Deck 14: Nonparametric Statistics

Question 1

A real estate agent surmises that the median rent for a one-bedroom apartment in a beach community is at least $1900 per month. The rents for a random sample of 15 one-bedroom apartments are listed below. Test the agent's claim. Use α = .01.

Answer 1

critical value 2.33; test statistic z = 0; fail to reject H0; There is not sufficient evidence to reject the claim.

Question 2

Nonparametric tests are useful for qualitative data that can be ranked.

Answer 2

True

Question 3

For a sign test to be valid, a large sample must be selected from a population which is approximately normal.

Answer 3

False

Question 4

Calculate or use a table to find the binomial probability . Also use the normal approximation to calculate the probability.

Question 5

The sign test can not be used when an outlier is present.

Question 6

Eight SmartCars were randomly selected and the highway mileage of each was noted. The highway mileage per gallon for each car is shown below:

Question 7

What is referred to as nonparametrics?

Question 8

Calculate or use a table to find the binomial probability $P ( x \geq 7 ) \text { when } n = 8 \text { and } p = .5$ .

A) .035
B) .004
C) .965
D) .996

Question 9

If a distribution is approximately normal, the t-test is a more powerful test about the central tendency of the distribution than the sign test.

Question 10

What are distribution-free tests?

Question 11

The sign test provides inferences about the population median rather than the population mean.

Question 12

Nonparametric methods focus on the location of the probability distribution, rather than on specific parameters of the population.

Question 13

Calculate or use a table to find the binomial probability .

Question 14

A convenience store owner believes that the median number of lottery tickets sold per day is 79. A random sample of 20 days yields the data below. Test the owner's claim. Use α = .05.

Question 15

The median household income of a community is reported to be $62,000. A random sample of six households in the community yielded the following incomes. Does the sample provide sufficient evidence to refute the reported median household income? Perform a sign test using α = .10.

Question 16

Which of the following is unknown, in general?

A) the probability that a randomly selected observation exceeds the mean of a nonnormal distribution
B) the probability that a randomly selected observation exceeds the median of a nonnormal distribution
C) the probability that a randomly selected observation exceeds the mean of a normal distribution
D) the probability that a randomly selected observation exceeds the median of a normal distribution

Question 17

Eight SmartCars were randomly selected and the highway mileage of each was noted. The highway mileage per gallon for each car is shown below:

Question 18

What are rank statistics (or rank tests)?

Question 19

Fifteen SmartCars were randomly selected and the highway mileage of each was noted. It was desired to determine if the median miles per gallon of all SmartCars exceeded 35 miles per gallon. What type of nonparametric analysis should be used?

A) The Sign Test
B) The Wilcoxon Rank Sum Test
C) The Wilcoxon Signed Rank Test
D) The Kruskal-Wallis H-Test
E) $\mathrm { F } _ { \mathrm { r } } \text {-Test }$

Question 20

The only assumption necessary to ensure the validity of the sign test is that the probability distribution of measurements is continuous.

Question 21

A government agency claims that the median hourly wages for workers at fast food restaurants in the western U.S. is $8.20. In a random sample of 100 workers, 68 were paid less than $8.20, 10 were paid $8.20, and the rest more than $8.20. Test the government's claim. Use α = .05.

Question 22

Verbal SAT scores for students randomly selected from two different schools are listed below. Use the Wilcoxon rank sum procedure to test the claim that there is no difference in the scores from each school. Use α = .05.

Question 23

A consumer protection organization claims that a new car model gets less than 29 miles per gallon of gas. Ten cars are tested. The results are given below. Test the organization's claim. Use α = .05.

Question 24

When performing a rank test comparing two populations, we rank the sample observations from both populations as though they were drawn from the same population.

Question 25

A researcher wants to know if the time spent in prison for a particular type of crime is the same for men and women. A random sample of men and women were each asked to give the length of sentence received. The data, in years, are listed below. Use the Wilcoxon rank sum procedure to test the claim that there is no difference in the sentence received by each gender. Use α = .05.

Question 26

A marketing study was conducted to compare the mean age of male and female purchasers of a certain product. Random and independent samples were selected for both male and female purchasers of the product. It was desired to test to determine if the distribution of ages of all female purchasers was shifted to the right of the distribution of ages of all male purchasers. What type of nonparametric analysis should be used?

A) The Sign Test
B) The Wilcoxon Rank Sum Test
C) The Wilcoxon Signed Rank Test
D) The Kruskal-Wallis H-Test
E) $\text { The Friedman } \mathrm { F } _ { \mathrm { r } } \text {-Test }$

Question 27

Online classes are becoming more and more prevalent at the college level. A statistics instructor randomly sampled ten students from his traditional face-to-face class and ten students from his online class to compare their comprehension of the material that was taught in the class. He administered the same final exam to each student and wants to use the Wilcoxon Rank Sum test to compare their exam scores. The results are shown below:

Question 28

Test the hypothesis that the median age of statistics teachers is 51 years. A random sample of 60 statistics teachers found 25 above 51 years and 35 below 51 years. Use α = .01.

Question 29

At six different randomly chosen times, a researcher measures the temperature (in degrees Fahrenheit) of a pint of milk from a supermarket's shelf. The measurements are shown below.

Question 30

Specify the rejection region for the Wilcoxon rank sum test in the following situation. H0: Two probability distributions, 1 and 2, are identical Ha: Probability distribution of population 1 is shifted to the right or left of the probability distribution for population 2

Question 31

Online classes are becoming more and more prevalent at the college level. A statistics instructor randomly sampled ten students from his traditional face-to-face class and ten students from his online class to compare their comprehension of the material that was taught in the class. He administered the same final exam to each student and wants to use the Wilcoxon Rank Sum test to compare their exam scores. The results are shown below: $\begin{array}{c}\begin{array}{lll}\text { Traditional Class }\\82 \quad 91 \quad 75 \quad 68 \\93 \quad 85\quad 74 \quad 70 \\56 \quad 82 & & \end{array}\begin{array}{lll}\text { Online Class }\\67 \quad 66 \quad 72 \quad 73 \\77 \quad 76 \quad 48 \quad 81 \\86 \quad 92 & &\end{array}\end{array}$
What alternative hypothesis should the instructor test to show that the students in the traditional class outperformed the students in the online class?

A) The distribution of exams scores for the traditional class is identical to the distribution of exams scores for the online class.
B) The distribution of exams scores for the traditional class is shifted to the left of the distribution of exams scores for the online class.
C) The distribution of exams scores for the traditional class is shifted to the right of the distribution of exams scores for the online class.
D) The distribution of exams scores for the traditional class is shifted to the right or to the left of the distribution of exams scores for the online class.

Question 32

The results of independent random samples from two populations are shown below. Sample A: 11, 15, 18, 21 Sample B: 9, 12, 15, 17, 20, 23 Calculate the rank sum for each sample. Which would be used as the test statistic in a Wilcoxon rank sum test?

Question 33

The results of independent random samples from two populations are shown below. Sample A: 35, 38, 42, 43, 45, 47, 49, 58 Sample B: 36, 41, 44, 53, 57 Calculate the rank sum for each sample. Which would be used as the test statistic in a Wilcoxon rank sum test?

Question 34

The Wilcoxon rank sum test is recommended for comparing discrete distributions.

Question 35

Specify the rejection region for the Wilcoxon rank sum test in the following situation. n1 = 7, n2 = 5, α = .05 H0: Two probability distributions, 1 and 2, are identical Ha: Probability distribution of population 1 is shifted to the right of the probability distribution for population 2

Question 36

For the Wilcoxon rank sum test to be valid, the number of ties in the measurements needs to be small relative to the sample sizes.

Question 37

The Wilcoxon rank sum test is used to test the hypothesis that the probability distributions associated with two populations are equivalent.

Question 38

Online classes are becoming more and more prevalent at the college level. A statistics instructor randomly sampled ten students from his traditional face-to-face class and ten students from his online class to compare their comprehension of the material that was taught in the class. He administered the same final exam to each student and wants to use the Wilcoxon Rank Sum test to compare their exam scores. The results are shown below:

Question 39

Suppose you want to compare two treatments, A and B. In particular, you wish to determine whether the distribution for population B is shifted to the right of the distribution for population A. You plan to use the Wilcoxon rank sum test. a. Specify the null and alternative hypotheses you would test. b. Suppose you obtained the following independent random samples of observations on experimental units subjected to the two treatments. Conduct the test of hypotheses described above, using α = .05. Sample A: 1.2, 1.5, 2.3, 3.2, 3.7, 4.1 Sample B: 2.5, 2.8, 3.6, 4.2, 4.5

Question 40

Independent random samples from two populations are shown in the table. Use the Wilcoxon rank sum test to determine whether the data provide sufficient evidence to indicate a shift in the locations of the probability distributions of the sampled populations. Use α = .05.

Question 41

Do popular diets really help customers in the long-run? Many diets help customers lose weight initially only to have the customer add the weight back on after the first few months on the diet. A commercially successful diet company conducted a study of twelve randomly selected customers and measured their weight after before starting the diet and then again after staying with the diet for a year. The company hoped that the initial weight loss would be sustained after the first year on the diet. The data is shown below:

Question 42

The Kruskal-Wallis test is valid only when each sample contains five or fewer measurements.

Question 43

A physician claims that a person's diastolic blood pressure can be lowered, if, instead of taking a drug, the person listens to a relaxation tape each evening. Ten subjects are randomly selected. Their blood pressures, measured in millimeters of mercury, are listed below. The 10 patients are given the tapes and told to listen to them each evening for one month. At the end of the month, their blood pressures are taken again. The data are listed below. Use the Wilcoxon signed rank test to test the physician's claim. Use α = .05.

Question 44

A certain manufacturer is interested in evaluating two alternative manufacturing plans consisting of different machine layouts. Because of union rules, hours of operation vary greatly for this particular manufacturer from one day to the next. Twenty-eight random working days were selected and each plan was monitored and the number of items produced each day was recorded. Some of the collected data is shown below:

Question 45

Do popular diets really help customers in the long-run? Many diets help customers lose weight initially only to have the customer add the weight back on after the first few months on the diet. A commercially successful diet company conducted a study of twelve randomly selected customers and measured their weight after before starting the diet and then again after staying with the diet for a year. The company hoped that the initial weight loss would be sustained after the first year on the diet. The data is shown below:

Question 46

Do popular diets really help customers in the long-run? Many diets help customers lose weight initially only to have the customer add the weight back on after the first few months on the diet. A commercially successful diet company conducted a study of twelve randomly selected customers and measured their weight after before starting the diet and then again after staying with the diet for a year. The company hoped that the initial weight loss would be sustained after the first year on the diet. The data is shown below: $$\begin{array} { c | c | c } \text { Customer } & \text { Initial } & \text { One Year } \\\hline 1 & 185 & 175 \\2 & 192 & 193 \\3 & 125 & 115 \\4 & 148 & 146 \\5 & 165 & 170 \\6 & 220 & 175 \\7 & 265 & 170 \\8 & 177 & 174 \\9 & 302 & 266 \\10 & 112 & 116 \\11 & 154 & 148 \\12 & 230 & 256\end{array}$$ What alternative hypothesis should be tested in order to determine if the diet is successful at helping customers lose and keep weight off after one year?

A) The distribution of initial weight is identical to the distribution of one-year weight.
B) The distribution of initial weight is shifted to the right of the distribution of one-year weight.
C) The distribution of initial weight is shifted to the left of the distribution of one-year weight.
D) The distribution of initial weight is shifted to the right or to the left of the distribution of one-year weight.

Question 47

When performing a signed rank test in a paired difference experiment, tied absolute value differences are assigned the average of the ranks they would receive if they were unequal, but successive measurements.

Question 48

For a valid signed rank test, the probability distribution from which the sample of paired differences is drawn must be continuous.

Question 49

A weight-lifting coach claims that a weight-lifter can increase strength by taking vitamin E. To test the theory, the coach randomly selects 9 athletes and gives them a strength test using a bench press. The results are listed below. Thirty days later, after regular training supplemented by vitamin E, they are tested again. The new results are listed below. Use the Wilcoxon signed rank test to find the critical value to test the claim that the vitamin E supplement is effective in increasing the athletes' strength. Use α = .05.

Question 50

A scientist is hoping to compare the levels of DDT toxin found in three species of fish in a local river. He randomly samples 7 of each species to use in the analysis. For each fish, he measures the amount of DDT toxin present. The data is shown below: $$\begin{array}{l}\text { Species }\\\begin{array} { c | c | c } 1 & 2 & 3 \\\hline 8.4 & 4.5 & 0.58 \\15 & 4.2 & 2 \\25 & 3 & 2.2 \\5.6 & 2.3 & 7.4 \\4.6 & 2.5 & 0.35 \\8.2 & 6.8 & 1.9 \\6.1 & 5.1 & 2.7\end{array}\end{array}$$ Ideally he will be able to determine if a difference exists in the population of DDT toxins found in the three fish species. Identify the test statistic for the Kruskal-Wallis H-Test.

A) H = 17.0
B) H = 10.286
C) H = 11.718
D) H = 5.714

Question 51

For a valid Kruskal-Wallis test, the probability distribution from which the samples are drawn must be continuous.

Question 52

Nine students took the SAT test. Their scores are listed below. Later, they took a test preparation course and retook the SAT. Their new scores are listed below. Use the Wilcoxon signed rank test to test the claim that the test preparation had no effect on their scores. Use α = .05.

Question 53

In a study of the effectiveness of physical exercise on weight loss, 20 people were randomly selected to participate in a program for 30 days. Use the Wilcoxon signed rank test to test the claim that exercise has no bearing on weight loss. Use α = .02.

Question 54

When applying the Wilcoxon signed rank test, the number of ties should be small relative to the number of observations to ensure the validity of the test.

Question 55

A scientist is hoping to compare the levels of DDT toxin found in three species of fish in a local river. He randomly samples 50 of each species to use in the analysis. For each fish, he measures the amount of DDT toxin present. Ideally he will be able to determine if a difference exists in the population of DDT toxins found in the three fish species. What type of nonparametric analysis should be used?

A) The Sign Test
B) The Wilcoxon Rank Sum Test
C) The Wilcoxon Signed Rank Test
D) The Kruskal-Wallis H-Test
E) $\text { The Friedman } \mathrm { F } _ { \mathrm { r } } \text {-Test }$

Question 56

Specify the rejection region for the Wilcoxon signed rank test in the following situation. n = 25, α = .05 H0: Two probability distributions, 1 and 2, are identical Ha: Probability distribution of population 1 is shifted to the right of the probability distribution for population 2

Question 57

Specify the rejection region for the Wilcoxon signed rank test in the following situation. n = 35, α = .10 H0: Two probability distributions, 1 and 2, are identical Ha: Probability distribution of population 1 is shifted to the right or left of the probability distribution for population 2

Question 58

A pharmaceutical company wishes to test a new drug with the expectation of lowering cholesterol levels. Ten subjects are randomly selected and their cholesterol levels are recorded. The results are listed below. The subjects were placed on the drug for a period of 6 months, after which their cholesterol levels were tested again. The results are listed below. (All units are milligrams per deciliter.) Use the Wilcoxon signed rank test to test the company's claim that the drug lowers cholesterol levels. Use α = .05.

Question 59

A local school district is concerned about the number of school days missed by its teachers due to illness. A random sample of 10 teachers is selected. The numbers of days absent in one year is listed below. An incentive program is offered in an attempt to decrease the number of days absent.The number of days absent in one year after the incentive program is listed below. Use the Wilcoxon signed rank test to test the claim that the incentive program cuts down on the number of days missed by teachers. Use α = .05. Days Absent

Question 60

The Wilcoxon signed rank test for large samples can be used when $n \geq 10$ .

Question 61

A scientist is hoping to compare the levels of DDT toxin found in three species of fish in a local river. He randomly samples 7 of each species to use in the analysis. For each fish, he measures the amount of DDT toxin present. The data is shown below: $\quad$ $\quad$$\text { Species }$
$\begin{array}{c|c|c}1 & 2 & 3 \\\hline 8.4 & 4.5 & 0.58 \\15 & 4.2 & 2 \\25 & 3 & 2.2 \\5.6 & 2.3 & 7.4 \\4.6 & 2.5 & 0.35 \\8.2 & 6.8 & 1.9 \\6.1 & 5.1 & 2.7\end{array}$
Ideally he will be able to determine if a difference exists in the population of DDT toxins found in the three fish species. The test statistic was determined to be H = 11.718. State the appropriate conclusion when testing at ? = 0.025.

A) There is sufficient evidence to indicate that the distributions of DDT toxins for the three species are identical.
B) There is insufficient evidence to indicate that the distributions of DDT toxins for the three species are identical.
C) There is sufficient evidence to indicate that the distributions of DDT toxins for the three species differ in location.
D) There is insufficient evidence to indicate that the distributions of DDT toxins for the three species differ in location.

Question 62

The grade point averages of students participating in sports at a college are to be compared. The data are listed below. Use the Kruskal-Wallis test to test the claim that there is no difference in the distribution of the populations. Use α = .05.

Question 63

Fading of wood is a problem with wooden decks on boats. Three varnishes used to retard this aging process were tested to see whether there were any differences among them. Samples of 10 different types of wood were treated with each of the three varnishes and the amount of fading was measured after three months of exposure to the sun. The data are listed below. Is there evidence of a difference in the probability distributions of the amounts of fading among the three types of varnish? Use α = .05.

Question 64

$$\mathrm { P } \left( \mathrm { r } _ { \mathrm { s } } > 0.608 \right) \text { when } \mathrm { n } = 19$$

A) 0.005
B) 0.01
C) 0.995
D) 0.05

Question 65

A researcher wishes to determine whether there is a difference in the average age of elementary school, high school, and community college teachers. Teachers are randomly selected. Their ages are recorded below. Use the Kruskal-Wallis H-test to test the claim that there is no difference in the distribution of the populations. Use α = .05.

Question 66

In a study comparing the effects of four energy drinks on running speed, eight runners were timed (in seconds) running four miles. On each day, they were given a single energy drink. The data are listed below. Is there evidence of a difference in the probability distributions of the running times among the four drinks? Use α = .025.

Question 67

Four brands of baseball bats were tested to determine which bat allowed hitters to hit a baseball farthest. Eight different batters were thrown 25 pitches while hitting with each of the four bats (assigned in random order). The median distance of the five longest hits using each bat is shown in the table below. The goal is to determine if a difference exists in the population of median distances for each of the four brands of baseball bat
$$\begin{array} { c | c | c | c | c } \text { Batter } & \text { Brand 1 } & \text { Brand 2 } & \text { Brand 3 } & \text { Brand 4 } \\\hline 1 & 307 & 315 & 300 & 275 \\2 & 310 & 317 & 305 & 285 \\3 & 335 & 335 & 335 & 302 \\4 & 325 & 328 & 320 & 300 \\5 & 300 & 305 & 295 & 270 \\6 & 345 & 350 & 340 & 310 \\7 & 312 & 315 & 308 & 300 \\8 & 298 & 302 & 295 & 288\end{array}$$
What type of nonparametric analysis should be used?

A) The Sign Test
B) The Wilcoxon Rank Sum Test
C) The Wilcoxon Signed Rank Test
D) The Kruskal-Wallis H-Test
E) The Friedman Fr-Test

Question 68

When there is a perfect negative rank correlation between two variables, the value of the test statistic rs in Spearman's test is -1.

Question 69

A medical researcher wishes to try three different techniques to lower blood pressure of patients with high blood pressure. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six weeks, the reduction in each subject's blood pressure is recorded. Use the Kruskal-Wallis H-test to test the claim that there is no difference in the distribution of the populations. Use α = .05.

Question 70

The time (in minutes) it takes to assemble a computer component for three different machines is listed below. Workers are randomly selected. Use the Kruskal-Wallis test to test the claim that there is no difference in the distribution of the populations. Use α = .05.

Question 71

A realtor wishes to compare the square footage of houses in 4 different cities, all of which are priced approximately the same. The data are listed below. Use the Kruskal-Wallis test to test the claim that there is no difference in the distribution of the populations. Use α = .05. 14.6 Comparing Three or More Populations: Randomized Block Design 1 Understand the Friedman

Question 72

A scientist is hoping to compare the levels of DDT toxin found in three species of fish in a local river. He randomly samples 7 of each species to use in the analysis. For each fish, he measures the amount of DDT toxin present. The data is shown below:

Question 73

For a valid Spearman's test, the probability distribution for just one of the two variables must be continuous.

Question 74

if a difference exists in the population of median distances for each of the four brands of baseball bat. 2 Perform Friedman Fr-Test
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Suppose you have used a randomized block design to compare the efficacy of three different doses of an experimental drug. You used seven patients in your study. The data is listed below. Do the data indicate that a particular dosage is more effective than any other dosage? Use α = .10.

Question 75

Four different types of fertilizers are used on raspberry plants. The number of raspberries on each randomly selected plant is given below. Use the Kruskal-Wallis H-test to test the claim that there is no difference in the distribution of the populations. Use α = .05.

Question 76

The advantage of the Friedman $F _ { r } \text {-test }$ is that the measurements within each block do not need to be ranked.

Question 77

Is there evidence that at least two of the treatment probability distributions differ in location? Explain. 14.7 Rank Correlation 1 Understand Spearman's Rank Correlation Test

Question 78

Four brands of baseball bats were tested to determine which bat allowed hitters to hit a baseball farthest. Eight different batters were thrown 25 pitches while hitting with each of the four bats (assigned in random order). The median distance of the five longest hits using each bat is shown in the table below. The goal is to determine if a difference exists in the population of median distances for each of the four brands of baseball bat.
$$\begin{array} { c | c | c | c | c } \text { Batter } & \text { Brand 1 } & \text { Brand 2 } & \text { Brand 3 } & \text { Brand 4 } \\\hline 1 & 307 & 315 & 300 & 275 \\2 & 310 & 317 & 305 & 285 \\3 & 335 & 335 & 335 & 302 \\4 & 325 & 328 & 320 & 300 \\5 & 300 & 305 & 295 & 270 \\6 & 345 & 350 & 340 & 310 \\7 & 312 & 315 & 308 & 300 \\8 & 298 & 302 & 295 & 288\end{array}$$
Use the Friedman Fr-Test and find the rejection region used to determine if at least two of the distance
distributions for the four baseball bats differ in location. Use ? = 0.025.

A) Reject H0 if Fr > 9.28
B) Reject H0 if Fr > 29.46
C) Reject H0 if Fr > 15.10
D) Reject H0 if Fr > 15.44

Question 79

Four brands of baseball bats were tested to determine which bat allowed hitters to hit a baseball farthest. Eight different batters were thrown 25 pitches while hitting with each of the four bats (assigned in random order). The median distance of the five longest hits using each bat is shown in the table below. The goal is to determine if a difference exists in the population of median distances for each of the four brands of baseball bat
$$\begin{array} { c | c | c | c | c } \text { Batter } & \text { Brand 1 } & \text { Brand 2 } & \text { Brand 3 } & \text { Brand 4 } \\\hline 1 & 307 & 315 & 300 & 275 \\2 & 310 & 317 & 305 & 285 \\3 & 335 & 335 & 335 & 302 \\4 & 325 & 328 & 320 & 300 \\5 & 300 & 305 & 295 & 270 \\6 & 345 & 350 & 340 & 310 \\7 & 312 & 315 & 308 & 300 \\8 & 298 & 302 & 295 & 288\end{array}$$
Use the Friedman Fr-Test to calculate the test statistic used to determine if at least two of the distance
distributions for the four baseball bats differ in location.

A) Fr = 3.125
B) Fr = 4.781
C) Fr = 22.950
D) Fr = 25.828

Question 80

For a valid Friedman Fr-test, the probability distributions from which the samples within each block are drawn must be continuous.