1) To compare salaries of major league baseball players at four different positions on the field, five teams were randomly chosen and the salaries (in millions of dollars) obtained for the positions of shortstop, third baseman, right fielder, and catcher. The objective is to test whether there is a difference in salary between the four positions, teams were used because of likely salary differences by team. Here are the data:
Yankees SS 16.7 Yankees 3B 29.0 Yankees RF 6.5 Yankees C 0.5 Phillies SS 11.0 Phillies 3B 18.3 Phillies RF .75 Phillies C 5.0 Twins SS .5 Twins 3B .5 Twins RF .5 Twins C 23.0 Mariners SS 3.25 Mariners 3B .5 Mariners RF .5 Mariners C 1.5 Rangers SS 5.0 Rangers 3B 16.0 Rangers RF 10.3 Rangers C 7.5
a) What design has been used? Write the model equation.
b) Analyze the data to test the null hypothesis that salary does not differ
among
the four positions.
c) Test for interaction of positions and teams.
2) Refer again to the data from Table 6.13 in the text, but this time suppose
that temperature and
density are fixed effects and salinity is a random effect.
a) Write down the general expression ( like F = MSA/MSE, etc) that can be used
to test for the
main effects of density and salinity. Also show the table of expected mean
squares that you used
to arrive at these expressions.
b) Using the printout of sums of squares and mean squares, calculate the value
of
the F statistics in part a) by hand. Also calculate their degrees of freedom.
c) Compare the degrees of freedom for the denominator of the F test for density
under this model
versus a model where all three factors are fixed effects. Give a brief
explanation (in non-technical
terms) of why the denominator degrees of freedom are less for this model than
for the fixed
effects model.
3) Suppose that the study from question 1 also assessed
the effect of years of major league service
on salary. Suppose that data from the same 5 teams was used, with two players
for each of the four
positions, one with less than six years major league experience, and one with
more than six years
major league experience, giving a total of 40 player salaries. It is now of
interest to test
for the effects of position, years of service, and their interaction, while
adjusting for team salary differences.
a) Name this model and write down the model equation.
b) Fill out the 'Source' and 'Degrees of Freedom' columns for the ANOVA table
for this design.