1. ## Bootstrapping

Hey all,

Final exam is tomorrow and we were just given some examples of a recent topic, which knowing the professor will be all over the final. Was hoping someone could tell me how to solve any of the following since we never solved one in class.

1. Suppose there is a process P that produces measurements with an unknown (or unanalyzable) distribution. You’re given a simple random sample of n = 30 measurements drawn from P and asked to construct a level C = 95% for the the mean μP of the process. The means for 1, 000 resamples of size 30 have the following estimated values for μP at the given percentiles:
Percentile .5% 1% 2.5% 5% 10% 90% 95% 97.5% 99% 99.5%
μP 0.734 0.749 0.789 0.834 0.906 1.349 1.427 1.477 1.559 1.632
Use these percentile estimates to construct the requested confidence interval.

2. Suppose that for the process of 1, the qq-normal plot of the empirical distribution X30e suggests it is close enough to a normal distribution to use a t-approximation. The sample mean is ¯x = 1.127 and its standard deviation is s = 0.988. The estimate of the t29 distribution is given by 1, 000 resamples of size 30 from the data, and have the following values for t29e:
.5% 1% 2.5% 5% 10% 90% 95% 97.5% 99% 99.5%
t 29e −0.052 −0.047 −0.040 −0.034 −0.026 0.022 0.028 0.036 0.045 0.049
Use the t confidence interval method along with this estimated t29 table to make a C = 99% confidence interval for μP . This is known as a studentized confidence interval.

3. Suppose that for the process of 1, the null hypothesis is H0 : μP = 1.0 versus the alternative hypothesis H1 : μP > 1.0, and you find that ¯x = 1.480. Use the empirical distribution given in the percentile table of 1 to do a level = 5% test of H0, and report your conclusions. Whatever that conclusion, assume that H0 is true to report the p-value of ¯x.

4. Explain why the bootstrap procedure is not useful for analyzing the distribution of estimates of the minimum and maximum values of a distribution. Hint: What are differences between distribution of the extreme values of 1, 000 resamples (~se)i, i = 1, . . . , 1000 of a given ~s, and the distribution the extreme values of 1, 000 samples {~si}, i = 1, . . . , 1000.

2. Originally Posted by lteece89
Hey all,

Final exam is tomorrow and we were just given some examples of a recent topic, which knowing the professor will be all over the final. Was hoping someone could tell me how to solve any of the following since we never solved one in class.

1. Suppose there is a process P that produces measurements with an unknown (or unanalyzable) distribution. You’re given a simple random sample of n = 30 measurements drawn from P and asked to construct a level C = 95% for the the mean μP of the process. The means for 1, 000 resamples of size 30 have the following estimated values for μP at the given percentiles:
Percentile .5% 1% 2.5% 5% 10% 90% 95% 97.5% 99% 99.5%
μP 0.734 0.749 0.789 0.834 0.906 1.349 1.427 1.477 1.559 1.632
Use these percentile estimates to construct the requested confidence interval.
If my algebra is right this is:

$\displaystyle [2\widehat{\mu}-\mu_{97.5},2\widehat{\mu}-\mu_{2.5}]$

where $\displaystyle \widehat{\mu}$ is the mean of our original sample.

In the case of an approximately symmetric distribution for the bootstrap samples is close to

$\displaystyle [\mu_{2.5},\mu_{97.5}]$

2. Suppose that for the process of 1, the qq-normal plot of the empirical distribution X30e suggests it is close enough to a normal distribution to use a t-approximation. The sample mean is ¯x = 1.127 and its standard deviation is s = 0.988. The estimate of the t29 distribution is given by 1, 000 resamples of size 30 from the data, and have the following values for t29e:
.5% 1% 2.5% 5% 10% 90% 95% 97.5% 99% 99.5%
t 29e −0.052 −0.047 −0.040 −0.034 −0.026 0.022 0.028 0.036 0.045 0.049
Use the t confidence interval method along with this estimated t29 table to make a C = 99% confidence interval for μP . This is known as a studentized confidence interval.

3. Suppose that for the process of 1, the null hypothesis is H0 : μP = 1.0 versus the alternative hypothesis H1 : μP > 1.0, and you find that ¯x = 1.480. Use the empirical distribution given in the percentile table of 1 to do a level = 5% test of H0, and report your conclusions. Whatever that conclusion, assume that H0 is true to report the p-value of ¯x.

4. Explain why the bootstrap procedure is not useful for analyzing the distribution of estimates of the minimum and maximum values of a distribution. Hint: What are differences between distribution of the extreme values of 1, 000 resamples (~se)i, i = 1, . . . , 1000 of a given ~s, and the distribution the extreme values of 1, 000 samples {~si}, i = 1, . . . , 1000.