Exam 21: Nonparametric Techniques

Use the 5% significance level to test the hypotheses $H _ { 0 } :$ The two population locations are the same $H _ { 1 } :$ The location of population A is to the right of the location of population B, given that the data below are drawn from two independent samples: Sample A: 9 11 9 10 12 8 5 Sample B: 8 7 5 7 9 5 6 8

The sign test and the Wilcoxon signed rank sum test require matched pairs.

The following statistics are drawn from two independent samples: $T _ { A } = 800$ , $n _ { A } = 25$ , $T _ { B } = 1100$ , $n _ { B } = 28$ . Test at the 5% significance level to determine whether the two population locations differ.

In a normal approximation to the sign test, the standardised test statistic is calculated as z = 2.07. If the alternative hypothesis states that the location of population 1 is to the right of the location of population 2, then the p-value of the test is 0.0192.

The Kruskal-Wallis test can be used to test for a difference between two populations. It will produce the same outcome as the two-tailed Wilcoxon rank sum test.

A Wilcoxon rank sum test for comparing two independent samples involves two samples of sizes 6 and 9. The alternative hypothesis is that the location of population 1 is to the left of the location of population 2. Using the 0.05 significance level, the appropriate critical values are 31 and 65.

In a Wilcoxon signed rank sum test for matched pairs with n = 35, the rank sums of the positive and negative differences are 380 and 225, respectively. The value of the standardised test statistic z is:

Because of the cost of producing television shows and the profits associated with successful shows, television network executives are keenly interested in public opinion. A network has recently developed three comedy series. The pilot of each series is shown to 10 randomly selected people, who evaluate each show on a 9-point scale where 1 = terrible and 9 = excellent. The results are shown below. $\quad$ $\quad$ $\quad$$\text { Assessments of Television Shows }$ Person Show 1 Show 2 Show 3 1 6 4 7 2 5 4 5 3 7 6 8 4 7 7 8 5 9 9 9 6 5 7 6 7 4 4 5 8 6 3 7 9 5 6 6 10 7 7 8 Use Excel to find the exact p-value for this test.

The Wilcoxon signed rank sum test is applied to compare two populations when the samples are matched pairs and the data are interval but not normally distributed.

The restaurant critic on a newspaper claims that the hamburgers one gets at the hamburger chain restaurants are all equally bad, and that people who claim to like one hamburger over others are victims of advertising. In fact, he claims that if there were no differences in appearance, then all hamburgers would be rated equally. To test the critic's assertion, ten teenagers are asked to taste hamburgers from three different fast-food chains. Each hamburger is dressed in the same way (mustard, relish, tomato and pickle) with the same type of bun. The teenagers taste each hamburger and rate it on a 9-point scale with 1 = bad and 9 = excellent. The data are listed below. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { Ham burger Ratings }$ Teenager Chain 1 Chain 2 Chain 3 1 7 5 6 2 5 3 4 3 6 4 5 4 9 8 8 5 4 3 2 6 4 5 4 7 6 5 5 8 5 4 5 9 8 7 9 10 9 8 7 Use Excel to find the exact p-value for this test.

A non-parametric method to compare two populations, when the samples consist of matched pairs of observations and the data are either ordinal or interval, and where the normality requirement necessary to perform the parametric test is not satisfied, is the:

A movie critic wanted to determine whether or not moviegoers of different age groups evaluated a movie differently. With this objective, he commissioned a survey that asked people their ratings of their most recently watched movies. The rating categories were: 1 = terrible. 2 = fair. 3 = good. 4 = excellent. Each respondent was also asked to categorise his or her age as either: 1 = teenager. 2 = young adult (20-34). 3 = middle age (35-50). 4 = senior (over 50). The results are shown below. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { Movie Ratings }$ Teenager Young Adult Middle Age Senior 3 2 3 3 4 3 2 4 3 3 1 4 3 2 2 3 3 2 2 3 4 1 3 4 2 3 1 4 4 2 4 3 Using the appropriate statistical table, what statement can be made about the p-value for the test?

Each year the personnel department in a large corporation assesses the performance of all of its employees. Each employee is rated for various aspects of his or her job on a 7-point scale where 1 = very unsatisfactory and 7 = satisfactory. The president of the company believes that the assessment scores this year are lower than last year's. To examine the validity of this belief, she draws a random sample of six employees' scores from last year and another six employees' scores this year. Do the data listed below allow the president to conclude at the 5% significance level that her belief is correct? Employees Ratings Scores This Year Last Year 5 5 6 5 4 3 5 3 5 4 4 3

Given the following statistics from a matched pairs experiment, perform the Wilcoxon signed rank sum test to determine whether we can infer at the 10% significance level that the two population locations differ. $T ^ { + } = 520$ $T ^ { - } = 700$ , n = 50

A movie critic wanted to determine whether or not moviegoers of different age groups evaluated a movie differently. With this objective, he commissioned a survey that asked people their ratings of their most recently watched movies. The rating categories were: 1 = terrible. 2 = fair. 3 = good. 4 = excellent. Each respondent was also asked to categorise his or her age as either: 1 = teenager. 2 = young adult (20-34). 3 = middle age (35-50). 4 = senior (over 50). The results are shown below. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { Movie Ratings }$ Teenager Young Adult Middle Age Senior 3 2 3 3 4 3 2 4 3 3 1 4 3 2 2 3 3 2 2 3 4 1 3 4 2 3 1 4 4 2 4 3 Do these data provide sufficient evidence to infer at the 5% significance level that there were differences in ratings among the different age categories?

The Wilcoxon rank sum test (like most of the non-parametric tests presented in your textbook) actually tests to determine whether the population distributions have identical:

The Friedman test is the non-parametric counterpart of the randomised block experimental design of the analysis of variance.

The marketing manager of a pizza chain is in the process of examining some of the demographic characteristics of her customers. In particular, she would like to investigate the belief that the ages of the customers of pizza parlours, hamburger emporiums and fast-food chicken restaurants are different. As an experiment, the ages of eight customers of each of the restaurants are recorded and listed below. From previous analysis we know that the ages are not normally distributed. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { Customer Age }$ Pizza Hamburger Chicken 23 26 25 19 20 28 25 18 36 17 35 23 36 33 39 25 25 27 28 19 38 31 17 31 Use Excel to find the exact p-value for this test of whether there are differences in age among the customers of the three restaurants.

In a Wilcoxon rank sum test for independent samples, the two sample sizes are 4 and 6, and the value of the Wilcoxon test statistic is T = 25. If the test is two-tailed and the level of significance is 0.05, then the null hypothesis will be rejected.

In a Friedman test for comparing four populations, provided that there are eight blocks, the test statistic is calculated as F = 10.98. If the test is conducted at the 5% significance level, the conclusion and p-value will be: