Exam 12: Tests of Goodness of Fit and Independence

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 $\alpha$ = 0.05,test to see if the methods of payment have changed during the Christmas rush.

A manager believes that the shelf life of apple juice is normally distributed.A sample of 30 containers of juice was taken and the shelf life was recorded.You are given the results below.The average shelf life in the sample was 23.07 days with a standard deviation of 4.29 days. a.State the null and alternative hypotheses. b.Compute the test statistic for the goodness of fit test. c.At 95% confidence using the p-value approach,test the hypotheses.What do you conclude about the distribution?

Exhibit 12-3 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. We are interested in determining whether or not the medication was effective in curing the common cold. -Refer to Exhibit 12-3.The expected frequency of those who received medication and were cured is

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. a.Determine the critical value of the chi-square $\chi$ ² for this test of independence. b.Calculate the value of the test statistic. c.What is the conclusion for this test? Let $\alpha$ = .05.

Exhibit 12-1 When individuals in a sample of 150 were asked whether or not they supported capital punishment,the following information was obtained. 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 12-1.The conclusion of the test (at 95% confidence)is that the

Exhibit 12-5 The table below gives beverage preferences for random samples of teens and adults. We are asked to test for independence between age (i.e. ,adult and teen)and drink preferences. -Refer to Exhibit 12-5.The test statistic for this test of independence is

An insurance company has gathered the following information regarding the number of accidents reported per day over a period of 100 days. Using the critical value approach test to see if the above data have a Poisson distribution.Let $\alpha$ = 0.05.

In the last presidential election,before the candidates started their major campaigns,the percentages of registered voters who favored the various candidates were as follows. After the major campaigns began,a random sample of 400 voters showed that 172 favored the Republican candidate;164 were in favor of the Democratic candidate;and 64 favored the Independent candidate.We are interested in determining whether the proportion of voters who favored the various candidates had changed. a.Compute the test statistic. b.Using the p-value approach,test to see if the proportions have changed. c.Using the critical value approach,test the hypotheses.

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: 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?

The personnel department of a large corporation reported sixty resignations during the last year.The following table groups these resignations according to the season in which they occurred: Test to see if the number of resignations is uniform over the four seasons. Let $\alpha$ = 0.05.

Before the start of the Winter Olympics,it was expected that the percentages of medals awarded to the top contenders to be as follows. Midway through the Olympics,of the 120 medals awarded,the following distribution was observed. We want to test to see if there is a significant difference between the expected and actual awards given. a.Compute the test statistic. b.Using the p-value approach,test to see if there is a significant difference between the expected and the actual values.Let $\alpha$ = .05. c.At 95% confidence,test for a significant difference using the critical value approach.

Exhibit 12-5 The table below gives beverage preferences for random samples of teens and adults. We are asked to test for independence between age (i.e. ,adult and teen)and drink preferences. -Refer to Exhibit 12-5.The p-value is

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. At $\alpha$ = 0.05,test the manufacturer's claim.

Exhibit 12-4 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 12-4.The expected frequency for the Business College 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. 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 12-1 When individuals in a sample of 150 were asked whether or not they supported capital punishment,the following information was obtained. 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 12-1.The calculated value for the test statistic equals

We want to determine if the following sample comes from a normal distribution. a.Compute the mean and the standard deviation. b.Compute the test statistic.Hint: divide the distribution into 10 equal intervals. c.At 95% confidence using the critical value approach,test to determine if the sample comes from a normal population. d.Compute the p-value.

Exhibit 12-7 You want to test whether or not the following sample of 30 observations follows a normal distribution.The mean of the sample equals 11.83,and the standard deviation equals 4.53. -Refer to Exhibit 12-7.The number of intervals or categories used to test the hypothesis for this problem is

Exhibit 12-7 You want to test whether or not the following sample of 30 observations follows a normal distribution.The mean of the sample equals 11.83,and the standard deviation equals 4.53. -Refer to Exhibit 12-7.The p-value is

During the first few weeks of the new television season,the evening news audience proportions were recorded as ABC- 31%,CBS- 34%,and NBC- 35%.A sample of 600 homes yielded the following viewing audience data. We want to determine whether or not there has been a significant change in the number of viewing audience of the three networks. a.State the null and alternative hypotheses to be tested. b.Compute 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.