Exam 11: Comparisons Involving Proportions and a Test of Independence

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 expected frequency for each group is

Exhibit 11-7 The results of a recent poll on the preference of shoppers regarding two products are shown below. Shoppers Favoring Product Shoppers Surveyed This Product A 800 560 B 900 612 -Refer to Exhibit 11-7. The 95% confidence interval estimate for the difference between the populations favoring the products is

A group of 500 individuals were asked to cast their votes regarding a particular issue of the Equal Rights Amendment. The following contingency table shows the results of the votes: Sex Favor Undecided Oppose TOTAL Female 180 80 40 300 Male 150 20 30 200 TOTAL 330 100 70 500 At α = .05 using the p-value approach, test to determine if the votes cast were independent of the sex of the individuals.

Prior to the start of the season, it was expected that audience proportions for the four major news networks would be CBS 28%, NBC 35%, ABC 22% and BBC 15%. A recent sample of homes yielded the following viewing audience data. Observed frequencies fi CBS 850 NBC 980 670 500 We want to determine whether or not the recent sample supports the expectations for the number of homes of the viewing audience of the four networks. a. State the null and alternative hypotheses to be tested. b. Compute the test statistic. c. The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test. d. What do you conclude?

In order not to violate the requirements necessary to use the chi-square distribution, each expected frequency in a goodness of fit test must be

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 calculated value for the test statistic equals

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 pooled proportion is

A comparative study of organic and conventionally grown produce was checked for the presence of E. coli. Results are summarized below. Is there a significant difference in the proportion of E. Coli in organic vs. conventionally grown produce? Test at ? = 0.10. Sample Size E. Coli Prevalence Organic 200 3 Connentional 500 20

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 calculated value for the test statistic equals

The results of a recent study regarding smoking and three types of illness are shown in the following table. Illness Non-Smokers Smokers Totals Emphysema 50 150 200 Heart problem 50 150 200 Cancer 100 500 600 Totals 200 800 1000 We are interested in determining whether or not illness is independent of smoking. a. State the null and alternative hypotheses to be tested. b. Show the contingency table of the expected frequencies and determine the test statistic. c. The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test d. What do you conclude?

The results of a recent study regarding smoking and three types of illness are shown in the following table. Illness Non-Smokers Smokers Totals Emphysema 30 90 120 Heart problem 70 210 280 Cancer 100 300 400 Totals 200 600 800 We are interested in determining whether or not illness is independent of smoking. a. State the null and alternative hypotheses to be tested. b. Show the contingency table of the expected frequencies and determine the test statistic. c. The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test. d. What do you conclude?

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 test statistic for this test of independence 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 2.0. The p-value for this test is

In 2013, forty percent of the students at a major university were Business majors, 35% were Engineering majors and the rest of the students were majoring in other fields. In a sample of 600 students from the same university taken in 2014, two hundred were Business majors, 220 were Engineering majors and the remaining students in the sample were majoring in other fields. At 95% confidence, test to see if there has been a significant change in the proportions between 2013 and 2014.

The results of a recent study regarding smoking and three types of illness are shown in the following table. Illness Nom-Smokers Smokers Totals Emphysema 20 60 80 Heart problem 70 80 150 Cancer 30 40 70 Totals 120 180 300 We are interested in determining whether or not illness is independent of smoking. a. State the null and alternative hypotheses to be tested. 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. What do you conclude? e. Determine the p-value and perform the test.

In a random sample of 200 Republicans, 160 opposed the new tax laws. While in a sample of 120 Democrats, 84 opposed the new tax laws. Determine a 95% confidence interval estimate for the difference between the proportions of Republicans and Democrats opposed to this new law.

A lottery is conducted that involves the random selection of numbers from 0 to 4. To make sure that the lottery is fair, a sample of 250 was taken. The following results were obtained: Value Frequency 0 40 1 45 2 55 3 60 4 50 a. State the null and alternative hypotheses to be tested. b. Compute the test statistic. c. The null hypothesis is to be tested at the 5% level of significance. Determine the critical value from the table. d. What do you conclude about the fairness of this lottery?

In a sample of 100 Republicans, 60 favored the President's anti-drug program. While in a sample of 150 Democrats, 84 favored his program. At 95% confidence, test to see if there is a significant difference in the proportions of the Democrats and the Republicans who favored the President's anti-drug program.

The data below represents the fields of specialization for a randomly selected sample of undergraduate students. We want to determine whether there is a significant difference in the fields of specialization between regions of the country. Northeast Midwest South West Total Business 54 65 28 93 240 Engineering 15 24 8 33 80 Liberal Arts 65 84 33 98 280 Fine Arts 13 15 7 25 60 Health Sciences 150 200 80 270 700 a. Determine the critical value of the chi-square ?2 for this test of independence. b. Calculate the value of the test statistic. c. What is the conclusion for this test? Let ? = .05.

The results of a recent poll on the preference of voters regarding the presidential candidates are shown below. Voters Surveyed Voters Favoring This Candidate Candidate A 200 150 Candidate B 300 195 a. Develop a 90% confidence interval estimate for the difference between the proportion of voters favoring each candidate. b. Does your confidence interval provide conclusive evidence that one of the candidates is favored more? Explain.