Exam 17: Additional Tests for Nominal Data: Chi-Squared Tests

An Australian tyre manufacturer operates a plant in Melbourne and another plant in Adelaide. Employees at each plant have been evenly divided among three issues (wages, working conditions and super benefits) in terms of which one they feel should be the primary issue in the upcoming contract negotiations. The secretary of the union has recently circulated pamphlets among the employees, attempting to convince them that super benefits should be the primary issue. A subsequent survey revealed the following breakdown of the employees according to the plant at which they worked and the issue that they felt should be supported as the primary one. Issues Plant Location Very interesting Fairly interesting Not interesting Melbourne 60 62 78 Adel aide 70 56 74 Do the data indicate at the 5% significance level that there are differences between the two plants regarding which issue should be the primary one?

An Australian firm has been accused of engaging in prejudicial hiring practices against women who have young children, where a young child is defined to be a child less than 12 years of age. According to the most recent census, the percentages of men, women without young children and women with young children in a certain community are 72%, 10% and 18%, respectively. A random sample of 200 employees of the firm revealed that 165 were men, 14 were women without young children and 21 were women with young children. Do the data provide sufficient evidence to conclude at the 10% level of significance that the firm has been engaged in prejudicial hiring practices?

A chi-squared test of a contingency table with 4 rows and 4 columns shows that the value of the test statistic is 23. Which of the following best describes the p-value for this test? A. p -value <0.010 B. p -value <0.005 C. p -value >0.10 D. 0.005

In a chi-squared test of independence, the value of the test statistic was $\chi ^ { 2 } =$ 15.652, and the critical value at $\alpha = 0.025$ was 11.1433. Thus we must reject the null hypothesis at $\alpha = 0.025$ .

If each element in a population is classified into one and only one of several categories, the population is a: A. normal population. B. multinomial population. C. chi-squared population. D. binomial population.

A study of education levels of 500 voters and their political party affiliations in a particular state in Australia showed the following results. Party Affliation Education level completed Liberal Other Labour Primary education 40 20 80 Secondary education 70 30 60 Tertiary education 90 50 60 Is education level independent of political affiliation? Test at the 5% level of significance.

One type of Chi-squared goodness of fit test is a Chi-squared test for normality.

The vice-chancellor of a university collected data from students concerning the building of a new library, and classified the responses into different categories (strongly agree, agree, undecided, disagree, strongly disagree) and according to whether the student was male or female. Which of the following tests should be used to test whether responses and gender are independent? A. Chi-squared test for goodness-of-fit. B. Chi-squared test for normality. C. Chi-square test of a multinomial experiment. D. Chi-squared test of a contingency table.

The personnel manager of a consumer product 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 gender. Response Gender Very interesting Fairly interesting Not interesting Male 70 41 9 Female 35 34 11 Do the data provide sufficient evidence to conclude that the level of job satisfaction is related to gender? Use $\alpha =$ 0.10.

In chi-squared tests, the conventional and conservative rule - known as the rule of five - is to require that the: A. expected frequency for each cell be at least 5 . B. number of degrees of freedom for the test be at least 5 . C. each expected and observed frequency be at least 5 . D. difference between the observed and expected frequency for each cell be at least 5 .

Which of the following uses the Chi-square distribution? A. Test of independence between two categorical variables. B. Test of a multinomial experiment with one categorical variable. C. Test about a single population variance with one numerical variable. D. All of these choices are correct.

A university lecturer had the following semester 1 student's final grades for the course that they teach: 8% High Distinction, 35% Distinction, 40% Credit, 12% Pass, and 5% Failed. A sample of 100 second semester final grades for the same course showed 12 High distinctions, 30 Distinctions, 35 Credits, 15 Passes, and 8 Fails. Test at the 10% significance level to determine whether the semester one grades differ significantly from the semester two grades.

Suppose that a random sample of 100 observations was drawn from a population. After calculating the mean and standard deviation, each observation was standardised and the number of observations in each of the intervals below was counted. Can we infer at the 10% significance level that the data were drawn from a normal population? Intervals Frequency Z\leq-1 12 -11 20

A psychologist claims that people who exercise regularly are more content with their body image. A random sample of 102 people was taken, where they were asked if they regularly exercise and if they were content with their body image. Not content Content Exercised regularly 20 30 Did not exercise regularly 28 24 Is there significant evidence to support this claim? Test at the 1% significance level.

A Chi-squared test can be used to test the equality of two population means.

In a goodness-of-fit test, suppose that a sample showed that the observed frequency $f _ { i }$ and expected frequency $e _ { i }$ were equal for each cell i. Which of the following best describes the decision for the hypothesis test? A. Reject $H_{0}$ at $\alpha$ of 0.05 but retain $H_{0}$ at $\alpha$ of 0.025 . B. Reject $\mathrm{Ho}$ at any level of $\alpha$ . C. Retain $\mathrm{Ho}$ at any level of $\alpha$ . D. Retain $\mathrm{Ho}$ at $\alpha$ of 0.05 but reject $\mathrm{Ho}$ at $\alpha$ of 0.025 .

Consider a multinomial experiment involving n = 200 trials and k = 5 cells. The observed frequencies resulting from the experiment are shown in the following table: Cell 1 2 3 4 5 Frequency 8 22 28 24 18 The null hypothesis to be tested is as follows. $H _ { 0 } : p _ { 1 } = 0.10$ , $p _ { 2 } = 0.25,$ $p _ { 3 } = 0.30,$ $p _ { 4 } = 0.20,$ $p _ { 5 } = 0.15$ $p _ { 5 } = 0.15$ .

Which of the following best describes the approach taken if the expected frequency $e _ { i }$ for any cell i is less than 5? A. We must choose another sample of five or more observations. B. We should use the normal distribution instead of the chi-squared distribution. C. We should combine the cells such that each observed frequency $f_{i}$ is 5 or more. D. We must increase the number of degrees of freedom for the test by 5 .

In a chi-squared test of a contingency table, the value of the test statistic was $\chi ^ { 2 } = 12.678$ , the significance level was $\alpha$ = 0.05 and the degrees of freedom was 6. Thus: A. we fail to reject the null hyp othesis at $\alpha = 0.05$ . B. we reject the null hypothesis at $\alpha = 0.05$ . C. we don't have enough evidence to accept or reject the null hypothesis at $\alpha = 0.05$ . D. we should increase the level of significance in order to reject the null hypothesis.

A caterer proposes to serve four main courses. For planning purposes, the caterer expects that the proportions of each that will be selected by customers will be: Selection Proportion Roast beef 0.50 Chicken 0.20 Fish 0.10 Vegeterian 0.20 Of the first 100 customers, 44 select roast beef, 24 select chicken, 13 select fish, and 19 select the vegetarian meal. Should the caterer revise the estimates? Use $\alpha$ = 0.05.