Exam 14: Nonparametric Statistics Available Online

When there is a perfect negative rank correlation between two variables, the value of the test $\text { statistic } r _ { S } \text { in Spearman's test is } - 1 \text {. }$

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

What are distribution-free tests?

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.

What are rank statistics (or rank tests)?

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

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 $\alpha = .10$ . Dosage Patient 1 44 46 53 2 48 48 48 3 46 47 47 4 45 50 51 5 44 43 45 6 49 47 51 7 45 47 50

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 $\alpha = .05$ . Group 1 Group 2 Group 3 13 10 8 14 7 14 11 4 6 17 5 10 15 6 11 10 2 6

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: 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 rejection region that should be used to conduct the Kruskal-Wallis H-Test at $\alpha = 0.05$ .

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

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

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.

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

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.

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.

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 $\alpha = 0.05$ .

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?

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

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 $\alpha = .05$ . Subject 1 2 3 4 5 6 7 8 9 10 Before 217 249 226 257 183 187 259 196 199 182 After 202 244 234 247 178 187 229 178 197 167