7-4 Bootstrapping: Using Technology for Estimates 355 where k is the number of degrees of freedom and za>2 is the critical z score described in Section 7-1. Use this approximation to find the critical values x2 L and x2 R for Exercise 9 “Body Temperature,” where the sample size is 106 and the confidence level is 95%. How do the results compare to the actual critical values of x2 L = 78.536 and x2 R = 135.247? 24.Finding Sample Size Instead of using Table 7-2 for determining the sample size required to estimate a population standard deviation s, the following formula can also be used n = 1 2 a za>2 d b 2 where za>2 corresponds to the confidence level and d is the decimal form of the percentage error. For example, to be 95% confident that s is within 15% of the value of s, use za>2 = 1.96 and d = 0.15 to get a sample size of n = 86. Find the sample size required to estimate the standard deviation of IQ scores of data scientists, assuming that we want 98% confidence that s is within 5% of s. Key Concept The preceding sections presented methods for estimating population proportions, means, and standard deviations (or variances). All of those methods have certain requirements that limit the situations in which they can be used. When some of the requirements are not satisfied, we can often use the bootstrap method to estimate a parameter with a confidence interval. The bootstrap method typically requires the use of software such as Statdisk. The bootstrap resampling method described in this section has these requirements: 1. The sample must be collected in an appropriate way, such as a simple random sample. (If the sample is not collected in an appropriate way, there’s a good chance that nothing can be done to get a usable confidence interval estimate of a parameter.) 2. When generating statistics using bootstrap resampling, the distribution of those statistics should be approximately symmetric. The preceding methods of this chapter also have these requirements: ■ CI for Proportion (Section 7-1): There are at least 5 successes and at least 5 failures, or np Ú 5 and nq Ú 5. ■ CI for Mean (Section 7-2): The population is normally distributed or n 7 30. ■ CI for S or S 2 (Section 7-3): The population must have normally distributed values, even if the sample is large. When the above requirements are not satisfied, we should not use the methods presented in the preceding sections of this chapter, but we can use the bootstrap method instead. The bootstrap method does not require large samples. This method does not require the sample to be collected from a normal or any other particular distribution, and so the bootstrap method is called a nonparametric or distribution-free method; other nonparametric methods are included in Chapter 13. 7-4 Bootstrapping: Using Technology for Estimates
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