Exam 13: Nonparametric Tests

An instructor gives a test before and after a lesson and results from randomly selected students are given below. At the 0.05 level of significance, test the claim that the lesson has no effect on the grade. Use the sign test. Before 54 61 56 41 38 57 42 71 88 42 36 23 22 46 51 After 82 87 84 76 79 87 42 97 99 74 85 96 69 84 79

Critical values for the runs test for randomness can be calculated by listing all possible sequences. Using the elements $\mathrm { B } , \mathrm { B } , \mathrm { B } , \mathrm { R } , \mathrm { R } , \mathrm { R }$ list the 20 different possible sequences. Find the number of runs for each sequence. Are you able to find $5 \%$ cutoff values for G? What do you conclude?

If test A has an efficiency rating of 0.94 as compared to test B, explain what that efficiency rating means. Do comparable nonparametric or parametric tests have higher efficiency ratings?

Define rank. Explain how to find the rank for data which repeats (for example, the data set: 4, 5, 5, 5, 7, 8, 12, 12, 15, 18).

11 runners are timed at the 100-meter dash and are timed again one month later after following a new training program. The times (in seconds) are shown in the table. Use Wilcoxon's signed-ranks test and a significance level of 0.05 to test the claim that the training has no effect on the times. Before 12.1 12.4 11.7 11.5 11.0 11.8 12.3 10.8 12.6 12.7 10.7 After 11.9 12.4 11.8 11.4 11.2 11.5 12.0 10.9 12.0 12.2 11.1

SAT scores for students selected randomly from two different schools are shown below. Use a significance level of 0.05 to test the claim that the scores for the two schools are from populations with the same median.

Which of the following terms is sometimes used instead of "non-parametric test"?

Describe the sign test. What types of hypotheses is it used to test? What is the underlying concept?

The Mann-Whitney U test is equivalent to the Wilcoxon rank-sum test for independent samples in the sense that they both apply to the same situations and always lead to the same conclusions. In the Mann-Whitney U test we calculate $\mathrm { z } = \frac { \mathrm { U } - \frac { \mathrm { n } _ { 1 } \mathrm { n } _ { 2 } } { 2 } } { \sqrt { \frac { \mathrm { n } _ { 1 } \mathrm { n } _ { 2 } \left( \mathrm { n } _ { 1 } + \mathrm { n } _ { 2 } + 1 \right) } { 12 } } } , \text { where } \mathrm { U } = \mathrm { n } _ { 1 } \mathrm { n } _ { 2 } + \frac { \mathrm { n } _ { 1 } \left( \mathrm { n } _ { 1 } + 1 \right) } { 2 } - \mathrm { R }$ For the sample data below, use the Mann-Whitney U test to test the null hypothesis that the two independent samples come from populations with the same median. State the hypotheses, the value of the test statistic, the critical values, and your conclusion. Test scores (men): $\quad 70,96,77,90,81,45,55,68,74,99,88$ Test scores (women): $89,92,60,78,84,96,51,67,85,94$

Use the rank correlation coefficient to test for a correlation between the two variables. Use the sample data below to find the rank correlation coefficient and test the claim of correlation between math and verbal scores. Use a significance level of 0.05. Mathematics 347 440 327 456 427 349 377 398 425 Verbal 285 378 243 371 340 271 294 322 385

When applying the runs test for randomness above and below the median for 10 scores on a final exam, the test statistic is G = 2. What does that value tell us about the data?

When performing a rank correlation test, one alternative to using the Critical Values of Spearman's Rank Correlation Coefficient table to find critical values is to compute them using this approximation: $r _ { \mathrm { S } } = \pm \sqrt { \frac { t ^ { 2 } } { t ^ { 2 } + n - 2 } }$ where $t$ is the t-score from the $t$ Distribution table corresponding to $n - 2$ degrees of freedom. Use this approximation to find critical values of $\mathrm { r } _ { \mathrm { S } }$ for the case where $\mathrm { n } = 17$ and $\alpha = 0.05$ .

Use the rank correlation coefficient to test for a correlation between the two variables. Ten trucks were ranked according to their comfort levels and their prices. Make Comfort Price A 1 6 B 6 2 C 2 3 D 8 1 E 4 4 F 7 8 G 9 10 H 10 9 I 3 5 J 5 7 Find the rank correlation coefficient and test the claim of correlation between comfort and price. Use a significance level of 0.05.

Use the rank correlation coefficient to test for a correlation between the two variables. Given that the rank correlation coefficient, $r _ { \mathrm { S } }$ for 37 pairs of data is 0.324, test the claim of correlation between the two variables. Use a significance level of 0.01.

The waiting times (in minutes) of 28 randomly selected customers in a bank are given below. Use a significance level of 0.05 to test the claim that the population median is equal to 5.3 minutes. 8.2 8.0 10.5 3.8 6.4 5.3 7.8 2.9 6.0 7.7 6.1 5.9 1.2 10.4 7.3 6.9 5.8 5.1 6.2 3.1 5.8 11.7 4.5 6.5 9.8 7.4 2.3 7.8

A pollster interviews voters and claims that her selection process is random. Listed below is the sequence of voters identified according to gender. At the 0.05 level of significance, test her claim that the sequence is random according to the criterion of gender. ,,,,,,,,,,,,,,, ,,,,,,,,,,,,,,,

Use the Wilcoxon rank-sum approach to test the claim that the sample student grade averages at two colleges come from populations with the same median. The sample data is listed below. Use a 0.05 level of significance, and assume that the samples were randomly selected. College A 3.2 4.0 2.4 2.6 2.0 1.8 1.3 0.0 0.5 1.4 2.9 College B 2.4 1.9 0.3 0.8 2.8 3.0 3.1 3.1 3.1 3.5 3.5

In a study of the effectiveness of physical exercise in weight reduction, 12 subjects followed a program of physical exercise for two months. Their weights (in pounds) before and after this program are shown in the Table. Use Wilcoxon's signed-ranks test and a significance level of 0.05 to test the claim that the exercise Program has no effect on weight. Before 162 190 188 152 148 127 195 164 175 156 180 136 After 157 194 179 149 135 130 183 168 168 148 170 138 What would be the signed rank for the column with values of 175 and 168?

A rank correlation coefficient is to be calculated for a collection of paired data. The values lie between -10 and 10. Which of the following could affect the value of the rank correlation coefficient? I: Multiplying every value of one variable by 3 II: Interchanging the two variables III: Adding 2 to each value of one variable IV: Replacing every value of one variable by its absolute value

When performing a rank correlation test, one alternative to using the Critical Values of Spearman's Rank Correlation Coefficient table to find critical values is to compute them using this approximation: $r _ { S } = \pm \sqrt { \frac { t ^ { 2 } } { t ^ { 2 } + n - 2 } }$ where $t$ is the t-score from the $t$ Distribution table corresponding to $n - 2$ degrees of freedom. Use this approximation to find critical values of $\mathrm { r } _ { \mathrm { S } }$ for the case where $\mathrm { n } = 7$ and $\alpha = 0.05$ .