Exam 14: Nonparametric Statistics

A government agency claims that the median hourly wages for workers at fast food restaurants in the western U.S. is $8.20. In a random sample of 100 workers, 68 were paid less than $8.20, 10 were paid $8.20, and the rest more than $8.20. Test the government's claim. Use α = .05.

What are rank statistics (or rank tests)?

For a valid signed rank test, the probability distribution from which the sample of paired differences is drawn must be continuous.

A consumer protection organization claims that a new car model gets less than 29 miles per gallon of gas. Ten cars are tested. The results are given below. Test the organization's claim. Use α = .05. 23.8 21.6 27.8 22.9 26 28.2 31.3 25.9 20.7 27

A researcher wishes to determine whether there is a difference in the average age of elementary school, high school, and community college teachers. Teachers are randomly selected. Their ages are recorded 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. Elementary School Teachers High School Teachers Community College Teachers 8 47 50 39 52 56 38 49 47 63 58 72 48 53 56 36 42 46

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.

Verbal SAT scores for students randomly selected from two different schools are listed below. Use the Wilcoxon rank sum procedure to test the claim that there is no difference in the scores from each school. Use α = .05. School 1 School 2 530 500 750 470 420 660 460 730 510 410 690 570 560 760 590 670 530 510 570 710 730 610 620 520

For the Wilcoxon rank sum test to be valid, the number of ties in the measurements needs to be small relative to the sample sizes.

At six different randomly chosen times, a researcher measures the temperature (in degrees Fahrenheit) of a pint of milk from a supermarket's shelf. The measurements are shown below. 36 38 39 39 41 43 a. Suppose the researcher is interested in determining whether the median temperature exceeds $40 ^ { \circ } \mathrm { F }$ . Set up the null and alternative hypotheses of interest. b. How many of the measurements exceed $40 ^ { \circ } \mathrm { F }$ ? c. Assuming that $p = .5$ , find the binomial probability that at least two measurements exceed $40 ^ { \circ } \mathrm { F }$ . d. What do you conclude about $H _ { 0 }$ and $H _ { a }$ ?

The temperatures on randomly chosen days during a summer class and the number of absences from class on those days are listed below. Can you conclude that there is a correlation between the temperature and the number absent? Use α = .01. Temp 65 78 84 83 81 91 68 93 73 Absences 2 6 9 9 7 14 3 14 4

In a study of the effectiveness of physical exercise on weight loss, 20 people were randomly selected to participate in a program for 30 days. Use the Wilcoxon signed rank test to test the claim that exercise has no bearing on weight loss. Use α = .02. Weight Before Program (in Pounds) 178 210 156 188 193 225 190 165 168 200 Weight After Program (in Pounds) 182 205 156 190 183 220 195 155 165 200 Weight Before Program (in Pounds) 186 172 166 184 225 145 208 214 148 174 Weight After Program (in Pounds) 180 173 165 186 240 138 203 203 142 170

The Wilcoxon signed rank test for large samples can be used when $n \geq 10$ .

Online classes are becoming more and more prevalent at the college level. A statistics instructor randomly sampled ten students from his traditional face-to-face class and ten students from his online class to compare their comprehension of the material that was taught in the class. He administered the same final exam to each student and wants to use the Wilcoxon Rank Sum test to compare their exam scores. The results are shown below: Traditional Class 82917568 93857470 5682 Online Class 67667273 77764881 8692 What alternative hypothesis should the instructor test to show that the students in the traditional class outperformed the students in the online class?

The Wilcoxon rank sum test is used to test the hypothesis that the probability distributions associated with two populations are equivalent.

What is referred to as nonparametrics?

For a valid Friedman Fr-test, the probability distributions from which the samples within each block are drawn must be continuous.

The table below lists the verbal and math SAT scores of 10 students selected at random. Test the hypothesis of no correlation between verbal and math SAT scores. Use α = .05. Verbal 535 620 625 530 610 Math 620 690 715 650 700 Verbal 640 540 590 660 550 Math 665 750 670 540 550