Exam 14: Analysis of Categorical Data

A chi-square goodness-of-fit test is always conducted as a two-tailed test.

The rejection region of the chi-square goodness-of-fit test has k - 1 degrees of freedom, where k is the number of categories (called cells).

If we wanted to conduct a one-tailed test of a population proportion, which of the following tests could we employ?

A librarian knows from old records that the percentages of checked-out materials that 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.

A survey of 500 respondents produced these cell counts in a 2 3 contingency table: Columns a. If you wish to test the null hypothesis of "independence"-that the probability that a response falls in any one row is independent of the column it falls into-and you plan to use a chi-square test, how many degrees of freedom will be associated with the chi-square statistic? b. Find the value of the test statistic. c. Find the rejection region for = 0.01. d. Conduct the test and state your conclusions. e. Find the approximate p-value for the test.

Average Casino Dice Rolls Narrative A casino customer is concerned about whether the dice the casino is using are "fair" (i.e., each face is equally likely to appear). The customer obtains a die that is being used by the casino and rolls it 240 times. The customer records the following data: -Refer to Average Casino Dice Rolls Narrative. State the null and alternative hypotheses.

Suppose that a response can fall into one of k = 5 categories with probabilities , and that n = 250 responses produced these category counts: The categories represent five car salespersons, and the observed frequencies represent the number of cars sold in a six-month period. -Refer to Car Sales Narrative. Test the hypotheses at = 0.05, and write your conclusion.

Peony Plants Narrative A peony plant with red petals was crossed with another plant having streaky petals. A geneticist states that 80% of the offspring from this cross will have red flowers. To test this claim, 100 seeds from this cross were collected and germinated, and 65 plants had red petals. -Refer to Peony Plants Narrative. Use the large-sample z test to test the hypothesis of the previous question p = 0.80. Verify that the squared value of the test statistic from the previous question.

Ice Cream Flavours Narrative The four most popular flavours of a particular brand of ice cream are French vanilla, natural vanilla, chocolate chip, and caramel almond swirl. The question arose as to whether each of these flavours is equally preferred. To answer that question, a random sample of 160 sales was selected, yielding the following data: -Refer to Ice Cream Flavours Narrative. State the null and alternative hypotheses.

A chi-square test of independence is applied to a contingency table with four rows and five columns for two qualitative variables. What are the degrees of freedom for this test?

A left-tailed area in the chi-square distribution equals 0.90. Taking this into consideration, for 7 degrees of freedom, what does the table value equal?

When either the row or the column totals in a contingency table are fixed, the test of independence of classifications becomes a test of the homogeneity of cell probabilities for several multinomial experiments.

Of the values for a chi-square test statistic listed below, which one is likely to lead to rejecting the null hypothesis in a goodness-of-fit test?

Seat Belts On School Buses Narrative A study was conducted to determine whether opinion concerning the addition of seat belts in school buses is independent of the population density in which a person resides. The following data were recorded: Local Use the following output generated using statistical software to answer the questions below. Chi-Square Test Expected counts are printed below observed counts Chi-Sq = 0.003 + 0.402 + 0.393 + 0.099 + 0.000 + 0.101 + 0.152 + 2.016 + 0.837 = 4.003 DF = 4, P-Value = 0.406 -Refer to Seat Belts On School Buses Narrative. What is the p-value associated with the test?

Ages of National College Students Narrative A national survey states that 67% of college students are under the age of 25, 21% are between the ages of 25 and 30, 8% are between 30 and 40, and 4% are over 40. A random sample of 250 students at Vanier College in Montreal yielded the following data: -Refer to Ages of National College Students Narrative. What is the appropriate conclusion?

Ink Pen Colours Narrative A national survey stated that 30% of the population prefer to use a pen with black ink, 30% prefer blue ink, 25% prefer red ink, and 15% prefer some other colour. A statistics professor took a random sample of 80 students and asked them to state their ink colour preference. The following data was recorded: -Refer to Ink Pen Colours Narrative. What is the appropriate conclusion?

In a chi-square test of independence, the value of the test statistic was = 15.652, and the critical value at was 11.1433. Thus, we must reject the null hypothesis at .

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? Justify your answer. (Use 0.10.)

A chi-square goodness-of-fit test is always conducted as a two-tailed test.

Average Casino Dice Rolls Narrative A casino customer is concerned about whether the dice the casino is using are "fair" (i.e., each face is equally likely to appear). The customer obtains a die that is being used by the casino and rolls it 240 times. The customer records the following data: -A total of 200 people in each of three grocery stores were asked if they favoured, opposed, or were indifferent to the sale of lottery tickets in grocery stores. The results of the survey are summarized below. Use the statistical software computer output shown here to conduct a chi-square test of independence at = 0.05. Chi-Square Test Expected counts are printed below observed counts Chi-Sq = 29.188 + 2.903 + 13.680 + 21.892 + 0.007 + 21.117 + 0.500 + 2.347 + 0.681 = 92.315 DF = 4