HW 64 ? C Nick My Home Courses MindTap – Cengage Learning ng.cengage.com/static/nb/ui/evo/index.html?deploymentid=5991401881038963484592120786&ISBN=9780357131466&id=1798221981&snapshotid=3481785& CENGAGE MINDTAP Chapter 13 Assignment Assignment: Chapter 13 Assignment ? Q Search this course 0 X Assignment Save Submit Assignment for Grading ? Catalog and Study Tools Questions Exercise 13.05 Algo (Analysis of Variance and the Completely Randomized Design) Question 2 of 8 A-Z Rental Options College Success Tips Hint(s) Check My Work (3 remaining) Career Success Tips In a completely randomized design, 12 experimental units were used for the first treatment, 15 for the second treatment, and 20 for the third treatment. Complete the following analysis of variance (to 2 decimals, if necessary). If the answer is zero enter “0”. Source of Variation Sum of Squares Degrees of Freedom Mean Square F p-value Help Give Feedback Treatments 1000 Error 1800 Total At a 0.05 level of significance, is there a significant difference between the treatments? The
-value is Select your answer What is your conclusion? Select your answer Exercise 13.05 Algo (Analysis of Variance and the Completely Randomized Design) ? Hint(s) Check My Work (3 remaining) Question 2 of 8 Save Submit Assignment for Grading 88°F Sunny 3:34 PM 7/13/2023
The Correct Answer and Explanation is:
Here are the completed values for the Analysis of Variance (ANOVA) table and the conclusion based on the hypothesis test.
Completed ANOVA Table:
| Source of Variation | Sum of Squares | Degrees of Freedom | Mean Square | F | p-value |
| Treatments | 1000 | 2 | 500.00 | 27.50 | 0.00 |
| Error | 800 | 44 | 18.18 | ||
| Total | 1800 | 46 |
Hypothesis Test:
- The p-value is: < 0.01
- What is your conclusion?: Reject the null hypothesis. There is a significant difference between the treatments.
Explanation
This problem requires completing an Analysis of Variance (ANOVA) table to determine if there is a statistically significant difference between the means of three treatments.
First, we determine the degrees of freedom (df). The number of treatments (k) is 3, and the total number of experimental units (n) is 12 + 15 + 20 = 47.
- df for Treatments: This is calculated as k – 1 = 3 – 1 = 2.
- df for Total: This is calculated as n – 1 = 47 – 1 = 46.
- df for Error: This is the difference between the total df and the treatment df: 46 – 2 = 44.
Next, we calculate the remaining Sum of Squares (SS) and the Mean Squares (MS).
- Sum of Squares for Error (SSE): This is the difference between the Total SS and the Treatment SS: 1800 – 1000 = 800.
- Mean Square for Treatments (MST): This is the Treatment SS divided by its df: 1000 / 2 = 500.00.
- Mean Square for Error (MSE): This is the Error SS divided by its df: 800 / 44 ≈ 18.18.
With the Mean Squares calculated, we can find the F-statistic.
- F-statistic: This is the ratio of the Mean Square for Treatments to the Mean Square for Error: F = MST / MSE = 500 / 18.18 ≈ 27.50.
Finally, we use the F-statistic (27.50) and the degrees of freedom (df1=2, df2=44) to find the p-value and make a conclusion at a 0.05 significance level. The p-value is the probability of obtaining an F-statistic at least as extreme as ours if the null hypothesis (that all treatment means are equal) were true. The calculated p-value is extremely small (approximately 0.0000000115), which rounds to 0.00.
Since our p-value (≈0.00) is less than the significance level (α = 0.05), we reject the null hypothesis. This indicates that there is strong statistical evidence to conclude that not all treatment means are equal; a significant difference exists between the treatments
