Exam 11: Chi-Square and Analysis of Variance

Suppose you are to test for equality of four different population means, with H0: µA = µB $= \mu _ { C } = \mu _ { D }$ . Write the hypotheses for the paired tests. Use methods of probability to explain why the process of ANOVA has a higher degree of confidence than testing each of the pairs separately.

An observed frequency distribution of exam scores is as follows: Exam Score under 60 60-69 70-79 80-89 90-100 Frequency 36 75 85 70 34 i) Assuming a normal distribution with $\mu = 75$ and $\sigma = 15$ , find the probability of a randomly selected subject belonging to each class. (Use boundaries of $59.5,69.5,79.5,89.5$ , 100.) ii) Using the probabilities found in part (i), find the expected frequency for each category. iii) Use a $0.05$ significance level to test the claim that the exam scores were randomly selected from a normally distributed population with $\mu = 75$ and $\sigma = 15$ .

A researcher wishes to test whether the proportion of college students who smoke is the same in four different colleges. She randomly selects 100 students from each college and records the number that smoke. The results are shown below. College A College B College C College D Smoke 17 26 11 34 Don't smoke 83 74 89 66 Use a 0.01 significance level to test the claim that the proportion of students smoking is the same at all four colleges.

At the same time each day, a researcher records the temperature in each of three greenhouses. The table shows the temperatures in degrees Fahrenheit recorded for one week. Greenhouse \#1 Greenhouse \#2 Greenhouse \#3 73 71 67 72 69 63 73 72 62 66 72 61 68 65 60 71 73 62 72 71 59 i) Use a 0.05 significance level to test the claim that the average temperature is the same in each greenhouse. ii) How are the analysis of variance results affected if the same constant is added to every one of the original sample values?

Four independent samples of 100 values each are randomly drawn from populations that are normally distributed with equal variances. You wish to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 }$ $= \mu _ { 4 }$ i) If you test the individual claims $\mu _ { 1 } = \mu _ { 2 } , \mu _ { 1 } = \mu _ { 3 } , \mu _ { 1 } = \mu _ { 4 } , \ldots , \mu _ { 3 } = \mu _ { 4 }$ , how many ways can you pair off the 4 means? ii) Assume that the tests are independent and that for each test of equality between two means, there is a $0.99$ probability of not making a type I error. If all possible pairs of means are tested for equality, what is the probability of making no type I errors? iii) If you use analysis of variance to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 }$ at the $0.01$ level of significance, what is the probability of not making a type I error?

The following table shows the number of employees who called in sick at a business for different days of a particular week. Day Sun Mon Tues Wed Thurs Fri Sat Number sick 8 12 7 11 9 11 12 i) At the 0.05 level of significance, test the claim that sick days occur with equal frequency on the different days of the week. ii) Test the claim after changing the frequency for Saturday to 152. Describe the effect of this outlier on the test.

Describe the null hypothesis for the test of independence. List the assumptions for the X² test of independence. What is the major difference between the assumptions for this test and the assumptions for the previous tests we have studied?

In studying the responses to a multiple-choice test question, the following sample data were obtained. At the 0.05 significance level, test the claim that the responses occur with the same frequency. Response A B C D E Frequency 12 15 16 18 19

Identify the value of the test statistic. Source DF SS MS F p Factor 3 30 10.00 1.6 0.264 Error 8 50 6.25 Total 11 80

Use a 0.01 significance level to test the claim that the proportion of men who plan to vote in the next election is the same as the proportion of women who plan to vote. 300 men and 300 women were randomly selected and asked whether they planned to vote in the next election. The results are shown below. Men Women Plan to vote 170 185 Do not plan to vote 130 115

Using the data below and a 0.05 significance level, test the claim that the responses occur with percentages of 15%, 20%, 25%, 25%, and 15% respectively. Response A B C D E Frequency 12 15 16 18 19

Define the term "treatment". What other term means the same thing? Give an example.

Fill in the missing entries in the following partially completed one-way ANOVA table. Source df SS MS=SS/df F-statistic Treatment 3 11.16 Error 13.72 0.686 Total

Describe a goodness-of-fit test. What assumptions are made when using a goodness-of-fit test?

Use a significance level of 0.01 to test the claim that workplace accidents are distributed on workdays as follows: Monday 25%, Tuesday: 15%, Wednesday: 15%, Thursday: 15%, and Friday: 30%. In a study of 100 workplace accidents, 26 occurred on a Monday, 15 occurred on a Tuesday, 17 occurred on a Wednesday, 17 occurred on a Thursday, and 25 occurred on a Friday.

A researcher wishes to test the effectiveness of a flu vaccination. 150 people are vaccinated, 180 people are vaccinated with a placebo, and 100 people are not vaccinated. The number in each group who later caught the flu was recorded. The results are shown below. Vaccinated Placebo Control Caught the flu 8 19 21 Did not catch the flu 142 161 79 Use a 0.05 significance level to test the claim that the proportion of people catching the flu is the same in all three groups.

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. -Identify the value of the test statistic. Source DF SS MS F P Factor 3 13.500 4.500 5.17 0.011 Error 16 13.925 0.870 Total 19 27.425

Use a $\chi ^ { 2 }$ test to test the claim that in the given contingency table, the row variable and the column variable are independent. -160 students who were majoring in either math or English were asked a test question, and the researcher recorded whether they answered the question correctly. The sample results are given below. At the 0.10 significance level, test the claim that response and major are independent. Correct Incorrect Math 27 53 English 43 37

A survey conducted in a small business yielded the results shown in the table. Men Women Health insurance 41 22 No health insurance 34 24 i) Test the claim that health care coverage is independent of gender. Use a 0.05 significance level. ii) Using Yates' correction, replace $\sum \frac{(O-E)^{2}}{E}\text { with } \sum \frac{(|O-E|-0.5)^{2}}{E}$ and repeat the test. What effect does Yates' correction have on the value of the test statistic?

In studying the occurrence of genetic characteristics, the following sample data were obtained. At the 0.05 significance level, test the claim that the characteristics occur with the same frequency. Characteristic A B C D E F Frequency 28 30 45 48 38 39