Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

312 Some Elementary Statistical Inferences

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Frequency

z*

Figure 4.9.3: Histogram of the 3000 bootstrapz∗s discussed in Example 4.9.3.

The bootstrapp-value is the area (relative to the total area) under the histogram

and to the right of the 95.27.

4.9.3. The mean of these 3000 values is 89.96, which is quite close to 90. Of these
3000 values, 563 exceededx=95.27; hence, thep-value of the bootstrap test is
0 .188. The fraction of the total area that is to the right of 95.27 in Figure 4.9.3 is
approximately equal to 0.188. Such a highp-value is usually deemed nonsignificant;
hence, the null hypothesis would not be rejected.
For comparison, the reader is asked to show in Exercise 4.9.12 that the value of
the one-samplet-test ist=0.84, which has ap-value of 0.20. A test based on the
median is discussed in Exercise 4.9.13.

EXERCISES

4.9.1.Consider the sulfur dioxide concentrations data discussed in Example 4.1.3.

Use the R functionpercentcibootto obtain a bootstrap 95% confidence interval

for the true mean concentration. Use 3000 bootstraps and compare it with the

t-confidence interval for the mean.

4.9.2.Letx 1 ,x 2 ,...,xnbe the values of a random sample. A bootstrap sample,

x∗′=(x∗ 1 ,x∗ 2 ,...,x∗n), is a random sample ofx 1 ,x 2 ,...,xndrawn with replacement.

(a)Show thatx∗ 1 ,x∗ 2 ,...,x∗nare iid with common cdfF̂n, the empirical cdf of

x 1 ,x 2 ,...,xn.

(b)Show thatE(x∗i)=x.

(c)Ifnis odd, show that median{x∗i}=x((n+1)/2).

(d)Show thatV(x∗i)=n−^1

∑n

i=1(xi−x)

(^2).