16.21 Bootstrapping a Normal data set. The following data are “really Normal.” They are an SRS from the standard Normal distribution N(0, 1), produced by a software Normal random number generator.

0.01 −0.04 −1.02 −0.13 −0.36 −0.03 −1.88 0.34 −0.00 1.21

−0.02 −1.01 0.58 0.92 −1.38 −0.47 −0.80 0.90 −1.16 0.11 0.23 2.40 0.08 −0.03 0.75 2.29 −1.11 −2.23 1.23 1.56

−0.52 0.42 −0.31 0.56 2.69 1.09 0.10 −0.92 −0.07 −1.76 0.30 −0.53 1.47 0.45 0.41 0.54 0.08 0.32 −1.35 −2.42 0.34 0.51 2.47 2.99 −1.56 1.27 1.55 0.80 −0.59 0.89

−2.36 1.27 −1.11 0.56 −1.12 0.25 0.29 0.99 0.10 0.30 0.05 1.44 −2.46 0.91 0.51 0.48 0.02 −0.54

(a) Make a histogram and Normal quantile plot.

Do the data appear to be “really Normal”? From the histogram, does the N(0, 1) distribution appear to describe the data well? Why?

(b) Bootstrap the mean. Why do your bootstrap results suggest that t confidence intervals are appropriate?

(c) Give both the bootstrap and the formula-based standard errors for x. Give both the bootstrap and usual t 95% confidence intervals for the population mean μ.