A study designed to see what type of incentive might be most effective in encouraging people to exercise. In the study, 281 overweight or obese people were assigned the goal to walk 7000 steps a day, and their activity was tracked for 100 days. The response variable is the number of days (out of 100) that each participant met the goal. The participants were randomly assigned to one of four different incentive groups: for each day they met the goal, participants in the first group got only praise, participants in the second group got entered into a lottery, and participants in the third group received cash (about \(\$ 1.50\) per day). In the fourth group, participants received all the money up front and lost money (about \(\$ 1.50\) per day) if they didn't meet the goal. (The overall financial effect for participants in the third and fourth conditions is identical, but the psychological effect between winning money and losing money is potentially quite different.) The summary statistics for the four conditions and overall are shown in Table 8.7.

(a) In the sample, which incentive had the most success in helping participants meet the goal? Which incentive had the least success?

(b) Do the conditions for using the F-distribution appear to be met?

(c) Test to see if the data provide evidence of a difference in mean success rates depending on the incentive used. Show all details of the test, including showing the ANOVA table.

Table 8.7 Number of days meeting an exercise goal

\begin{array}{lccc}
\hline Condition & \boldsymbol{n} & Mean & St.Dev. \\
\hline Praise & 70 & 30.0 & 32.0 \\
Lottery & 70 & 35.0 & 29.9 \\
Get money & 70 & 36.0 & 29.4 \\
Lose money & 71 & 45.0 & 30.1 \\
\hline Overall & 281 & 36.5 & 30.6865 \\
\hline \end{array}