  ## Exercises - Nonparametric Models

1. An American karate studio plans to advertise but is unsure as to which of three ads to use. The ads are tested on randomly selected consumers and the reactions measured on an ordinal scale that produces the following data: $$\begin{array}{l|l} \textrm{Red} & 80, 80, 78, 81, 72, 85, 96, 84, 71, 75, 98\\\hline \textrm{White} & 75, 55, 98, 92, 86, 78, 87, 79, 88, 87, 85, 94, 99\\\hline \textrm{Blue} & 72, 76, 70, 77, 68, 82, 85, 81, 65, 69\\ \end{array}$$ Test the claim that reactions are the same for the three different ads. Perform the Kruskal Wallis Test. If appropriate, follow with Wilcoxon tests and interpret your results.

$H_0 :$ All three ads are liked equally by consumers.
Red ranks: 16.5 16.5 13.5 18.5  7.5 23.0 31.0 21.0  6.0  9.5 32.5
(sum = 195.5)

White ranks: 9.5  1.0 32.5 29.0 25.0 13.5 26.5 15.0 28.0 26.5 23.0 30.0 34.0
(sum = 293.5)

Blue ranks: 7.5 11.0  5.0 12.0  3.0 20.0 23.0 18.5  2.0  4.0
(sum = 106)

Test statistic is $H = 8.188$, critical value is $7.378$ for $2$ degrees of freedom at $\alpha = 0.025$. Ads are not equally liked by consumers. The Wilcoxon tests produced the following test statistics: $z = -1.79$ for comparison between blue and red; $z = -1.30$ for comparison between red and white; $z = 2.69$ for comparison between blue and white. There is no difference between blue and red or between red and white. There is a significant difference between blue and white, with a $p$-value of $0.0072$ or critical value of $2.33$ for $\alpha = 0.01$. White probably should be given the contract.

2. A test for independence of judgment was administered to three different groups of students. The higher the score, the more independent the person is in judgment (the person is not easily influenced by another person's opinions). Group A includes premedical students ($n=11$), Group B includes English majors ($n=10$), and Group C includes psychology majors ($n=11$). $$\begin{array}{l|l} \textrm{A} & 69, 78, 90, 92, 68, 77, 85, 96, 87, 71, 93\\\hline \textrm{B} & 44, 36, 41, 90, 51, 42, 38, 52, 87, 46\\\hline \textrm{C} & 36, 97, 84, 72, 75, 64, 62, 79, 63, 66, 58 \end{array}$$ The scores represent ordinal data. Are there any significant differences among the groups? Specifically: (a) Perform a Kruskal Wallis test. (b) If the Kruskal Wallis test is significant, follow with the appropriate Wilcoxon tests. (remember to re-rank your groups). (c) Interpret the results from (a) and (b). Are there any significant differences among the groups? Which group is more independent, if any?

(a) $H_0$ : There is no difference in independence of judgment across student types. Finding ranks, we have:

A: 16.0 21.0 27.5 29.0 15.0 20.0 24.0 31.0 25.5 17.0 30.0
B: 6.0  1.5  4.0 27.5  8.0  5.0  3.0  9.0 25.5  7.0
C: 1.5 32.0 23.0 18.0 19.0 13.0 11.0 22.0 12.0 14.0 10.0

$H = 11.1$, reject the null hypothesis at $\alpha = 0.005$ since the chi square value for $2$ degrees freedom is $7.879$ for $\alpha = 0.005$. There is at least one significant difference. (FYI: using low value to high value for ranking: $R_A = 256, R_B = 96.5, R_C = 175.5$).

(b) The null hypothesis is that there is no difference between groups A and B (or A and C, or B and C, respectively). Between groups A and B, $z = -2.89$ with $p$-value of $0.0038$, reject the null. A is more independent than B. Between groups A and C, $z = -2.20$ with a $p$-value of $0.0278$, reject the null. A is more independent than C. Between groups B and C, $z = 1.94$ with a $p$-value of $0.0524$, fail to reject the null. There is no significant difference between B and C.

(c) A is significantly more independent than B or C. There is no significant difference between B and C. The premedical students seem to be more independent than English or Psychology majors.

3. A movie producer wishes to test the audience response to three different possible endings for a movie. Three audiences were randomly selected and each was shown the movie with one of the possible endings. Each audience member was asked to rate the movie on a scale of 1 to 100 (with 100 representing the best rating). Assume this is ordinal data. Using the information provided, test whether there is a significant difference in the audience response to the endings. If there is a difference, determine where that difference lies. Show your hypothesis testing steps clearly. $$\begin{array}{|c|c|c|} \textrm{Ending A} & \textrm{Ending B} & \textrm{Ending C}\\\hline 35 & 42 & 12\\\hline 40 & 50 & 20\\\hline 54 & 52 & 28\\\hline 60 & 55 & 35\\\hline 64 & 57 & 40\\\hline 67 & 60 & 45\\\hline 70 & 60 & 50\\\hline 72 & 62 & 51\\\hline 75 & 65 & 53\\\hline 78 & 70 & 64\\\hline 80 & 73 & 70\\\hline 84 & 77 & 75\\\hline \end{array}$$

Kruskal Wallis test produces $H=7.786$ with critical value of $5.991$ ($\alpha = 0.05$ and $2$ degrees of freedom). Reject the null hypothesis, the populations have the same distributions. Therefore, there is a significant difference in the audience response to the endings. Three Wilcoxon Rank-Sum tests give the following test statistics (remember to re-rank for each Wilcoxon Test): 1.18 (between A and B), 2.45 (between A and C), and 2.08 (between B and C). Critical value is $\pm1.96$ for $\alpha = 0.05$. There is a significant difference between A and C, and a significant difference between B and C. Based on the study, it seems that the producer should not choose ending C.