Example 8.2 (Bootstrap estimate of standard error). The law school data set law in the bootstrap [286] package is from Efron and Tibshirani [91]. The data frame contains LSAT (average score on law school admission test score) and GPA (average undergraduate grade point average) for 15 law schools. LSAT 576 635 558 578 666 580 555 661 651 605 653 575 545 572 594 GPA 339 330 281 303 344 307 300 343 336 313 312 274 276 288 296 This data set is a random sample from the universe of 82 law schools in law82 (bootstrap). Estimate the correlation between LSAT and GPA scores, and compute the bootstrap estimate of the standard error of the sample correla- tion. 1. For each bootstrap replicate, indexed 6 = 1,..., B: (a) Generate sample () = xi,..., by sampling with replacement from the observed sample 21, ...,xn. 216 Statistical Computing with R (b) Compute the 6th replicate () from the 6th bootstrap sample, where Ô is the sample correlation R between (LSAT, GPA). 2. The bootstrap estimate of se(R) is the sample standard deviation of the replicates (1),..., (B) = R(1), library(bootstrap) R(B). # for the law data print (cor (law$LSAT, law$GPA)) [1] 0.7763745 print (cor (law82$LSAT, law82$GPA)) [1] 0.7599979 The sample correlation is R = 0.7763745. The correlation for the universe of 82 law schools is R = 0.7599979. Use bootstrap to estimate the standard error of the correlation statistic computed from the sample of scores in law. # set up the bootstrap B <- 200 n < nrow(law) R < numeric(B) #number of replicates #sample size #storage for replicates #bootstrap estimate of standard error of R for (b in 1:B) { } #randomly select the indices 1 < sample (1:n, size n, replace=TRUE) LSAT < law$LSAT [i] GPA < law$GPA [i] R[b] print(se.R <- sd (R)) [1] 0.1358393 > hist (R, prob = TRUE) #i is a vector of indices The bootstrap estimate of se(R) is 0.1358393. The normal theory estimate for standard error of R is 0.115. The jackknife-after-bootstrap method of es- timating se(se()) is covered in Section 9.1. The histogram of the replicates of R is shown in Figure 8.1.

How is an array stored in main memory? How is a linked list stored in main memory? What are their comparative advantages and disadvantages? Give examples of data that would be best stored as an array and as a linked list.

What happens to the existing rows of a table if the DEFAULT value of a column is changed?

(For thought) a. A token of a computer language is any sequence of characters, with no intervening characters or white space, that taken as a unit has a unique meaning. Using this definition of a token, determine whether escape sequences, function names, and the keywords listed in Table 2.1 are tokens of the C++ language. b. Discuss whether adding white space to a message alters the message and whether messages can be considered tokens of C++. c. Using the definition of a token in Exercise 4a, determine whether the following statement is true: “Except for tokens of the language, C++ ignores all white space.”