1. In a quality control test of parts manufactured at Dabco Corporation, an engineer sampled parts produced on the first, second, and third shifts. The research study was designed to determine if the population proportion of good parts was the same for all three shifts. The test to be used is a chi-square test of homogeneity using a significance level =0.05=0.05 . Sample data follow.

   Hint: Refer to Section 12.1 to help answer this question.

   Observed	Production Shift
   Quality	First	Second	Third
   Good	285	368	176
   Defective	15	32	24

   1. (1) State the hypotheses.

   2. (2) Use Excel to calculate the expected frequencies under the assumption H₀ is true and use these and the observed frequencies to calculate the test statistic.

   3. (1) Use Excel to find the p-value.

      Hint: Use CHISQ.DIST.RT

   4. (2) What is your hypothesis test decision?

      Hint: There are two possible decisions: reject H₀ in favour of H_a or fail to reject H₀.

   5. (2) What is your conclusion in the context of the application?

      Hint: The conclusion should relate back to the question.

   6. (4) If the conclusion in part (e) is that the population proportions are not all equal, use the Marascuilo Pairwise Comparison Procedure to determine which pairs of shifts differ in terms of quality. Which shift(s) need to improve the quality of parts produced?

2. Bura Research Group conducted a survey about church attendance. The survey respondents were asked about their church attendance and to indicate their age. Use the sample data to determine whether church attendance is independent of age using a chi-square test of independence using a significance level =0.05=0.05 . Sample data follow.

   Hint: Refer to Section 12.2 to help answer this question.

   Observed	Age
   Church Attendance	20 to 29	30 to 39	40 to 49	50 to 59
   Yes	31	63	94	72
   No	69	87	106	78

   1. (1) State the hypotheses.

   2. (2) Use Excel to calculate the expected frequencies under the assumption H₀ is true and use these and the observed frequencies to calculate the test statistic.

   3. (1) Use Excel to find the p-value.

      Hint: Use CHISQ.DIST.RT

   4. (2) What is your hypothesis test decision?

      Hint: There are two possible decisions: reject H₀ in favour of H_a or fail to reject H₀.

   5. (2) What is your conclusion in the context of the application?

      Hint: The conclusion should relate back to the question.

   6. (2) If the conclusion in part (e) is that church attendance is dependent on age, briefly describe how church attendance changes as individuals grow older.

3. Based on sales over a six-month period, the five top-selling compact cars are Chevy Cruze, Ford Focus, Hyundai Elantra, Honda Civic, and Toyota Corolla (Motor Trend, November 2, 2011). Based on total sales, the market shares for these five compact cars were Chevy Cruze 24%, Ford Focus 21%, Hyundai Elantra 20%, Honda Civic 18%, and Toyota Corolla 17%. A sample of 400 compact cars sales in Chicago showed the following number of vehicles sold: Chevy Cruze 108, Ford Focus 92, Hyundai Elantra 64, Honda Civic 84, and Toyota Corolla 52. Use a goodness of fit test with a significance level =0.05=0.05 to determine if the sample data indicate that the market shares for the five compact cars in Chicago are different than the market shares reported by Motor Trend.

   Hint: Refer to Section 12.3 to help answer this question.

   1. (1) State the hypotheses.

   2. (2) Use Excel to calculate the expected frequencies under the assumption H₀ is true and use these and the observed frequencies to calculate the test statistic.

   3. (1) Use Excel to find the p-value.

      Hint: Use CHISQ.DIST.RT

   4. (2) What is your hypothesis test decision?

      Hint: There are two possible decisions: reject H₀ in favour of H_a or fail to reject H₀.

   5. (2) What is your conclusion in the context of the application?

      Hint: The conclusion should relate back to the question.

   6. (2) If the conclusion in part (e) is that the market shares for the five compact cars in Chicago are different than the market shares reported by Motor Trend, briefly describe the market share differences in Chicago.

4. A salesperson makes four calls per day. A sample of 100 days gives the following frequencies of sales volumes.

   Number of Sales	Observed Frequency (days)
   0	30
   1	32
   2	25
   3	10
   4	3
   Total	100

   Records show sales are made to 30% of all sales calls. Assuming independent sales calls, the number of sales per day should follow a binomial probability distribution with n = 4 and p = 0.30. We can use a goodness of fit test with a significance level =0.05=0.05 to determine whether the number of sales per day follows a binomial probability distribution. Using the binomial probability function, we can calculate the probabilities for x = 0, 1, 2, 3, and 4 to be 0.2401, 0.4116, 0.2646, 0.0756, and 0.0081, respectively. We need to combine the categories for x = 3 or 4 to satisfy the requirement that the expected frequency is at least five, so the probabilities for x = 0, 1, 2, and 3 or 4 are 0.2401, 0.4116, 0.2646, and 0.0837, respectively. Given the overall sample size of 100, the expected frequencies for x = 0, 1, 2, and 3 or 4 are therefore 24.01, 41.16, 26.46, and 8.37, respectively.

   Hint: Refer to Section 12.3 to help answer this question.

   1. (1) State the hypotheses.

   2. (1) Use Excel to calculate the test statistic using the observed and expected frequencies given above.

   3. (1) Use Excel to find the p-value.

      Hint: Use CHISQ.DIST.RT. Since no parameters of the binomial probability distribution were estimated from the sample data, the degrees of freedom are k 1, where k is the number of categories.

   4. (2) What is your hypothesis test decision?

      Hint: There are two possible decisions: reject H₀ in favour of H_a or fail to reject H₀.

   5. (2) What is your conclusion in the context of the application?

      Hint: The conclusion should relate back to the question.

5. In a study conducted to investigate browsing activity by shoppers, each shopper was initially classified as a non-browser, light browser, or heavy browser. For each shopper, the study obtained a measure to determine how comfortable the shopper was in a store. Higher scores indicated greater comfort. The data collected are contained in the file Browsing.xlsx. Use a significance level =0.05=0.05 to test for differences among mean comfort levels for the three types of browsers.

   Hint: Refer to Sections 13.2 and 13.3 to help answer this question.

   1. (1) State the hypotheses.

   2. (2) Use XLSTAT to compute the F test statistic and the p-value.

      Hint: Select Modeling data > ANOVA. Select cells B1:B25 for Y / Dependent variables: Quantitative and select cells A1:A25 for X / Explanatory variables: Qualitative. The F test statistic and the p-value are in the last two columns of the Analysis of variance table in the XLSTAT output.

   3. (2) What is your hypothesis test decision?

      Hint: There are two possible decisions: reject H₀ in favour of H_a or fail to reject H₀.

   4. (2) What is your conclusion in the context of the application?

      Hint: The conclusion should relate back to the question.

   5. (4) If the conclusion in part (d) is that the population means are not all equal, use Fishers Least Significant Difference (LSD) Procedure to determine which pairs of browser types differ in terms of comfort.

      Hint: Calculate the sample means for each browser type and the absolute pairwise differences between the means. Then, calculate Fishers LSD critical value using the formula LSD=t/2MSE(1ni+1nj)LSD=t/2MSE(1ni+1nj) , where t/2t/2 is a critical value from a t-distribution with nTk=243=21nTk=243=21 degrees of freedom, MSE can be found in the Analysis of variance table in the XLSTAT output, and ni=nj=8ni=nj=8 . Then, see if any of the absolute pairwise differences between the means exceeds the calculated value of LSD.

   6. (4) Suppose we are not willing to assume that the populations have a normal distribution with the same standard deviations. Then, we should not use a parametric analysis of variance F-test, but we can instead use a nonparametric Kruskal-Wallis test for differences among median comfort levels for the three types of browsers. Use XLSTAT to compute the H test statistic and the p-value. Is the p-value similar to the corresponding value you obtained in part (b)?

      Hint: Select Nonparametric tests > Comparison of k samples (Kruskal-Wallis, Friedman, ). Make sure One column per variable is checked, select cells B1:B25 for Data, select cells A1:A25 for Sample identifiers, and select Kruskal-Wallis test. Under Options make sure Asymptotic p-value is selected. The H test statistic is labelled K (Observed value) in the XLSTAT output.