Exam 14: Analysis of Categorical Data

A cafeteria proposes to serve 4 main entrees. For planning purposes, the manager expects that the proportions of each that will be selected by his customers will be: Of the first 100 customers, 44 selected chicken, 24 selected roast beef, 13 select steak, and 10 selected fish. Should the manager revise his estimates? Use = 0.01. = ______________ Critical Value(s) = ______________ Conclusion: ______________ The manager ______________ revise his estimates.

A contingency table classifies data with respect to two qualitative variables that are each divided into two or more categories.

A study of the purchase decisions of three stock portfolio managers, A, B, C, was conducted to compare the numbers of stock purchases that resulted in profits over a time period less than or equal to 1 year. One hundred randomly selected purchases were examined for each of the managers. Do the data provide evidence of differences among the rates of successful purchases for the three managers? Compute = ______________ The p-value is ______________. Conclude that there ______________ enough information to conclude that the proportion of successful purchases will differ among the managers.

Which of the following statements about a contingency table test is false?

A chi-square test for independence with 10 degrees of freedom results in a test statistic of 17.894. Using the chi-square table, the most accurate statement that can be made about the p-value for this test is that 0.05 < p-value < 0.10.

The personnel manager of a consumer products company asked a random sample of employees how they felt about the work they were doing. The following table gives a breakdown of their responses by age. Is there sufficient evidence to conclude that the level of job satisfaction is related to age? Use = 0.10. What is ? ______________ Critical Value = ______________ Conclusion: ______________ Job satisfaction ______________ related to age.

The degrees of freedom associated with a chi-square test of independence where data are summarized in a contingency table with r rows and c columns equal the number of rows times the number of columns in the table; that is, rc.

The degrees of freedom associated with a goodness-of-fit test equal the number of rows times the number of columns in the table.

The area to the right of a chi-square value is 0.05. For 9 degrees of freedom, the table value is:

For a chi-square distributed random variable with 10 degrees of freedom and a level of significance of 0.025, the chi-square table value is 20.4831. The computed value of the test statistics is 16.857. This will lead us to reject the null hypothesis.

In a chi-square test of independence with 6 degrees of freedom and a level of significance of 0.05, the critical value from the chi-square table is 12.5916. The computed value of the test statistics is 11.264. This will lead us to reject the null hypothesis.

Because the chi-square random variable is a sum of squares, it cannot take on negative values.

Suppose you wish to test the null hypothesis that three binomial parameters , , and are equal versus the alternative hypothesis that at least two of the parameters differ. Independent random samples of 100 observations were selected from each of the populations. The data are shown in the table: Write the null and alternative hypotheses for testing the equality of the three binomial proportions. : ______________ Test statistic: = ______________ What is the p-value? ______________ ______________ . There is ______________ sufficient evidence to indicate that the proportions depend upon the population from which they were drawn.

If we want to perform a two-tail test for differences between two populations of nominal data with exactly two categories, we can employ either the z-test of , or the chi-square test of homogeneity, since squaring the value of the z statistic yields the value of statistics.

A librarian knows from old records that the percentages of checked-out materials which are fiction, nonfiction, children's books, books on tape, and magazines are 35%, 25%, 20%, 10%, and 10%, respectively. The material checked out in each of the categories in a recent survey of 600 randomly chosen library items is listed below. Use these data to assess whether there has been a change in the distribution of checked-out library materials. State the hypotheses to be tested and use = 0.01 to interpret your results. The hypotheses are: : = _______, = _______, = _______, = _______, = _______ : ______________ from the specified value(s). Compute : ______________ Thus we ______________ . Conclude that ______________ from those specified in .

If we want to conduct a two-tail test of a population proportion, we can employ:

Five brands of orange juice are displayed side by side in several supermarkets in the city of Chicago. It was noted that in one day, 180 customers purchased orange juice. Of these, 30 picked Brand A, 40 picked Brand B, 25 picked Brand C, 35 picked Brand D, and 50 picked brand E. In Chicago, can you conclude at the 5% significance level that there is a preferred brand of orange juice? Compute : ______________ Reject when > ______________. Conclude that there ______________ preferred brand of orange juice in Chicago.

A sample survey of 850 people in Chicago was conducted in 2001 to see how people get to work. The results are shown in the frequency table below. A study of the same nature was conducted in 1990 and revealed 12% walk, 67% drive an automobile, 18% take a bus, and 3% ride a bicycle. Do the current data support that the proportions are the same as in 1990? Use . Compute : ______________ One should ______________ . Conclude that the current data ______________ that the proportions in 2001 are the same as in 1990.

The number of Americans who visit fast-food restaurants regularly has grown steadily over the last decade. For this reason, marketing experts are interested in the demographics of fast-food customers. Is a customer's preference for a fast- food chain affected by the age of the customer? If so, advertising might need to target a particular age group. Suppose a random sample of 400 fast-food customers aged 16 and older was selected, and their favorite fast-food restaurants along with their age groups were recorded, as shown in the table: Use an appropriate method to determine whether or not a customer's fast-food preference is dependent on age. Compute : ______________ What is the p-value? ______________ Conclude that there ______________ a dependence between age group and favorite fast food restaurant. Write a short paragraph presenting your statistical conclusions and their practical implications for marketing experts. ________________________________________________________

The chi-square technique has numerous applications, including: