350 CHAPTER 7 Estimating Parameters and Determining Sample Sizes Bootstrap Resampling for Constructing Confidence Intervals Section 7-4 describes the method of bootstrap resampling for constructing a confidence interval estimate of a population parameter. A technology such as Statdisk can be used to “resample” the sample data many times (such as 1000), then use the sorted list of 1000 results to find the confidence interval. If we repeat Example 2 using the bootstrap resampling method, here is a typical result: 7.2 bpm 6 s 6 11.5 bpm. Because of the randomness used in the procedure, the resulting confidence interval may differ somewhat. Rationale for the Confidence Interval See Figure 7-9 on page 347 to make sense of this statement: If we select random samples of size n from a normally distributed population with variance s 2, there is a probability of 1 - a that the statistic 1n - 12s2>s 2 will fall between the critical values of x2 L and x2 R. It follows that there is a 1 - a probability that both of the following are true: 1n - 12s2 s 2 6 x2 R and 1n - 12s2 s 2 7 x2 L Multiply both of the preceding inequalities by s 2, then divide each inequality by the appropriate critical value of x2, so the two preceding inequalities can be expressed in these equivalent forms:1n - 12s2 x2 R 6 s 2 and 1n - 12s2 x2 L 7 s 2 Using Table A-4 If using Table A-4 , we first use the sample size of n = 22 to find degrees of freedom: df = n - 1 = 21. In Table A-4 , refer to the row corresponding to 21 degrees of freedom, and refer to the columns with areas of 0.975 and 0.025. (For a 95% confidence level, we divide a = 0.05 equally between the two tails of the chi-square distribution, and we refer to the values of 0.975 and 0.025 across the top row of Table A-4 .) The critical values are x2 L = 10.283 and x2 R = 35.479 (as shown in Example 1). Step 3: Using the critical values of x2 L = 10.283 and x2 R = 35.479, the sample standard deviation of s = 9.91959 and the sample size of n = 22, we construct the 95% confidence interval by evaluating the following: 1n - 12s2 x2 R 6 s 2 6 1n - 12s2 x2 L 122 - 1219.9195922 35.479 6 s 2 6 122 - 1210.9195922 10.283 Step 4: Evaluating the expression above results in 58.2419 6 s 2 6 200.9495. Finding the square root of each part and rounding the result to one decimal place, we get this 95% confidence interval estimate of the population standard deviation s: 7.6 bpm 6 s 6 14.2 bpm. INTERPRETATION Based on this result, we have 95% confidence that the limits of 7.6 bpm and 14.2 bpm contain the true value of s. The confidence interval can also be expressed as (7.6 bpm, 14.2 bpm), but it cannot be expressed in a format of s { E. YOUR TURN. Find the confidence interval in Exercise 5 “Nicotine in Menthol Cigarettes.”
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