Exam 11: Chi-Square and Analysis of Variance

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

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.

Describe the null hypothesis for the test of independence. List the assumptions for the $\chi ^ { 2 }$ test of independence.

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.

The table in number 18 is called a two-way table. Why is the terminology of two-way table used?

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

Use a $\chi ^ { 2 }$ test to test the claim that in the given contingency table, the row variable and the column variable are independent. Responses to a survey question are broken down according to employment status and the sample results are given below. At the $0.10$ significance level, test the claim that response and employment status are independent. Yes No Undecided Employed 30 15 5 Unemployed 20 25 10

Select the null hypothesis for a test of independence.

At a high school debate tournament, half of the teams were asked to wear suits and ties and the rest were asked to wear jeans and t-shirts. The results are given in the table below. In Order to test the claim at the 0.05 level that the proportion of wins is the same for teams Wearing suits as for teams wearing jeans, what would the null hypothesis be? Win Loss Suit 22 28 T-Shirt 28 22

According digits that follow the distribution shown in the table below. Test for goodness-of-fit with Benford's Law. Leading Digit 1 2 3 4 5 6 7 8 9 Benford's law: distribution of leading digits 30.1\% 17.6\% 12.5\% 9.7\% 7.9\% 6.7\% 5.8\% 5.1\% 4.6\% When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be $0,12,0,73,482,186,8,23$ , and 0 , and those digits correspond to the leading digits of $1,2,3,4,5,6,7,8$ , and 9 , respectively. If the observed frequenciesare substantially different from the frequencies expected with Benford's Law, the check amounts appear to result from fraud. Use a $0.05$ significance level to test for goodness-of-fit with Benford's Law. What is the value of the test statistic? Does it appear that the checks are the result of fraud?

Use a $\chi ^ { 2 }$ test to test the claim that in the given contingency table, the row variable and the column variable are independent. The table below shows the age and favorite type of music of 668 randomly selected people. Rock Pop Classical 50 85 73 68 91 60 35-45 90 74 77 Use a 5 percent level of significance to test the null hypothesis that age and preferred music type are independent.

At a high school debate tournament, half of the teams were asked to wear suits and ties and the rest were asked to wear jeans and t-shirts. The results are given in the table below. Test the claim at the 0.05 level that the proportion of wins is the same for teams wearing suits As for teams wearing jeans. Win Loss Suit 22 28 T-Shirt 28 22 What is your conclusion about the null hypothesis?

A table summarizes the success and failures when subjects used different methods (yoga, acupuncture, and chiropractor)to relieve back pain. If we test the claim at a 5% level of Significance that success is independent of the method used, technology provides a P-value Of 0.0355. What does the P-value tell us about the claim?

Describe the test of homogeneity. What characteristic distinguishes a test of homogeneity from a test of independence?

For a test of independence, the population that the data has come from must be normally distributed.

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.

Discuss the three characteristics of a chi-square distribution.

A researcher wishes to test whether the proportion of college students who smoke is the same at 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.

Which statement is not true for goodness-of -fit tests?

Discuss the three characteristics of a chi-square distribution.