Exam 11: Comparisons Involving Proportions and a Test of Independence

A poll was taken this year asking college students if they considered themselves overweight. A similar poll was taken five years ago. Results are summarized below. Has the proportion increased significantly? Let α = 0.05. Sample Size Number Considered Themselves Overweight Present Sample 300 150 Previous sample 275 121

Five hundred randomly selected automobile owners were questioned on the main reason they had purchased their current automobile. The results are given below. Styling Engineering Fuel Economy Total Male 70 130 150 350 Female Total 100 150 250 500 a. State the null and alternative hypotheses for a contingency table test. b. State the decision rule for the critical value approach. Let ? = .01. c. Calculate the ?2 test statistic. d. Give your conclusion for this test.

In a sample of 40 Democrats, 6 opposed the President's foreign policy, while of 50 Republicans, 8 were opposed to his policy. Determine a 90% confidence interval estimate for the difference between the proportions of the opinions of the individuals in the two parties.

Exhibit 11-1 When individuals in a sample of 150 were asked whether or not they supported capital punishment, the following information was obtained. Do you support capital punishment? Do you support capital punishment? Number of individuals Yes 40 No 60 No Opinion 50 We are interested in determining whether or not the opinions of the individuals as to Yes, No, and No Opinion) are uniformly distributed. -Refer to Exhibit 11-1. The conclusion of the test at 95% confidence) is that the

Exhibit 11-6 The following shows the number of individuals in a sample of 300 who indicated they support the new tax proposal. Political Party Support Democrats 100 Republicans 120 Independents 80 We are interested in determining whether or not the opinions of the individuals of the three groups are uniformly distributed. -Refer to Exhibit 11-6. The expected frequency for each group is

Exhibit 11-2 Last school year, the student body of a local university consisted of 30% freshmen, 24% sophomores, 26% juniors, and 20% seniors. A sample of 300 students taken from this year's student body showed the following number of students in each classification. Freshmen 83 Sophomores 68 Juniors 85 Seniors 64 We are interested in determining whether or not there has been a significant change in the classifications between the last school year and this school year. -Refer to Exhibit 11-2. The p-value is

Exhibit 11-8 An insurance company selected samples of clients under 18 years of age and over 18 and recorded the number of accidents they had in the previous year. The results are shown below. Under Age of 18 Over Age of 18 =500 =600 Number of accidents =180 Number of accidents =150 We are interested in determining if the accident proportions differ between the two age groups. -Refer to Exhibit 11-8. The test statistic is

Exhibit 11-4 In the past, 35% of the students at ABC University were in the Business College, 35% of the students were in the Liberal Arts College, and 30% of the students were in the Education College. To see whether or not the proportions have changed, a sample of 300 students was taken. Ninety of the sample students are in the Business College, 120 are in the Liberal Arts College, and 90 are in the Education College. -Refer to Exhibit 11-4. The hypothesis is to be tested at the 5% level of significance. The critical value from the table equals

Exhibit 11-6 The following shows the number of individuals in a sample of 300 who indicated they support the new tax proposal. Political Party Support Democrats 100 Republicans 120 Independents 80 We are interested in determining whether or not the opinions of the individuals of the three groups are uniformly distributed. -Refer to Exhibit 11-6. The calculated value for the test statistic equals

Shown below is 3 x 2 contingency table with observed values from a sample of 1,500. At 95% confidence, test for independence of the row and column factors. Column Factor Row Factor Total 450 300 750 300 300 600 100 50 150 Total 850 650 1500 Row Factor Total A 450 300 750 300 300 600 C 100 50 150 Total 850 650 1500

Exhibit 11-4 In the past, 35% of the students at ABC University were in the Business College, 35% of the students were in the Liberal Arts College, and 30% of the students were in the Education College. To see whether or not the proportions have changed, a sample of 300 students was taken. Ninety of the sample students are in the Business College, 120 are in the Liberal Arts College, and 90 are in the Education College. -Refer to Exhibit 11-4. The p-value is

The following table shows the results of recent study regarding gender of individuals and their selected field of study. Field of study Male Female TOTAL Medicine 80 40 120 Business 60 20 80 Engineering 160 40 200 TOTAL 300 100 400 We want to determine if the selected field of study is independent of gender. a. Compute the test statistic. b. Using the p-value approach at 90% confidence, test to see if the field of study is independent of gender. c. Using the critical method approach at 90% confidence, test for the independence of major and gender.

The degrees of freedom for a contingency table with 6 rows and 3 columns is

If we are interested in testing whether the proportion of items in population 1 is larger than the proportion of items in population 2, the

A population where each element of the population is assigned to one and only one of several classes or categories is a

A goodness of fit test is always conducted as a

Exhibit 11-5 The table below gives beverage preferences for random samples of teens and adults. Teens Adults Total Coffee 50 200 250 Tea 100 150 250 Soft Drink 200 200 400 Other 50 50 100 400 400 600 1,000 We are asked to test for independence between age i.e., adult and teen) and drink preferences. -Refer to Exhibit 11-5. The expected number of adults who prefer coffee is

From a poll of 800 television viewers, the following data have been accumulated as to their levels of education and their preference of television stations. We are interested in determining if the selection of a TV station is independent of the level of education. Educational Level High School Bachelor Graduate TOTAL Public Broadcasting 50 150 80 280 Commercial Stations 150 250 120 520 TOTAL 200 400 200 800 a. State the null and the alternative hypotheses. b. Show the contingency table of the expected frequencies. c. Compute the test statistic. d. The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test. e. Determine the p-value and perform the test.

Exhibit 11-9 The results of a recent poll on the preference of teenagers regarding the types of music they listen to are shown below. Teenagers Favoring Music Type Teenagers Surveyed This Type Pop 800 384 Rap 900 450 -Refer to Exhibit 11-9. The standard error of is

Exhibit 11-9 The results of a recent poll on the preference of teenagers regarding the types of music they listen to are shown below. Teenagers Favoring Music Type Teenagers Surveyed This Type Pop 800 384 Rap 900 450 -We are interested in testing the following hypotheses. H0: P1- P2 ? 0 Ha: P1- P2 > 0 The test statistic Z is computed to be 0.58. The p-value for this test is