Exam 14: Nonparametric Statistics

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 Calculate the test statistic 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. A) $T _ { - } = 19$ B) $T _ { + } = 59$ C) $\mathrm { T } _ { + } = 78$ D) $T _ { + } = 26$

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 to calculate the test statistic used to determine if at least two of the distance distributions for the four baseball bats differ in location.

Six patients were each given four different pain killers and asked to rate each pain killer's effectiveness in reducing pain on a scale of 1 to 10. A Friedman $F _ { r }$ -test was performed on the results. A printout is shown below. Friedman Test: Response versus Treatment blocked by Patient $\mathrm { S } = 18.00 \mathrm { DF } = 3 \mathrm { P } = 0.000$ Est Sum of Treatment Median Ranks 1 6 7.500 24.0 2 6 4.500 12.0 3 6 6.000 18.0 4 6 2.500 6.0 Grand median $= 5.125$ 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

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

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

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 What type of nonparametric analysis should be used?

Eight SmartCars were randomly selected and the highway mileage of each was noted. The highway mileage per gallon for each car is shown below: 32 42 29 34 41 37 38 39 40 It was desired to determine if the median miles per gallon of all SmartCars exceeded 35 miles per gallon. Find A) $\mathrm { p } = 0.145$ B) $0.035$ C) $\mathrm { p } = 0.965$ D) $\mathrm { p } = 0.855$

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

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.

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. Group 1 Group 2 Group 3 15 12 10 16 9 16 13 6 8 19 7 12 17 8 13 12 4 8

Calculate or use a table to find the binomial probability $P ( x \geq 6 ) \text { when } n = 9 \text { and } p = .5$ .

The ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults are given below. Can you conclude that there is a correlation between age and blood pressure? Use α = .05. Age 38 41 45 48 51 53 57 61 65 Pressure 116 120 123 131 142 145 148 150 152

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. \ 2200 \ 2150 \ 1600 \ 1775 \ 1635 \ 2650 \ 2075 \ 1570 \ 2290 \ 2900 \ 1895 \ 1900 \ 1975 \ 1900 \ 1680

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

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

Consider the data set shown below. Find the Spearman's Rank Correlation for the variables x and y. 0 4 2 3 8 10 11 -2 0 2 4 6 8 10 A) $r _ { \mathrm { S } } = 0.8929$ B) $r _ { \mathrm { S } } = - 0.8929$ C) $r _ { \mathrm { s } } = 0.7972$ D) $\mathrm { r } _ { \mathrm { s } } = 1.000$

The final exam scores of 10 randomly selected statistics students and the number of hours they studied for the exam are given below. Can you conclude that there is a correlation between the scores on the test and the times spent studying? Use α = .01. Hours 7 9 6 12 6 8 8 9 10 7 Scores 69 84 64 92 70 82 89 94 94 75

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?

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

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. Fertilizer 1 Fertilizer 2 Fertilizer 3 Fertilizer 4 23 22 23 20 22 25 20 22 23 22 21 20 24 22 20 21 24 22 19 22 23 23 20 21