14A InClass

Question 1

1) A researcher is interested in comparing the average weight loss over a 12-week  period between individuals randomly assigned to one of four groups:

  • Diet only
  • Diet and assorted cardio routines four days a week
  • Diet and cycling activities four days a week
  • Diet and combination of strength training and cardio activities four days a week Explain why a one-way ANOVA should be considered for this situation.

Question 2

2) Write the null hypothesis for the weight loss situation. Be sure to define each  parameter.

Question 3

3) Write the alternative hypothesis for this situation.

Question 4

4) Before conducting a formal hypothesis test, the researcher would like to visually  assess the data. The following boxplots and dotplots compare the distributions of  each group. The sample means for each group, as well as the grand mean (e.g.,  17.1 pounds (lb) is the mean of all the data values), are provided.

Side-by-side box plots with “Weight Loss (lb)” on the horizontal axis and “Diet,” “Diet + Cardio,” “Diet + Cycling,” and “Diet + Strength Training + Cardio” on the vertical axis. There is a dash on the horizontal axis labeled “X bar = 17.1 lb.” For Diet, the low point is at approximately 11.5 and the high point is at approximately 14. The low end of the box is at approximately 12.25, the high end of the box is at approximately 13.75, and the middle line is at approximately 12.5. There is a label reading “X bar sub 1 = 12.8 lb.” For Diet + Cardio, the low point is at approximately 15 and the high point is at approximately 17.5. The low end of the box is at approximately 15.25, the high end of the box is at approximately 17, and the middle line is at approximately 16.25. There is a label reading “X bar sub 2 = 16.2 lb.” For Diet + Cycling, the low point is at approximately 17.25 and the high point is at approximately 19.5. The low end of the box is at approximately 17.5, the high end is at approximately 19.25, and the middle line is at approximately 18. There is a label reading “X bar sub 3 = 18.3 lb.” For Diet + Strength Training + Cardio, the low point is at approximately 19.75 and the high point is at approximately 22. The low end of the box is at approximately 20, the high end is at approximately 21.75, and the middle line is at approximately 21.7. There is also a label reading “x bar sub 4 equals 21.1 lb.”

Side-by-side dot plots with “Weight Loss (lb)” on the horizontal axis. The first plot is labeled “Diet” and has dots at approximately 11.5, 12.4, 12.6, 13.8, and 14.2. The second plot is labeled “Diet + Cardio” and shows dots at approximately 14.9, 15.3, 16.3, 17, and 17.5. The third plot is labeled “Diet + Cycling” and shows dots at approximately 17.3, 17.4, 17.9, 19.3, and 19.5. The last plot is labeled “Diet + Strength Training + Cardio” and has dots at approximately 19.8, 19.9, 21.6, 21.6, and 21.8.

Based on these graphs alone, does it appear there is visual evidence that the diets differ in average weight loss? That is, is there visual evidence to reject the null  hypothesis in favor of the alternative hypothesis? Explain.

Question 5

5) Suppose the results are different than those presented in Question 4. An alternative  result is reflected in the following boxplot and dotplot.

Side-by-side box plots with “Weight Loss (lb)” on the horizontal axis and “Diet,” “Diet + Cardio,” “Diet + Cycling,” and “Diet + Strength Training + Cardio” on the vertical axis. There is a dash on the horizontal axis labeled “X bar = 17.1 lb.” For Diet, the low point is at approximately 7.5 and the high point is at approximately 19.5. The low end of the box is at approximately 10, the high end of the box is at approximately 15, and the middle line is at approximately 12.5. There is a label reading “X bar sub 1 = 12.8 lb.” For Diet + Cardio, the low point is at approximately 9.5 and the high point is at approximately 22. The low end of the box is at approximately 12, the high end of the box is at approximately 20, and the middle line is at approximately 18. There is a label reading “X bar sub 2 = 16.2 lb.” For Diet + Cycling, the low point is at approximately 10.5 and the high point is at approximately 24. The low end of the box is at approximately 14, the high end is at approximately 23.5, and the middle line is at approximately 19.5. There is a label reading “X bar sub 3 = 18.3 lb.” For Diet + Strength Training + Cardio, the low point is at approximately 13 and the high point is at approximately 26.5. The low end of the box is at approximately 18, the high end is at approximately 25, and the middle line is at approximately 22.5. There is also a label reading “x bar sub 4 equals 21.1 lb.”

Side-by-side dot plots labeled “Weight Loss (lb)” on the horizontal axis. The first plot is labeled “Diet” and shows dots at approximately 6, 12, 13, 15, and 20. The second plot is labeled “Diet + Cardio” and has dots at approximately 9, 12, 17, 19, and 22. The third plot is labeled “Diet + Cycling” and has dots at approximately 10.5, 14.5, 19.5, 23.5, and 24. The last plot is labeled “Diet + Strength Training + Cardio” and has points at approximately 13, 20, 22, 25, and 26.5.

Based on these graphs alone, does it appear there is visual evidence that the diets  differ in average weight loss? That is, is there visual evidence to reject the null  hypothesis in favor of the alternative hypothesis? Explain.

Question 6

6) Compare and contrast the graphical displays in Questions 4 and 5.

Part A: How are they similar and how are they different?

Part B: Which one appeared to provide more convincing evidence? What might the differences suggest about making a conclusion about the null hypothesis in a  one-way ANOVA?

The test statistic and P-value are calculated by considering the ratio of variation within  each of the groups to the variation between each of the groups. That is, when the  variation between each of the groups is significantly greater than the variation within each of the groups, we will reject the null hypothesis and conclude that at least two of  the means are different (similar to what we saw in Question 4).

However, when there is a significant amount of variation within groups, relative to the  variation between groups (like in Question 5), we will have less evidence of a difference  and may fail to reject the null hypothesis.

The statistic measuring the variation within the groups is the error sum of squares.  This is calculated by summing the variation within each of the groups. The variation  within each of the groups is visualized in the boxplot by the size of the box and in the  dotplot as the spread of the dots within each group.

A statistic measuring the variation between the groups is the group sum of squares. This is calculated by summing the variation between each of the group means and the  grand mean (i.e., the mean of all the data values).

Question 7

7) Given the information about the error sum of squares, which data (Question 4 or 5)  do you think would have the greater error sum of squares value? Explain.

Question 8

8) Given the information about the group sum of squares, which data (Question 4 or 5)  do you think would have the greater group sum of squares value? Explain.