1. In baseball, are players at some positions better hitters than at other positions? To address this question a sample of major league baseball teams was drawn in late August and the batting averages of starting players was recorded for four positions (first base, shortstop, center fielder, and catcher). The data are given below. Please use R and comment for each problem and sub-problem.

data

team name position BA

Boston Bogaerts SS .298

Boston Hernandez CF .258

Boston Dalbec FB .236

Boston Vazquez C .255

Baltimore Mullins CF .306

Baltimore Severino C .239

Baltimore Galvis SS .249

Baltimore Mountcastle FB .259

Cleveland Rosario SS .284

Cleveland Hedges C .181

Cleveland Zimmer CF .251

Cleveland Bradley FB .219

Minnesota Polanco SS .274

Minnesota Sano FB .218

Minnesota Kepler CF .209

Minnesota Jeffers C .207

LosAngeles Iglesias SS .259

LosAngeles Walsh FB .260

LosAngeles Lagares CF .237

LosAngeles Stassi C .269

a. Use a boxplot to visualize potential effects of position on batting average.

b. Conduct an analysis of variance (using a one-way ANOVA) to test the null hypotheses of equality of batting averages by position (show the ANOVA table, including sums of squares, mean squares, F statistic, and P value. Be sure to include a line for total SS and df). b. Examine a residual by predicted plot and a normal plot of the residuals to assess model assumptions. c. Use the Box-Cox procedure to find a recommended transformation of the response. Would you reject the null hypothesis that the power parameter = 1? d. Select a transformation of the batting averages and repeat the ANOVA on the transformed data. Do the results change? Which analysis is more appropriate? 2. A future study is being planned to compare the batting averages at these positions. If we used our current data (problem 1, on the original scale) to estimate the variance of the batting averages, how many teams do we need to sample to detect a difference if the true mean batting averages are .22, .26, .23, and .30, at alpha = .01 for a power of 90%?

Data here:

team name position BA

Boston Bogaerts SS .298

Boston Hernandez CF .258

Boston Dalbec FB .236

Boston Vazquez C .255

Baltimore Mullins CF .306

Baltimore Severino C .239

Baltimore Galvis SS .249

Baltimore Mountcastle FB .259

Cleveland Rosario SS .284

Cleveland Hedges C .181

Cleveland Zimmer CF .251

Cleveland Bradley FB .219

Minnesota Polanco SS .274

Minnesota Sano FB .218

Minnesota Kepler CF .209

Minnesota Jeffers C .207

LosAngeles Iglesias SS .259

LosAngeles Walsh FB .260

LosAngeles Lagares CF .237

LosAngeles Stassi C .269