Exam 14: Nonparametric Statistics

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 Teacher 1 2 3 4 5 6 7 8 9 10 Days Absent Before Incentive 3 4 7 6 5 6 2 3 5 9 Days Absent After Incentive 1 3 7 4 4 4 0 4 3 9

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. Athlete 1 2 3 4 5 6 7 8 9 before 265 277 208 248 202 201 189 246 196 After 275 282 208 246 209 216 194 241 201

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

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

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

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: Customer Initial One Year 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 Find the rejection region for the Wilcoxon Signed Rank Test to test to determine if the distribution of initial weight is shifted to the right of the distribution of one-year weight. Test using $\alpha = 0.05$ . A) $T _ { - } \leq 14$ B) $T _ { - } \leq 10$ C) $T _ { - } \leq 7$ D) $T _ { - } \leq 17$

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

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. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\quad$ $\quad$$\text { Drink }$ Runner 1 2 3 4 1 1216 1217 1264 1235 2 1120 1026 1142 1150 3 1220 1348 1282 1294 4 1247 1208 1263 1258 5 1150 1112 1206 1211 6 1309 1286 1339 1335 7 1211 1252 1248 1234 8 1234 1125 1262 1239

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. Tennis Golf Swimming 3.3 1.9 2.8 2.7 2.2 3.1 2.6 3.4 2.9 3.6 2.0 2.6 3.2 2.4 2.6 2.2 2.1 2.5

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. City 1 City 2 City 3 City 4 2670 2300 2050 2920 2500 2060 2190 2870 2520 2210 2100 3120 2730 2170 2120 2670 2420 2220 2020 2520 2570 2270 2720 2170 2870 2770 14.6 Comparing Three or More Populations: Randomized Block Design 1 Understand the Friedman $F _ { r } \text {-test }$

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 }$ 1 2 3 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 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 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: Species 1 2 3 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 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 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. Subject 1 2 3 4 5 6 7 8 9 10 Before 172 228 207 270 204 178 248 194 239 245 After 157 223 215 260 199 178 218 176 237 230

if a difference exists in the population of median distances for each of the four brands of baseball bat. Batter Brand 1 Brand 2 Brand 3 Brand 4 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 Use the Friedman $\mathrm { F } _ { \mathrm { r } }$ -Test and find the rejection region used to determine if at least two of the distance distributions for the fotr baseball bats differ in location. Use $\alpha = 0.025$ . A) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { F } _ { \mathrm { r } } > 9.28$ B) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { F } _ { \mathrm { r } } > 29.46$ C) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { F } _ { \mathrm { r } } > 15.10$ D) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { F } _ { \mathrm { r } } > 15.44$ 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. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { Dosage }$ Patient A B C 1 55 57 64 2 59 59 59 3 57 58 58 4 56 61 62 5 55 54 56 6 60 58 62 7 56 58 61

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. Varnish Sample 1 2 3 1 5.4 5 5.7 2 8 7.2 7.8 3 4.2 4.3 4.2 4 7 6.1 6.7 5 9.2 7.6 8.6 6 6.4 5.6 6.5 7 4.5 4.5 4.9 8 6.9 7 7.2 9 5.7 4.9 5 10 6 5.8 5.9

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.

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: Customer Initial One Year 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 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?

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?

Specify the rejection region for the Wilcoxon rank sum test in the following situation. $n _ { 1 } = 6 , n _ { 2 } = 8 , \alpha = .10$ 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

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. Batter Brand 1 Brand 2 Brand 3 Brand 4 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 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.