Exam 13: Nonparametric Statistics

Two independent samples of tall and short students are selected, and their scores on a 25 point statistics quiz are recorded as shown in the table below. What is the rank of Wilt's s in most exams, higher values rank ahead of lower values. Tall 19 21 16 18 14 17 (Wilt) 12 21 14 16 Short 14 17 21 19 16 21 18 17 19 18

A professional golfer wanted to determine if there was a difference in the number of yards that golfers could hit a ball depending on the brand of driver (a particular type of club) used. Twenty-five golfers (of approximately the same ability) were divided into five groups , and each golfer in a group hit a ball with one particular brand of driver. The number of yards they hit the ball was recorded, and a rank was assigned to each value. The data are tabulated below with the number of yards hit listed first and the ranks listed in parentheses. Brand A Brand B Brand C Brand D Brand E 300(25) 190(5) 235(11) 275(17) 165(2) 298(23) 164(1) 296(21) 299(24) 175(3) 297(22) 239(13) 274(16) 185(4) 256(14) 260(15) 215(7) 280(18) 238(12) 224(9) 295(20) 225(10) 290(19) 196(6) 219(8) Determine the critical value for a Kruskal-Wallis test, then compare it to the test value anı decide if the null hypothesis should be rejected or not.

The Wilcoxon rank sum test is used to test dependent samples and the Wilcoxon signed-rank test is used for independent samples.

A researcher tests three different brands of $2 \%$ milk to see how many grams of fat are in e 8 ounce glass. The results are shown below along with the sums of the ranks of the differ brands. Brand A Brand B Brand C 4.68 4.64 4.70 4.66 4.67 4.68 4.64 4.68 4.71 4.71 4.66 4.69 4.67 4.69 4.70 $\boldsymbol { R } _ { \mathbf { A } } = 33 ; \boldsymbol { R } _ { \mathbf { B } } = 29 ; \boldsymbol { R } _ { \mathbf { C } } = 58$ What is the test value for the Kruskal-Wallis test?

It is believed that children can throw a ball further after lunch than before lunch. A resear، decided to investigate this hypothesis using the paired-sample sign test. Each of 14 childı a ball as far as they could before and after lunch. The distances in meters are tabulated be Subject Before Lunch 23 29 28 28 29 40 19 After Lunch 26 35 34 25 35 37 22 Subject Before Lunch 23 28 56 19 40 28 28 After Lunch 26 34 65 22 37 25 34 $\text { Test the researcher's hypothesis at } \alpha=0.05 \text {. }$ You may use the following abbreviated table of critical values for the sign test (one tailed $\alpha=0.05$ ), taken from Table $\mathrm{J}$ in the text: 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 critical value 2 2 3 3 3 4 4 5 5 5 6 6 7 7 7 her

The following data was collected as part of a study examining whether there is a differenc between the number of hours men and women watch television. The values represent the of hours a subject watched television on a designated Monday night. Lower values rank $\varepsilon$ higher ones. Men 2.0 1.5 3.0 2.5 2.0 1.0 0.0 2.0 1.5 2.5 2.0 2.0 Women 2.0 2.5 1.0 1.0 1.5 2.5 2.0 1.0 2.0 1.5 1.0 0.0 Calculate $\mu _ { R } , \sigma _ { R }$ and the $z$ test value for the Wilcoxon rank sum test.

When performing a sign test, one needs to consider whether or not data values are greater or less than a hypothesized median, as well as the magnitude of the differences between the values and the hypothesized median..

Find the critical value from Table $\mathrm { L }$ for the rank correlation coefficient, given sample $\operatorname { size } n = 9$ and $\alpha = 0.05$ .

A researcher collects data on the distance between college students' high schools and their colleges. He believes that the median distance is 200 miles. A random sample of 15 students yields the following number of miles between their high schools and colleges: 25 25 30 30 45 45 55 75 85 110 135 175 195 810 2170 Test the researcher's hypothesis at $\alpha = 0.05$ . You may use the following abbreviated table of critical values for the sign test (two-tailed $\alpha = 0.05$ ), taken from Table $J$ in the text: critical value 1 2 2 3 3 3 4 critical value 4 4 5 5 5 6 6 6

A landscaping company hypothesizes that the median number of lawns they mow in a we 20 . They tabulated the number of lawns mowed in a random sample of 14 weekends. 22 18 19 20 25 20 18 22 23 20 21 26 24 22 Use the test value, the critical value, and the appropriate table to decide if the landscaping company's hypothesis that they mow a median of 20 lawns per weekend valid at $\alpha$ . d is

Test the following sequence for randomness at $\alpha = 0.05$ . A A A A B B B A A A B B Table of Critical Values for the Number of Runs, $\alpha = 0.05$ Value of Value of 2 3 3 3 9 10 11 12 2 3 3 3 10 11 12 13 3 3 3 4 10 11 12 13

Six second grade students tried tossing a ball into a basket ten times each. Their teacher tl suggested a different way of tossing the ball, and the six students tried again. The numbel successful tosses, before and after the teacher's suggestion, are shown below. Student Before the suggestion (out of 10) 5 3 5 2 5 3 After the suggestion (out of 10) 3 3 3 3 3 3 Find the negative rank sum. [Recall, differences of zero are not ranked.]

A researcher believes that the median annual rainfall in the valley is about $32.0$ inches. A randomly selected sample of 20 years had the following numbers of inches of rainfall in tl valley: 15.2,16.8,22.5,25.1,27.1,27.3,31.0,37.2,37.7,39.0,41.1,42.4,45.7,49.0,45 ,50.9,53.1,55.3,57.2,61.8 Test the researcher's hypothesis at $\alpha = 0.05$ . You may use the following abbreviated table of critical values for the sign test (two tailed $\alpha = 0.05$ ), taken from Table $J$ in the text: 11 12 13 14 15 16 17 18 critical value 1 2 2 3 3 3 4 4 critical value 4 5 5 5 6 6 6

If a panel of two judges were asked to rank 10 science fair projects, and the sum of the squares of the differences in the ranks was 49.5, then the Spearman rank correlation coefficient would be 0.70.

A social studies teacher wanted to see whether the students exit the classroom randomly by gender. He observed the gender of students leaving the classroom in the order given below. Test for randomness at $\alpha = 0.05$ . M M M M M MFFFFFFFFFM MFFF

A researcher needs to conduct a suitable nonparametric test to compare two dependent variables. Which of the following should he use?

Six second grade students tried tossing a ball into a basket ten times each. Their teacher tl suggested a different way of tossing the ball, and the six students tried again. The number successful tosses, before and after the teacher's suggestion, are shown below. Student Before the suggestion (out of 10) 3 5 5 3 5 5 After the suggestion (out of 10) 5 5 2 5 2 5 In the ranking of the absolute values of the differences, child C's rank is_________ differences of zero are not ranked.]

Eight subjects were weighed before and after a new three-week "healthy" diet. At the $0.05$ level of significance, can it be concluded that a difference in weight resul (Weights are in pounds.) Subject A B C D E F G H Before 148 185 176 170 194 155 167 184 After 151 189 169 171 189 153 168 177

Nonparametric statistics can be applied to populations that are not normally distributed.

Given the following data set, what is the ranking of the data value 23? Presume that lower data values rank ahead of higher data values. 26, 26, 16, 22, 33, 28, 11, 23