7-3 Estimating a Population Standard Deviation or Variance 345 7-2 Beyond the Basics 33.Ages of Prisoners The accompanying frequency distribution summarizes sample data consisting of ages of randomly selected inmates in federal prisons (based on data from the Federal Bureau of Prisons). Use the data to construct a 95% confidence interval estimate of the mean age of all inmates in federal prisons. 34.Finite Population Correction Factor If a simple random sample of size n is selected without replacement from a finite population of size N, and the sample size is more than 5% of the population size 1n 7 0.05N2, better results can be obtained by using the finite population correction factor, which involves multiplying the margin of error E by 21N - n2>1N - 12. Refer to the weights of the M&M candies in Data Set 38 “Candies” in Appendix B. a. Use only the red M&Ms and treat that sample as a simple random sample from a very large population of M&Ms. Find the 95% confidence interval estimate of the mean weight of all M&Ms. b. Use only the red M&Ms and treat that sample as a simple random sample selected from the population of the 345 M&Ms listed in the data set. Find the 95% confidence interval estimate of the mean weight of all 345 M&Ms. Compare the result to the actual mean of the population of all 345 M&Ms. c. Compare the confidence intervals from parts (a) and (b). Key Concept This section presents methods for using a sample standard deviation s (or a sample variance s2) to estimate the value of the corresponding population standard deviation s (or population variance s 2). Here are the main concepts included in this section: ■ Point Estimate: The sample variance s 2 is the best point estimate (or single value estimate) of the population variance s 2. The sample standard deviation s is commonly used as a point estimate of s, even though it is a biased estimator, as described in Section 6-3. ■ Confidence Interval: When constructing a confidence interval estimate of a population standard deviation (or population variance), we construct the confidence interval using the x2 distribution. (The Greek letter x is pronounced “kigh.”) Chi-Square Distribution Here are key points about the x2 (chi-square or chi-squared) distribution: ■ In a normally distributed population with variance s 2, if we randomly select independent samples of size n and, for each sample, compute the sample variance s2, the sample statistic x2 = (n - 1) s2>s 2 has a sampling distribution called the chi-square distribution, as shown in Formula 7-5. 7-3 Estimating a Population Standard Deviation or Variance FORMULA 7-5 x2 = 1n - 12s2 s 2 Age (years) Number 16–25 13 26–35 61 36–45 66 46–55 36 56–65 14 Over 65 5
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