3.4 : Bootstrap Confidence Intervals using PercentilesIn the previous sections we looked at how to construct a 95% confidence interval, where we estimated the SE by using the sampling distribution or the bootstrap distribution. In this section we are going to look at how to create a confidence interval for any level of confidence using a bootstrap distribution.

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Example: Body Temperature Is normal body temperature really 98.6 â—¦F? A sample of body temperature for 50 healthy individuals was taken. Find this dataset in StatKey under â€śConfidence Interval for a Meanâ€ť or as BodyFat50 in the textâ€™s datasets.(a)What is the sample mean? What is the sample standard deviation?

(b) Generate a bootstrap distribution, using at least 1000 simulated statistics. What is the standard error?

(c)Use the standard error to find a 95% confidence interval. Show your work. Is 98.6 in the

(d) Using the same distribution, find a 95% confidence interval using the â€śTwo-tailâ€ť option on StatKey (or other technology to give percentiles from the bootstrap distribution).

(e) Compare the two 95% confidence intervals you found. Are they similar?

(f) Still using the same bootstrap distribution, give a 99% confidence interval.

(g) Is the 99% confidence interval wider or narrower than the 95% confidence interval?

(h) Clearly interpret the 99% confidence interval We are 99% sure that the mean body temperature for all healthy individuals is between98.017 and 98.544.Example: Problems with Bootstrap Distributions If a bootstrap distribution is not relatively symmetric, it is not appropriate to use the methods of this chapter to construct a confidence interval. Consider the following data set: 5, 6, 7, 8, 25, 100 (a) What is the standard deviation of this dataset?

(b) Use StatKey (or other technology) to create a bootstrap distribution for the standard deviationof this dataset. Describe the distribution. Is the distribution symmetric and bell-shaped?

(c) Is it appropriate to use the methods of this section to find a bootstrap confidence interval for this standard deviation?

(d) Discuss with a neighbor why the bootstrap distribution might look the way it does.