Exam 15: Nonparametric Tests

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:

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 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:

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 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:

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

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

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

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:

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 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:

The Kruskal-Wallis test is best described as:

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 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:

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:

The Wilcoxon signed rank test is best described as:

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:

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

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:

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: