Exam 15: Nonparametric Statistics Available on CD

Six patients were each given four different pain killers and asked to rate each pain killer's effectiveness in reducing pain on a scale of 1 to 10. A Friedman $\mathrm { F } _ { \mathrm { r } ^ { - } }$ test was performed on the results. A printout is shown below. Friedman Test: Response versus Treatment blocked by Patient $\mathrm { S } = 18.00 \mathrm { DF } = 3 \mathrm { P } = 0.000$ Est Sum of Treatment N Median Ranks 1 6 7.500 24.0 2 6 4.500 12.0 3 6 6.000 18.0 4 6 2.500 6.0 Grand median $= 5.125$ Is there evidence that at least two of the treatment probability distributions differ in location? Explain.

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

A researcher wants to know if the time spent in prison for a particular type of crime is the same for men and women. A random sample of men and women were each asked to give the length of sentence received. The data, in months, are listed below. Use the Wilcoxon rank sum procedure to test the claim that there is no difference in the sentences received by men and the sentences received by women. Use ? = 0.05. Men 34 46 40 42 43 50 Women 33 36 33 38 50 36 Men 38 46 36 43 47 48 Women 58 32 34 37 41 51

What are distribution-free tests?

A real estate agent believes that the median rent for a one-bedroom apartment in a beach community in southern California is greater than $\$ 1200$ per month. The rents for a random sample of 15 one-bedroom apartments are listed below. Test the agent's claim. Use $\alpha =$ $0.01$ . \ 1500 \ 1450 \ 900 \ 1075 \ 935 \ 1950 \ 1375 \ 870 \ 1590 \ 2200 \ 1195 \ 1200 \ 1275 \ 1200 \ 980

The median household income of a community is reported to be $62,000. A random sample of six households in the community yielded the following incomes. $\$ 42,000 \quad \$ 57,000 \quad \$ 65,000 \quad \$ 69,000 \quad \$ 75,000 \quad \$ 150,000$ Does the sample provide sufficient evidence to refute the reported median household income? Perform a sign test using $\alpha = 0.10$ .

Calculate or use a table to find the binomial probability $P ( x \geq 6 )$ when $n = 9$ and $\mathrm { p } = 0.5$ .

The table below lists the verbal and math SAT scores of 10 students selected at random. Calculate Spearman's rank correlation coefficient. Test the hypothesis of no correlation between verbal and math SAT scores. Be sure to specify the null and alternative hypotheses. Use $\alpha = 0.05 .$ Verbal 380 465 470 375 455 Math 465 535 560 495 545 Verbal 485 385 435 505 395 Math 510 595 515 385 395

Nine students took the SAT test. Later, they took a test preparation course and retook the SAT. Their original scores and new scores are shown below. Use the Wilcoxon signed rank test to test the claim that the test preparation had an effect on their scores. $\text { Use } \alpha = 0.05 \text {. }$ Student 1 2 3 4 5 6 7 8 9 Before Score 930 810 900 870 1120 910 940 980 1080 After Score 950 810 890 910 1150 920 930 1020 1100