Chi-square tests #1 - Questions and Answers

Question 1 Which of the following accurately describes the observed and expected frequencies in a chi-square test? a. Both the observed and expected frequencies will always be whole numbers. b. The observed frequencies are always whole numbers but the expected frequencies may be decimals or fractions. c. The expected frequencies are always whole numbers but the observed frequencies may be decimals or fractions. d. Both the observed and expected frequencies may be decimals or fractions. What does it mean to obtain a negative value for the chi-square statistic? a. The observed frequencies tend to be larger than the expected frequencies. b. The expected frequencies tend to be larger than the observed frequencies. c. There are large discrepancies between the observed and expected frequencies for most categories. d. The chi-square statistic can never be negative. Answer In the chi-square test, observed frequencies are given to us, and we calculate the expected frequencies from the observed frequencies using chi-square test. The expected frequencies are always whole numbers as we count the number of occurrences, While the observed frequencies are calculated, so they can be whole numbers or may have decimals. Hence option b is correct. As the chi-squared statistic is calculated using the formula Σ; (0;-;)² Hence it cannot be negative. option d is correct. E Question 2 State the null hypothesis for a chi-square test for independence. Answer Chi-square independence test is used to test whether there is relation between two- categorical variables or not in the study. It is a statistical method that tests whether at each level of the categorical variable the observed frequencies are similar to the expected frequencies or not. The null hypothesis for the chi-square test for independence is that, the two categorical variables are not related or independent or the observed frequency is equal to the expected frequency for two categorical variables. Question 3 A clothing manufacturer wants to determine whether customers prefer any specific color over other colors in shirts. He selects a random sample of 100 shirts sold and notes the color. The table below shows the result. At x= 0.05, is there a color preference? Color Number Sold White 43 Blue 22 Black 16 Red 9 Yellow 10 What test statistics will be used for this problem? Chi-square test of Independence Spearman rho Chi-square test of goodness fit Pearson ANOVA Answer Test is that whether there is a color preference in shirts. To test that whether there is a color preference in shirts, the appropriate test statistic to used is Chi-square test of goodness of fit. So the answer is 3. Chi-square test of goodness of fit. Question 4 A controversial issue in the sport of professional soccer is the use of instant replay for making difficult goal line decisions. Each person in a representative sample of 102 players, fans, coaches, and officials was asked his or her opinion about the use of instant replay for goal-line decisions. The data are summarized in the two-way frequency table below. Opinion Favor Use Oppose Use Category Players 22 2 Fans 18 6 Coaches 15 26 Officials 3 10 In testing to see whether opinion with respect to the use of instant replay is independent of the category of the person interviewed, a chi-square test statistic of 27.99 and a p-value less than 0.001 were calculated. Which of the following statements is correct? a. The number of degrees of freedom for the test is 8 -1 = 7. b. The chi-square test should not have been used because two of the counts in the table are less than 5. c. The null hypothesis states that there is an association between category and opinion about the use of instant replay, and the small p-value suggests that the null hypothesis should be rejected. d. The small p-value suggests that there is evidence of an association between category and opinion about the use of instant replay. e. The chi-square test shows that fans favor the use of instant replay. Answer 1 Given information: The two-way frequency table is provided as: Category Players Opinion Favor Use Oppose Use 22 2 Fans 18 6 Coaches 15 26 Officials 3 10 The chi-square test statistic is obtained as /x² = 27.99 with a p-value of less than 0.001. It is required to choose the correct one out of the provided statements regarding this problem. The correct statement is D; that is, the small p-value suggests that there is evidence of an association between category and opinion about the use of instant replay. Reason: The null and the alternative hypothesis for this problem are: H₀: There is no relation between category and opinion about the use of instant replay. Hₐ: There is an association between category and opinion about the use of instant replay. Since the p-value is less than 0.001, a very small p-value, also less than any level of significance (0.10, 0.05, or 0.01); therefore, the null hypothesis gets rejected. Hence, it indicates that the small p-value suggests that there is evidence of an association between category and opinion about the use of instant replay. Answer 2 Reasons for the wrong options: a. The number of degrees of freedom for the test is obtained as: df = (number of rows − 1)×(number of columns − 1) = (4 − 1)×(2 − 1) = 3 Hence, statement a is wrong. b. The chi-square test should not have been used because two of the counts in the table are less than 5. This statement is also wrong since the assumption of the Chi-square test is for the expected counts of each cell to be greater than 5, not for the observed counts of each cell. c. The null hypothesis states that there is an association between category and opinion about the use of instant replay, and the small p-value suggests that the null hypothesis should be rejected. This statement is also wrong because of the framing of the wrong null hypothesis. The correct null hypothesis states that there is no association between category and opinion about the use of instant replay. e. The chi-square test shows that fans favor the use of instant replay. This statement is also wrong since the chi-square test for this problem will only indicate whether there is an association between category and opinion about the use of instant replay or not. It will not provide any result regarding the statement: fans favor the use of instant replay. Question 5 A study was conducted to investigate a relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or not injured in a crash. There were five colors of helmets. For each color the number of persons wearing that color who were not injured in a crash and the number of people who were injured in a crash was recorded. What test would be most appropriate? Name of Test - ? Answer The chi-square test of independence is known as the method that is used to define whether there is a significant relationship between two variables. To test the claim about the relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or not injured in a crash, the chi-square test of independence will be the most appropriate test. Name of Test - chi-square test of independence 15.5s