Independent random samples from two populations are shown in the table. $\begin{array}{c} \hline\text { Sample 1 }\quad\text { Sample 2 }\\ \begin{array}{ccc} \hline 11 & 14 & 9 \\ 3 & 12 & 5 \\ 16 & & \\ \hline \end{array} \begin{array}{ccc} \hline 19 & 17 & 10 \\ 22 & 18 & 8 \\ 14 & 25 & \\ \hline \end{array} \end{array}$ 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 $\alpha = 0.05$

@#LAT-DLM& Two probability distributions, 1 and 2

A researcher wishes to determine whether physical exercise is effective in helping people to lose weight. 20 people were randomly selected to participate in an exercise program for 30 days. Use the Wilcoxon signed rank test to test the claim that exercise has an effect on weight. Use $\alpha = 0.02$ \[\begin{array} { l l l l l l l } \hline{ \text { Weight Before Program } } & & & & & \\ \text { (in Pounds) } & 178&210&156&188&193&225&190&165&168 & 200 \\ \hline \begin{array} { l } \text { Weight After Program } \\ \text { (in Pounds) } \end{array} & 182&205&156&190&183&220&195&155&165 & 200 \\ \hline\\ \hline{ \text { Weight Before Program } } & & & & & \\ \text { (in Pounds) } &186&172&166&184&225&145&208&214&148& 174\\ \hline \text { Weight After Program } \\ \text { (in Pounds) } &180&173&165&186&240&138&203&203&142&170\\ \hline \end{array}\]

@#LAT-DLM& : Probability distribution of weights b

Exam 15: Nonparametric Statistics Available on CD

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 for the three different types of varnish? Apply the Friedman Fr-test to the data. Be sure to specify the null and alternative hypotheses. $\text { Use } \alpha = 0.05$ Varnish Sample 1 2 3 1 5.2 4.8 5.5 2 7.8 7.0 7.6 3 4.0 4.1 4.0 4 6.8 5.9 6.5 5 9.0 7.4 8.4 6 6.2 5.4 6.3 7 4.3 4.3 4.7 8 6.7 6.8 7.0 9 5.5 4.7 4.8 10 5.8 5.6 5.7

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Question 1

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$\mathrm { H } _ { 0 }$ : The probability distributions of amounts of fading are identical for the three different types of varnish
$\mathrm { H } _ { \mathrm { a } }$ : At least two of the three types of varnish have distributions of fading amounts that differ in location
Critical value 5.991; $\mathrm { F } _ { \mathrm { r } } = 6.35$ ; reject $\mathrm { H } _ { 0 }$
There is enough evidence to conclude that at least two of the three types of varnish have distributions of fading amounts that differ in location

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 ? = 0.05. Patient 1 2 3 4 5 6 7 8 9 10 Before 80 91 87 99 86 81 95 84 93 94 After 77 85 87 91 80 70 98 74 89 78

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Question 2

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$\mathrm { H } _ { 0 }$ : Probability distribution of blood pressures before relaxation tapes is identical to probability distribution of blood pressures after relaxation tapes,
$\mathrm { H } _ { \mathrm { a } }$ : Probability distribution of blood pressures after relaxation tapes is shifted to the left of probability distribution of blood pressures before relaxation tapes; critical value 8 ; test statistic $\mathrm { T } = 1.5$ ; reject $\mathrm { H } _ { 0 }$ ; There is sufficient evidence to support the claim that the relaxation tapes are effective in lowering blood pressure.

The temperatures on randomly chosen days during a summer class and the number of absences from class on those days are listed below. Calculate Spearman's rank correlation coefficient. Can you conclude that there is a correlation between the temperature and the number absent? Use $\alpha = 0.01$ Temp 62 75 81 80 78 88 65 90 70 Absences 16 20 23 23 21 28 17 28 18

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Question 3

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critical values $\pm 0.833 \text {; test statistic } \mathrm { r } _ { \mathrm { S } } \approx 0.992 \text {; reject } \mathrm { H } _ { 0 } \text {; }$ ; There is enough evidence to
conclude that there is a significant correlation between the temperature and the
number of absences.

Independent random samples from two populations are shown in the table. Sample 1 Sample 2 11 14 9 3 12 5 16 19 17 10 22 18 8 14 25 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 $\alpha = 0.05$

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Question 4

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 $\alpha = 0.05$ 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

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Question 5

Specify the rejection region for the Wilcoxon signed rank test in the following situation. $\mathrm { n } = 25 , \alpha = 0.05$ $\mathrm { H } _ { 0 }$ : Two probability distributions, 1 and 2, are identical $\mathrm { H } _ { \mathrm { a } }$ : Probability distribution of population 1 is shifted to the right of the probability distribution for population 2

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Question 6

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 whether the distributions of the ages of teachers differ among the three types of school. Be sure to specify the null and alternative hypotheses. $\text { Use } \alpha = 0.05$ Elementary School Teachers High School Teachers Community College Teachers 30 41 44 33 46 50 32 43 41 57 52 66 42 47 50 30 36 40

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Question 7

A technician is interested in comparing the time it takes to assemble a certain computer component using three different machines. Workers are randomly selected and randomly assigned to one of the machines. The assembly times (in minutes) are shown in the table. Use the Kruskal-Wallis H-test to test whether the distributions of assembly times differ for the three different machines. Be sure to specify the null and alternative hypotheses. Use $\alpha = 0.05$ Machine 1 Machine 2 Machine 3 38 46 34 37 35 31 38 44 35 36 39 37 39 41 36 37 38 33 38 42 43

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Question 8

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.

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Question 9

The number of absences and the final grades of 9 randomly selected students from a statistics class are given below. Calculate Spearman's rank correlation coefficient. Can you conclude that there is a correlation between the final grade and the number of absences? Use $\alpha = 0.01$ Number of Absences 0 3 6 4 9 2 15 8 5 Final Grade 98 86 80 82 71 92 55 76 82

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Question 10

Calculate or use a table to find the binomial probability P(x $P ( x \geq 20 ) \text { when } n = 25 \text { and } p = 0.5 \text {. }$ Also use the normal approximation to calculate the probability.

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Question 11

Specify the rejection region for the Wilcoxon rank sum test in the following situation. $\mathrm { n } _ { 1 } = 6 , \mathrm { n } _ { 2 } = 8 , \alpha = 0.10$ $\mathrm { H } _ { 0 }$ : Two probability distributions, 1 and 2 , are identical $\mathrm { H } _ { \mathrm { a } }$ : Probability distribution of population 1 is shifted to the right or left of the probability distribution for population 2

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Question 12

The final exam scores of 10 randomly selected statistics students and the number of hours they studied for the exam are given below. Calculate Spearman's rank correlation coefficient. Can you conclude that there is a correlation between the scores on the test and the times spent studying? $\text { Use } \alpha = 0.01$ Hours 6 8 5 11 5 7 7 8 9 6 Scores 68 83 63 91 69 81 88 93 93 74

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Question 13

The Wilcoxon signed rank test for large samples can be used when n ≥ 10.

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Question 14

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 level $\text { Use } \alpha = 0.05$ Subject 1 2 3 4 5 6 7 8 9 10 Before 230 219 204 262 177 260 186 254 175 268 After 215 214 212 252 172 260 156 236 173 253

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Question 15

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. An incentive program is offered in an attempt to reduce absences. The number of days of absence in the year before the incentive program and in the year after the incentive program are shown below for each teacher. Use the Wilcoxon signed rank test to test the claim that the incentive program is effective in reducing absences. Use $\alpha = 0.05$ Teacher Days Absent Before Incentive Days Absent After Incentive 1 2 3 4 5 6 7 8 9 10 \ 4 3 7 8 5 8 2 10 4 5 2 2 7 6 4 6 0 11 2 5

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Question 16

Specify the rejection region for the Wilcoxon rank sum test in the following situation. $\mathrm { n } _ { 1 } = 7 , \mathrm { n } _ { 2 } = 5 , \alpha = 0.05$ $\mathrm { H } _ { 0 }$ : Two probability distributions, 1 and 2 , are identical $\mathrm { H } _ { \mathrm { a } }$ : Probability distribution of population 1 is shifted to the right of the probability distribution for population 2

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Question 17

Suppose you have used a randomized block design to compare the effects of four different energy drinks on running speeds. Eight athletes were randomly selected. Each Monday each athlete was assigned an energy drink and their time to run four miles was recorded. The results (in seconds) are shown below. Is there evidence of a difference in the probability distributions of the running times among the four different drinks? Apply the Friedman Fr-test to the data. Be sure to specify the null and alternative hypotheses. Use $\alpha = 0.025$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\text { Drink }$ Runner 1 2 3 4 1 1275 1276 1323 1294 2 1179 1085 1201 1209 3 1279 1407 1341 1353 4 1306 1267 1322 1317 5 1209 1171 1265 1270 6 1368 1345 1398 1394 7 1270 1311 1307 1293 8 1293 1184 1321 1298

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Question 18

The grade point averages of students participating in different sports at a college are to be compared. The GPAs of students randomly selected from three different groups are listed below. Use the Kruskal-Wallis H-test to test whether the distributions of GPAs differ among the three groups. Be sure to specify the null and alternative hypotheses. Use $\alpha = 0.05$ Tennis Golf Swimming 3.0 1.6 2.5 2.4 1.9 2.8 2.3 3.1 2.6 3.3 1.7 2.3 2.9 2.1 2.3 1.9 1.8 2.2

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Question 19

A researcher wishes to determine whether physical exercise is effective in helping people to lose weight. 20 people were randomly selected to participate in an exercise program for 30 days. Use the Wilcoxon signed rank test to test the claim that exercise has an effect on weight. Use $\alpha = 0.02$ Weight Before Program (in Pounds) 178 210 156 188 193 225 190 165 168 200 Weight After Program (in Pounds) 182 205 156 190 183 220 195 155 165 200 Weight Before Program (in Pounds) 186 172 166 184 225 145 208 214 148 174 Weight After Program (in Pounds) 180 173 165 186 240 138 203 203 142 170

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Question 20

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.

Calculate or use a table to find the binomial probability P(x $P ( x \geq 20 ) \text { when } n = 25 \text { and } p = 0.5 \text {. }$ Also use the normal approximation to calculate the probability.

The Wilcoxon signed rank test for large samples can be used when n ≥ 10.

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Random Variables and Probability Distributions

Sampling Distributions

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single Sample: 355 Tests of Hypotheses

Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Design of Experiments and Analysis of Variance

Categorical Data Analysis

Simple Linear Regression

Multiple Regression and Model Building

Methods for Quality Improvement: Statistical Process Control Available on CD

Time Series: Descriptive Analyses, Models, and Forecasting Available on CD

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Exam 15: Nonparametric Statistics Available on CD

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.

Calculate or use a table to find the binomial probability P(x P(x≥20) when n=25 and p=0.5. P ( x \geq 20 ) \text { when } n = 25 \text { and } p = 0.5 \text {. }P(x≥20) when n=25 and p=0.5. Also use the normal approximation to calculate the probability.

The Wilcoxon signed rank test for large samples can be used when n ≥ 10.

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Random Variables and Probability Distributions

Sampling Distributions

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single Sample: 355 Tests of Hypotheses

Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Design of Experiments and Analysis of Variance

Categorical Data Analysis

Simple Linear Regression

Multiple Regression and Model Building

Methods for Quality Improvement: Statistical Process Control Available on CD

Time Series: Descriptive Analyses, Models, and Forecasting Available on CD

Filters

Calculate or use a table to find the binomial probability P(x $P ( x \geq 20 ) \text { when } n = 25 \text { and } p = 0.5 \text {. }$ Also use the normal approximation to calculate the probability.