Exam 13: Nonparametric Tests

A standard aptitude test is given to several randomly selected programmers, and the scores are given below for the mathematics and verbal portions of the test. Use the sign test to test the claim that programmers do better on the mathematics portion of the test. Use a 0.05 level of significance. Mathematics 347 440 327 456 427 349 377 398 425 Verbal 285 378 243 371 340 271 294 322 385

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

Suppose that a Kruskal-Wallis Test is to be performed and that there are three samples each of size six. What is the largest possible value of the test statistic H?

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 $t$ n $- 2$ degrees of freedom. Use this approximation to find critical values of $\mathrm { r } _ { \mathrm { S } }$ for the case where $\mathrm { n } = 11$ and $\alpha = 0.01$ .

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 $\mathrm { n } = 40$ and $\alpha = 0.10$ .

How does the Wilcoxon rank-sum test compare to the corresponding t-test in terms of efficiency, ease of calculations and assumptions required? Are there any kinds of data for which the Wilcoxon rank-sum test can be used but the t-test cannot be used?

Find the critical value. Assume that the test is two-tailed and that n denotes the number of pairs of data. - $\mathrm { n } = 20 , \alpha = 0.01$

Use the Wilcoxon signed-ranks test to test the claim that the matched pairs have differences that come from a population with a median equal to zero. Eleven 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

Find the critical value. Assume that the test is two-tailed and that n denotes the number of pairs of data. -Given that the rank correlation coefficient, rs, for 72 pairs of data is -0.770, test the claim of correlation between the two variables. Use a significance level of 0.05.

Give at least two examples of nonparametric tests and their comparable parametric tests.

Four different judges each rank the quality of 20 different singers. What method can be used for agreement among the four judges?

Fourteen people rated two brands of soda on a scale of 1 to 5. Brand A 2 3 2 4 3 1 2 Brand B 1 4 5 5 1 2 3 Brand A 5 4 2 1 1 4 3 Brand B 4 5 5 2 4 5 4 At the 5 percent level, test the null hypothesis that the two brands of soda are equally popular.

What is the corresponding parametric test for the Kruskal-Wallis test?

Use the Wilcoxon rank-sum test to test the claim that the two independent samples come from populations with equal medians. 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.

A person who commutes to work is choosing between two different routes. He tries the first route 11 times and the second route 12 times and records the time of each trip. The results (in minutes) are shown below. Use a significance level of 0.01 to test the claim that the times for both routes come from populations with the same median. Assume the routes were tested on days which were randomly selected.

Find the critical value. Assume that the test is two-tailed and that n denotes the number of pairs of data. - $\mathrm { n } = 50 , \alpha = 0.05$

When applying the runs test for randomness above and below the median for 12 dollar/Euro exchange rate highs, the test statistic is G = 2. What does that value tell us about the data?

Find the critical value. Assume that the test is two-tailed and that n denotes the number of pairs of data. -A college administrator collected information on first-semester night-school students. A random sample taken of 12 students yielded the following data on age and GPA during the first semester. Do the data provide sufficient evidence to conclude that the variables age, x, and GPA, y, are correlated? Apply a rank-correlation test. Use $\alpha = 0.05$

Provide the appropriate response. Describe the Wilcoxon signed-ranks test. What types of hypotheses is it used to test? What assumptions are made for this test?

The table below shows the weights (in pounds) of 6 randomly selected women in each of three different age groups. Use a 0.01 significance level to test the claim that the 3 age-groups have the same median weight. 18-34 35-55 56 and older 119 123 140 134 147 128 114 135 59 125 110 134 153 154 120 138 163 116