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

The chi-squared test of a contingency table is used to determine if there is enough evidence to infer that two nominal variables are related, and to infer that differences exist among two or more populations of nominal variables.

Consider a multinomial experiment involving 100 trials and 4 categories (cells). The observed frequencies resulting from the experiment are shown in the accompanying table. Category 1 2 3 4 Frequency 18 30 25 27 Use the 5% significance level to test the hypotheses. H₀ : p₁ = 0.25, p₂ = 0.30, p₃ = 0.20, p₄ = 0.25. H₁ ;At least two proportions differ from their specified values.

Which of the following are the degrees of freedom used in a Chi-squared test of independence between gender and mode of transport to university, if the mode of transport choices are public transport, car, bicycle or other. A. 8 B. 5 C. 4 D. 3

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 4 11 14 12 9 The null hypothesis to be tested is as follows. $H _ { 0 } : p _ { 1 } = 0.10$ , $p _ { 2 } = 0.25,$ $p _ { 3 } = 0.30,$ $p _ { 4 } = 0.20,$ $p _ { 5 } = 0.15$ $p _ { 5 } = 0.15$ .

Five brands of orange juice are displayed side by side in several supermarkets in a large city. 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 this city, can you conclude at the 5% significance level that there is a preferred brand of orange juice?

The chi-squared test of a contingency table is based upon: A. two nominal variables. B. two numerical variables. C. three or more nominal variables. D. three or more numerical variables.

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 Can you infer at the 5% significance level that the proportional support by the Adelaide employees for the three issues has changed since the pamphlet was circulated?

Whenever the expected frequency of a cell is less than 5, one remedy for this condition is to increase the size of the sample.

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 $p _ { 1 } - p _ { 2 }$ , or the chi-squared test of a contingency table. (Squaring the value of the z-statistic yields the value of the $\chi ^ { 2 }$ -statistic.)

Last year, Brand A mobile android phones had 45% of the market, Brand B had 35% and Brand C had 20%. This year the makers of Brand C launched a heavy advertising campaign. A random sample of mobile phone stores shows that of the 10 000 android mobile phones sold, 4350 were Brand A, 3450 were Brand B, and 2200 were Brand C. Has the market changed? Test at $\alpha =$ 0.05.

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 Can you infer at the 5% significance level that the proportional support by the employees at both plants for the issues has changed since the pamphlet was circulated?

Which of the following should be used if we want to conduct a two-tail test of a population proportion? A. z -test of a popul ation proportion. B. Chi-squared test of a binomial experiment since = C. the chi-squared test of a contingency table. D. z -test of a popul ation proportion and Chi-squared test of a binomial experiment since =. E. Chi-squared test of a binomial experiment since = and the chi-squared test of a contingency table.

A biology professor claimed that the proportions of grades in his classes are the same. A sample of 100 students showed the following frequencies. Grade A B C D F Frequency 14 23 27 26 10 Determine the rejection region at the 5% significance level.

A sport preference poll showed the following data for men and women: Fanourite Sport Gender Rugby B asketball Football Golf Tennis Male 24 17 30 18 22 Female 21 20 22 12 28 Using the 5% level of significance, test to determine whether sport preferences depend on gender.

Which of the following best describes the sampling distribution of the test statistic for a goodness-of-fit test with k categories? A. Student $t$ -distribution with $k-1$ degrees of freedom. B. Normal distribution. C. Chi-squared distribution with $k-1$ degrees of freedom. D. Approximately Chi-squared distribution with $k-1$ degrees of freedom.

The area to the right of a chi-squared value is 0.01. For 8 degrees of freedom, the table value is 20.0902.

The number of degrees of freedom in a test of a contingency table with 7 rows and 5 columns is: A. 24. B. 35. C. 30. D. 28.

In 2003, the student body of a University in NSW consisted of 30% first-years, 25% second-years, 27% third-years, and 18% fourth-years. A sample of 400 students taken from the 2004 student body showed that there are 138 first-years, 88 second-years, 94 third-years, and 80 fourth-years. Test with 5% significance level to determine whether the student body proportions have changed.

Which of the following best describes the rejection region in a chi-squared test of independence? A. $\quad \chi^{2}=\chi^{2} \alpha(r-2)(c-2)$ B. $\left.\chi^{2}=\chi^{2} 1-\alpha,(r-1) c-1\right)$ C. $\chi^{2}=\chi_{\alpha}^{2}(r-1)(c-1)$ D. $\chi^{2}<\chi_{\alpha}^{2}(r-1)(c-1)$ E. None of these choices are correct.

Which of the following best describes a Chi-squared goodness-of-fit test? A. Two tailed test B. Left tailed test C. Right tailed test D. All of these choices are correct.