12.27 A study has been carried out to compare the United Way contributions made by clerical workers from three local corporations. A sample of clerical workers has been randomly selected from each firm, and the dollar amounts of their contributions are as follows. (Use data file XR12027.)
Firm 1 Firm 2 Firm 3
199 108 162
236 104 86
167 153 160
263 218 135
254 210 207
96 201
a. What are the null and alternative hypotheses for this test?
b. Use ANOVA and the 0.05 level of significance in testing the null hypothesis identified in part (a).
a. The null hypothesis (H0) for this test would be that there is no significant difference in the mean United Way contributions between the clerical workers from the three firms. The alternative hypothesis (Ha) would be that there is a significant difference in the mean contributions.
b. To test the null hypothesis, we can use ANOVA (Analysis of Variance). ANOVA allows us to compare more than two groups and determine if there are significant differences between them.
Running the ANOVA test on the given data, with a significance level of 0.05, would provide us with the p-value. If the p-value is less than 0.05, we would reject the null hypothesis and conclude that there is a significant difference in the mean contributions between the firms. If the p-value is greater than 0.05, we would fail to reject the null hypothesis and conclude that there is no significant difference.
a. The null hypothesis for this test is that there is no significant difference in the United Way contributions made by clerical workers from the three local corporations. The alternative hypothesis is that there is a significant difference in the contributions.
b. To test the null hypothesis, we will use ANOVA (Analysis of Variance). ANOVA tests whether the means of three or more groups are equal. We will use the 0.05 level of significance.
Step-by-step:
1. Collect the data and organize it in a table:
Firm 1: 199, 236, 167, 263, 254
Firm 2: 108, 104, 153, 218, 210, 96
Firm 3: 162, 86, 160, 135, 207, 201
2. Calculate the sum of squared deviations for each firm:
- For Firm 1:
Mean = (199 + 236 + 167 + 263 + 254) / 5 = 223.8
Sum of squared deviations = (199 - 223.8)^2 + (236 - 223.8)^2 + (167 - 223.8)^2 + (263 - 223.8)^2 + (254 - 223.8)^2 = 1697.6
- For Firm 2:
Mean = (108 + 104 + 153 + 218 + 210 + 96) / 6 = 147.5
Sum of squared deviations = (108 - 147.5)^2 + (104 - 147.5)^2 + (153 - 147.5)^2 + (218 - 147.5)^2 + (210 - 147.5)^2 + (96 - 147.5)^2 = 7467.5
- For Firm 3:
Mean = (162 + 86 + 160 + 135 + 207 + 201) / 6 = 149.8
Sum of squared deviations = (162 - 149.8)^2 + (86 - 149.8)^2 + (160 - 149.8)^2 + (135 - 149.8)^2 + (207 - 149.8)^2 + (201 - 149.8)^2 = 6742.8
3. Calculate the total sum of squared deviations:
Total sum of squared deviations = 1697.6 + 7467.5 + 6742.8 = 15907.9
4. Calculate the between-group sum of squared deviations:
Between-group sum of squared deviations = [(5 * (223.8 - 149.8)^2) + (6 * (147.5 - 149.8)^2) + (6 * (149.8 - 149.8)^2)] = 1000.1
5. Calculate the within-group sum of squared deviations:
Within-group sum of squared deviations = total sum of squared deviations - between-group sum of squared deviations = 15907.9 - 1000.1 = 14907.8
6. Calculate the degrees of freedom:
- Degrees of freedom between groups = number of groups - 1 = 3 - 1 = 2
- Degrees of freedom within groups = total number of observations - number of groups = 17 - 3 = 14
- Total degrees of freedom = degrees of freedom between groups + degrees of freedom within groups = 2 + 14 = 16
7. Calculate the mean squared deviations:
Mean squared deviations between groups = between-group sum of squared deviations / degrees of freedom between groups = 1000.1 / 2 = 500.05
Mean squared deviations within groups = within-group sum of squared deviations / degrees of freedom within groups = 14907.8 / 14 = 1064.84
8. Calculate the F-statistic:
F-statistic = mean squared deviations between groups / mean squared deviations within groups = 500.05 / 1064.84 = 0.47
9. Look up the critical value of F in the F-distribution table for the given level of significance (0.05) and degrees of freedom (2 and 14). The critical value is 3.682.
10. Compare the calculated F-statistic with the critical value:
Since the calculated F-statistic (0.47) is less than the critical value (3.682), we fail to reject the null hypothesis. There is not enough evidence to conclude that there is a significant difference in the United Way contributions made by clerical workers from the three local corporations.
Conclusion: Based on the ANOVA test, we do not have enough evidence to support the alternative hypothesis that there is a significant difference in the contributions.
a. The null hypothesis for this test would be that there is no difference in the mean United Way contributions between the three firms. The alternative hypothesis would be that there is a difference in the mean contributions between at least one pair of firms.
b. To conduct ANOVA and test the null hypothesis, we compare the variances between the groups (sample firms) with the variances within the groups (variation within each firm). Here are the steps to perform ANOVA:
1. Calculate the sum of squares (SS) for each group (firm). This can be done by finding the sum of squared deviations from the group mean for each observation.
2. Calculate the sum of squares within the groups (SSw). This can be done by finding the sum of squared deviations from the individual group mean for each observation, and then summing them across all groups.
3. Calculate the sum of squares between the groups (SSb). This can be done by subtracting the SSw from the total sum of squares (SStotal).
4. Calculate the degrees of freedom (df) for SSb and SSw. The degrees of freedom for SSb would be (number of groups - 1), and the degrees of freedom for SSw would be (total number of observations - number of groups).
5. Calculate the mean squares (MS) for SSb and SSw by dividing the sum of squares by the corresponding degrees of freedom.
6. Calculate the F-statistic by dividing the MSb by MSw.
7. Determine the critical value of the F-statistic for the desired level of significance (0.05 in this case) and the degrees of freedom for SSb and SSw.
8. Compare the calculated F-statistic with the critical value. If the calculated F-statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
In this case, you can use statistical software or a spreadsheet program to perform the ANOVA calculations. Enter the data for each firm into separate columns, and then run the ANOVA test. The software or spreadsheet program will provide you with the F-statistic and its corresponding p-value.
By comparing the p-value with the significance level (0.05), you can determine whether to reject or fail to reject the null hypothesis.
Remember, ANOVA assumes certain conditions are met, such as normality and homogeneity of variances. It's also important to consider the limitations of the test and interpret the results in the context of the study.