis of the two-sample t test and the corresponding nonparametric Wilcoxon rank sum test is:

A)the Wilcoxon test presumes equality between the medians of the two samples.
B)the Wilcoxon test presumes equality between the means of the two samples.
C)the Wilcoxon test presumes equality between the medians of the two populations.
D)the Wilcoxon test presumes equality between the means of the two populations.

Question 17

The mean and the standard deviation of W for the nonparametric Wilcoxon rank sum test:

A)depend entirely upon the characteristics of the two samples.
B)depend entirely upon the characteristics of the two populations.
C)depend on the number of items that are being ranked, but not on the values themselves.
D)depend on the number of items that are being ranked, as well as the values themselves.

Question 18

The use of carpeting in hospitals, while having aesthetic value, raises an obvious question: Are carpeted floors sanitary? An experiment comparing airborne bacteria levels in carpeted rooms with that in uncarpeted rooms was performed in an effort to answer this question. Specifically, for each of eight carpeted rooms and eight uncarpeted rooms in a hospital, the number of bacteria per cubic foot of air was measured. The data are given below. Assume that room sizes are equal, and that the 16 rooms in this experiment were assigned randomly to the carpet and no-carpet groups. The researcher is interested in testing H₀ : no difference in distribution of bacteria counts for carpeted and uncarpeted rooms, against the one-sided alternative H _a^: the carpeted group tends to have higher bacteria counts. If W, the Wilcoxon rank sum test statistic, is the sum of the ranks assigned to the carpeted group, then the P-value for the test is:

A)0.046.
B)0.248.
C)0.496.
D)0.752.

Question 19

A tool manufacturer tests three types of cutters used in a lathe operation. One type is a laminated steel cutter made of very hard high-carbon steel sandwiched between two softer steels. Another is a special high-speed steel cutter developed using powder metallurgy. The final one is made from cryogenically treated steel. Cutters of each type are tested to see how long they will last (in hours) in continuous operation until they need to be sharpened. The times are recorded for each cutter used in the study, and the results are given below. The data are to be analyzed with the Kruskal-Wallis test. The null hypothesis is that the time until sharpening is needed has the same distribution in all groups. The alternative is:

A)mean time until sharpening is needed is systematically higher in some groups than others.
B)not all three mean times until sharpening is needed are equal.
C)the mean times until sharpening are different in each of the three groups of cutter type.
D)the times until sharpening is needed are systematically longer for cryogenically treated steel cutters than for the other two types of cutters.

Question 20

In a study to investigate the effect of regular exercise on raising HDL (good cholesterol) levels, a random sample of five male subjects known to have low HDL levels had their HDL measured. After six months on a regular exercise schedule, it was measured again. The data are given in the following table. How do you know that the data should be analyzed using the Wilcoxon signed rank test?

A)There are three groups, and some of the observations are negative numbers.
B)There are three groups, with one observation in each group matched to a subject.
C)There are two groups, and some of the observations are negative numbers.
D)There are two groups, with one observation in each group matched to a subject.

Question 21

A study was conducted to assess the effects of caffeine on concentration. Fifteen subjects were recruited into the study and randomly divided into three groups of equal sample size. Group 1 was given an 8-ounce cup of coffee; group 2, an 8-ounce cup of black tea; and group 3, an 8-ounce cup of decaffeinated coffee. Each subject was then shown a set of 20 numbers for one minute. The subjects were then asked to enter five of the numbers they had been shown and the time it took was recorded. A statistician told the investigators they should use a nonparametric test, since their data was unlikely to be Normally distributed and their sample was small. The researchers consulted the literature on nonparametric tests and found several. An appropriate test to use here is:

A)the Wilcoxon signed rank test.
B)the Kruskal-Wallis test.
C)the Wilcoxon rank sum test.
D)the Mann-Whitney test.

Question 22

Which of the following is part of the signed rank test?

A)The absolute values of the differences of two measurements on each subject are ranked.
B)The ranks of subjects in two samples are calculated and the differences of all pairs of ranks are calculated.
C)Subjects are given one of two treatments in random order. The responses under treatment 1 are then ranked and the same is done for treatment 2. The differences in ranks are the computed.
D)All of the answer options are correct.

Question 23

Which of the following is false?

A)As the sample sizes n₁, n₂ increase, the distribution of the Wilcoxon Rank Sum test can be approximated by the Normal distribution.
B)There is a direct algebraic link between the Mann-Whitney test and the Wilcoxon Rank Sum test.
C)The Kruskal-Wallis test is used for paired data.
D)Zero differences in the Wilcoxon Signed Rank test do not contribute to the test statistic.

Question 24

Ten stores were selected to test the effects of two different designs for packaging a tablet on sale. Five stores were randomly assigned to design 1 and the other five stores were assigned to design 2. The data below provide the sales over a one-week period. The standard deviation of the Wilcoxon Rank Sum test is:

A)4.787.
B)22.92.
C)3.05.
D)9.3025.

Question 25

The Kruskal-Wallis test is the non-parametric equivalent of the analysis of variance comparison of three or more means. It is carried out by combining the samples, obtaining the ranks in the combined sample, and carrying out an analysis of variance on the ranks. However, unlike traditional analysis of variance, it is not necessary to calculate both SSG and SSE because:

A)the total sum of the ranks is a fixed number and the same for all studies with the same total number of observations.
B)the SSE is fixed and the same for all studies with the same number of observations.
C)the sum of the ranks is an approximate chi-square statistic and no standard error is needed.
D)All of the answer options are correct.

Question 26

The Wilcoxon Rank Sum test comparing two populations tests the hypothesis H₀: median₁ = median₂ when which of the following conditions holds?

A)The standard deviations ₁ = ₂.
B)The means ₁ = ₂.
C)Both populations are symmetric.
D)The shapes of both populations are the same.

Question 27

A cosmetics company wants to market a new suntan lotion. The developers recruit 10 volunteers for their study to test the effectiveness of the new lotion. Each volunteer was given the old lotion on one arm, randomly selected, and the other lotion on the other arm. The volunteer was then asked to expose both arms to the sun. The time was measured until there was visible redness in one arm, which was then shielded from the sun. The experiment ended when the other arm showed signs of redness. Below are the times for both arms for each subject. This study design is called:

A)a left vs. right design.
B)a same subject design.
C)a matched pairs design.
D)a before and after design.

Question 28

Fifteen stores were selected to test the effects of three different designs for packaging a tablet on sale. Five stores each were randomly assigned to design 1, 2, or 3. The data below provide the sales over a one-week period. The marketing team decided not to use ANOVA to test whether the three different designs had the same effect on sales or whether sales were increased by some designs over others. The hypotheses to be tested are:

A)H₀: ₁ = ₂ = ₃ vs. H_a: at least two means are different.
B)H₀: median₁ = median₂ = median₃ vs. H_a: at least two medians are different.
C)H₀: sales are not the same for all designs vs. H_a: sales are systematically higher for some designs over others.
D)None of the answer options is correct.