Question

The intrinsic \(\mathrm{R}\) function cor.test \((\mathrm{x}, \mathrm{y})\) computes the estimate of \(\rho\) and the confidence interval in Exercise 9.7.6. Recall the baseball data which is in the file bb.rda. (a) Using the baseball data, determine the estimate and the confidence interval for the correlation coefficient between height and weight for professional baseball players. (b) Separate the pitchers and hitters and for each obtain the estimate and confidence for the correlation coefficient between height and weight. Do they differ significantly? (c) Argue that the difference in the estimates of the correlation coefficients is the mle of \(\rho_{1}-\rho_{2}\) for two independent samples, as in Part (b).

Short answer

The estimate and confidence interval for the correlation coefficient between height and weight for professional baseball players can be obtained by using the R function cor.test on the relevant columns of the dataset. When split into groups of pitchers and hitters, these statistics can differ. The difference between the two correlation coefficients can be deemed as the maximum likelihood estimates of the difference in correlations for the two independent samples, assuming certain statistical properties and assumptions. Note that the concrete values can only be given when the data analysis is completed.

Step-by-step solution

Load the Data and Perform Correlation Test

First, load the baseball data into the R environment. Then, use the cor.test function to compute the correlation and confidence interval between height and weight. The syntax is: cor.test(data$height, data$weight).

Separate the Data for Pitchers and Hitters

Next, the data needs to be separated into two subsets: one for pitchers and one for hitters. This is done using a filtering operation on the player’s role. The correlation and confidence interval for each group is then computed with cor.test just as in Step 1.

Compare the Correlation Estimates

With the correlation estimates for each group, compare them to determine if they significantly differ. This can be performed through a hypothesis test where the null hypothesis states no significant difference exists.

Argument Regarding Difference in Correlation Estimates

Analysis must be done to argue that the difference in correlation estimates can be interpreted as the MLE of \(\rho_{1}-\rho_{2}\) for the two independent samples. It may involve underlying statistical principles and theories such as Maximum Likelihood Estimation and properties of correlation coefficients.