Exam 11: Comparisons Involving Proportions and a Test of Independence

An important application of the chi-square distribution 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 -For a one-tailed test lower tail) at 99.7% confidence, Z =

Exhibit 11-3 In order to determine whether or not a particular medication was effective in curing the common cold, one group of patients was given the medication, while another group received sugar pills. The results of the study are shown below. Patients Cured Patients Not Cured Received medication 70 10 Received sugar pills 20 50 We are interested in determining whether or not the medication was effective in curing the common cold. -Refer to Exhibit 11-3. The number of degrees of freedom associated with this problem is

Two hundred fifty managers with degrees in business administration indicated their fields of concentration as shown below. Major Top Management Middle Management TOTAL Management 65 60 125 Marketing 30 20 50 Accounting 25 50 75 TOTAL 120 130 250 At α = .01 using the p-value approach, test to determine if the position in management is independent of the major of concentration.

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 expected number of freshmen 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 -Refer to Exhibit 11-9. The 95% confidence interval for the difference between the two proportions is

Exhibit 11-3 In order to determine whether or not a particular medication was effective in curing the common cold, one group of patients was given the medication, while another group received sugar pills. The results of the study are shown below. Patients Cured Patients Not Cured Received medication 70 10 Received sugar pills 20 50 We are interested in determining whether or not the medication was effective in curing the common cold. -Refer to Exhibit 11-3. The p-value is

The sampling distribution for a goodness of fit test is the

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 expected frequency for the Business College is

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 p-value 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 conclusion of the test is that the

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 p-value is

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 p-value is

Last school year, in the school of Business Administration, 30% were Accounting majors, 24% Management majors, 26% Marketing majors, and 20% Economics majors. A sample of 300 students taken from this year's students of the school showed the following number of students in each major: Accounting 83 Management 68 Marketing 85 Economics 64 Total 300 We want to see if there has been a significant change in the number of students in each major. a. Compute the test statistic. b. Has there been any significant change in the number of students in each major between the last school year and this school year. Use the p-value approach and let α = .05.

A major automobile manufacturer claimed that the frequencies of repairs on all five models of its cars are the same. A sample of 200 repair services showed the following frequencies on the various makes of cars. Model of Car Frequency 32 45 43 34 46 At α = 0.05, test the manufacturer's claim.

The number of degrees of freedom for the appropriate chi-square distribution in a test of independence is

Shoppers were asked where they do their regular grocery shopping. The table below shows their responses of the sampled shoppers. We are interested in determining if the proportions females in the three categories are different from each other. Gender Grocery chain Discount store Membership warehouse Total Female 230 80 100 410 Male 80 50 60 190 Total 310 130 160 600 a. Provide the null and the alternative hypotheses. b. Determine the expected frequencies. c. Compute the sample proportions. d. Compute the critical values CVij). e. Give your conclusions by providing numerical reasoning.

Before the rush began for Christmas shopping, a department store had noted that the percentage of its customers who use the store's credit card, the percentage of those who use a major credit card, and the percentage of those who pay cash are the same. During the Christmas rush in a sample of 150 shoppers, 46 used the store's credit card; 43 used a major credit card; and 61 paid cash. With α = 0.05, test to see if the methods of payment have changed during the Christmas rush.

Exhibit 11-3 In order to determine whether or not a particular medication was effective in curing the common cold, one group of patients was given the medication, while another group received sugar pills. The results of the study are shown below. Patients Cured Patients Not Cured Received medication 70 10 Received sugar pills 20 50 We are interested in determining whether or not the medication was effective in curing the common cold. -Refer to Exhibit 11-3. The hypothesis is to be tested at the 5% level of significance. The critical value from the table equals