Exam 15: B:nonparametric Statistics

In testing the hypotheses, The two population locations are the same vs. The two population locations are different, the statistics , , and are calculated with data drawn from a matched pairs experiment. a. Which test is used for testing the hypotheses above? b. What is the value of the test statistic? c. What is the p-value of this test? d. Can we infer at the 5% significance level that the population locations differ?

Typing Speed Narrative Ten secretaries were selected at random from among the secretaries of a large university. The typing speed (number of words per minute) was recorded for each secretary on two different brands of computer keyboards. Assume that the typing speeds are not normally distributed. The following results were obtained. -Refer to Typing Speed Narrative. Which test is appropriate for this situation?

The following paired observations were obtained on two variables x and y: a. Calculate Spearman's rank correlation coefficient b. Do the data present sufficient evidence to indicate a correlation between x and y? Justify your response. Test using = 0.05.

Movie Ratings Narrative 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 the most recently watched movies. The rating categories were 1 = terrible, 2 = fair, 3 = good, and 4 = excellent. Each respondent was also asked to categorize his or her age as either 1 = teenager, 2 = young adult (20-34), 3 = middle age (35-50), and 4 = senior (over 50). The results are shown below. -Refer to Movie Ratings Narrative. Which test can the movie critic use in this situation?

In testing the hypotheses The two population locations are the same vs The location of population A is to the right of the location of population B, the statistics , , and are calculated with data drawn from a matched pairs experiment. a. Which test is used in testing the hypotheses above? b. What is the value of the test statistic? c. What is the p-value of this test? d. Can we infer at the 1% significance level that the location of population A is to the right of the location of population B? Explain.

Attitude Test Narrative Twenty students are given an attitude test before and after viewing a motion picture designed to change their attitudes favourably toward a new curriculum. A high score indicates a favourable attitude and a low score indicates an unfavourable attitude, with the scores ranging from 1 to 30. This will use the sign test on the data given below to see if we can conclude the motion picture was successful in improving attitudes. -Refer to Attitude Test Narrative. State the null and alternative hypotheses if the Wilcoxon signed-rank test is used on the data to see if we can conclude that the motion picture was successful in changing attitudes.

Standardized College Entrance Exam Narrative Two psychometricians (educators who are experts in the field of psychological test design) were asked to rank six designs for a new standardized college entrance exam. Design This uses Spearman's rank correlation coefficient to see if there is a positive relationship between the educators' rankings. -Refer to Standardized College Entrance Exam Narrative. Do the data present sufficient evidence to indicate a positive correlation in the rankings of the two educators? What does this result mean in the context of the ?

Three treatments were compared using a completely randomized design. The data are shown in the table. Do the data provide sufficient evidence to indicate a difference in location for at least two of the population distributions? Test using the Kruskal-Wallis H statistic with = 0.05.

Customers' Ages Narrative 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. Customers' Ages -Refer to Customers' Ages Narrative. Explain how to use the p-value for testing the hypotheses.

Reaction Times Narrative The reaction times to three stimuli were recorded for each of eight subjects. The data, recorded in seconds, are shown below. This uses Friedman's F_r test to determine if there is a difference among the population distributions of reaction times. -Refer to Reaction Times Narrative. State the null and alternative hypotheses.

In testing the hypotheses, The two population locations are the same, and The location of population 1 is to the left of the location of population 2, the statistics , , , and are calculated with data drawn from two independent samples. a. Which test is used for testing the hypotheses above? b. What is the value of the test statistic? c. What is the rejection region for this test at = 0.05? d. What is your conclusion at = 0.05?

A toy store manager was interested in determining whether the assembly time is the same for three models of baby strollers. The manager selected five employees at random and asked each of them to assemble each of the strollers. The assembly time, in minutes, was recorded as follows: The manager wasn't sure whether the assumptions for the usual analysis of variance were valid, so she decided to use a nonparametric procedure. Use the appropriate method to determine whether the assembly time is the same for the three models of baby strollers. Use = 0.05.

Heart Rates and Exercise Narrative An experiment was conducted to examine the effect of age on heart rate when a person is subjected to a specific amount of exercise. Ten men were randomly selected from each of four age groups: 10-19, 20-39, 40-59, and 60-69. Each man walked a treadmill at a fixed grade for a period of 12 minutes, and the increase in heart rate (the difference before and after exercise) was recorded (in beats per minute). The data are shown in the table. -Refer to Heart Rates and Exercise Narrative. Find the approximate p-value for the test in the previous question.

A professor was interested in the relationship between a student's rank on an oral exam and the student's rank on a written exam. The professor selected eight students at random and ranked their scores for both the oral exam and the written exam. The following data were recorded: Find and interpret the rank correlation between a student's rank on the oral exam and the student's rank on the written exam.

Teaching Methods Narrative Two different workbooks and two distinct teaching machines were to be evaluated on their effectiveness in teaching the concept of multiplication. A Grade 4 class of 24 subjects was randomly assigned to 4 groups, and each group in turn was randomly assigned to a teaching method. A test was given and the number of errors was recorded. This uses the Kruskal-Wallis H test to see if the number of errors differs from one teaching method to another. -Refer to Teaching Methods Narrative. Find the rejection region for $\alpha$ = 0.05.

Movie Ratings Narrative 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 the most recently watched movies. The rating categories were 1 = terrible, 2 = fair, 3 = good, and 4 = excellent. Each respondent was also asked to categorize his or her age as either 1 = teenager, 2 = young adult (20-34), 3 = middle age (35-50), and 4 = senior (over 50). The results are shown below. -Refer to Movie Ratings Narrative. Do these data provide sufficient evidence to infer at the 5% significance level that there were differences in ratings among the different age categories? Justify your response.

Teachers and Students' IQ Narrative. A school principal suspected that a teacher's attitude toward a Grade 1 student depended on his original judgment of the child's ability. The principal also suspected that much of that judgment was based on the Grade 1 student's IQ score, which was usually known to the teacher. After three weeks of teaching, a teacher was asked to rank the nine children in his class from 1 (highest) to 9 (lowest) as to his opinion of their ability. -Refer to Teachers and Students' IQ Narrative. Do the data provide sufficient evidence to indicate a positive correlation between the teacher's ranks and the ranks of the IQs? Justify your answer. Use = 0.05.

Ice Cream Narrative A supermarket chain has its own house brand of ice cream. The general manager claims that her ice cream is better than the ice cream sold by a well-known ice cream parlour chain. To test the claim, 40 individuals are randomly selected to participate in the following experiment. Each respondent is given the two brands of ice cream to taste (without any identification) and asked to judge which one is better. Suppose that 25 people judge the ice cream parlour brand as better, 4 say that the brands taste the same, and the rest claim that the supermarket brand is better. -Refer to Ice Cream Narrative. What is the p-value of the test in the previous question?

Suppose you wish to detect a difference in the locations of two population distributions based on a paired-difference experiment consisting of n = 35 pairs. a. Give the null and alternative hypotheses for the Wilcoxon signed-rank test. b. Give the test statistic. c. Give the rejection region for the test for = 0.05. d. If = 339, what are your conclusions? [Note: = n(n + 1)/2]. e. Conduct the test using the large-sample z test. Compare your results with the nonparametric test results in the previous question.

Teaching Methods Narrative Two different workbooks and two distinct teaching machines were to be evaluated on their effectiveness in teaching the concept of multiplication. A Grade 4 class of 24 subjects was randomly assigned to 4 groups, and each group in turn was randomly assigned to a teaching method. A test was given and the number of errors was recorded. This uses the Kruskal-Wallis H test to see if the number of errors differs from one teaching method to another. -Refer to Teaching Methods Narrative. Describe what the test statistic is. What is the value of in this ?