A researcher wishes to see whether there is any difference in the weight gains of athletes following one of three special diets. Athletes are randomly assigned to three groups and placed on the diet for 6 weeks. The weight gains (in pounds) are given. Assume weight gains are normally distributed and the variances are equal. At a $0.05$ significance level, can the researcher conclude that there is a difference in the diets?

- Diet A: 3, 6, 7, 4
- Diet B: 10, 12, 11, 14, 8, 6
- Diet C: 8, 3, 2, 5

Three classes of ten students each were taught using the following methodologies: traditional, inquiry-oriented and a mixture of both. At the end of the term, the students were tested, their scores were recorded and this yielded the following partial ANOVA table. Assume distributions are normal and variances are equal. $$\begin{array}{l|c|c|c|c|} & SS & df & MS & F\\\hline \textrm{Between} & 136 & & & \\\hline \textrm{Within} & 432 & & & \\\hline \textrm{Total} & & & & \\\hline \end{array}$$

Complete the above table and use it to test the claim that there is a difference in the mean score of the students taught with the three different methodologies. Use $\alpha = 0.05$.

Suppose that the students taught by the traditional method had a mean score of $80$ and the ones taught by the inquiry-oriented method had a mean score of $89$. Do a Scheffe test to determine whether the difference in means is significant when we compare the traditional method with the inquiry-based one.

The mean for the mixed approach is $85$. Test statistics are $F_S = 7.8125$ for between traditional and mixed methods and $F_S = 5.000$ for between inquiry oriented and mixed methods. Give an overall inference for the teaching methods.