334 CHAPTER 7 Estimating Parameters and Determining Sample Sizes Based on the preceding results, does it seem that the packages are being filled so that the total package weight is 340.2 g as indicated by the label? 8.639 8.689 8.548 8.980 8.936 9.042 SOLUTION a. The point estimate of the data is the sample mean x, which is 8.8057 g. b. REQUIREMENT CHECK Before constructing the 95% confidence interval, we must first verify that the requirements are satisfied. (1) The sample is a simple random sample. (2) Because the sample size is n = 6, the requirement that “the population is normally distributed or the sample size is greater than 30” can be satisfied only if the sample data appear to be from a normally distributed population, so we need to investigate normality. In the accompanying normal quantile plot, the points appear to fit a straight-line pattern, so the sample data appear to be from a normally distributed population. This second requirement is satisfied. Using Technology Technology can be used to automatically construct the confidence interval. (See instructions near the end of this section.) Shown here is the StatCrunch display resulting from the six weights. The display shows the lower confidence interval limit (L. Limit) and the upper confidence interval limit (U. Limit). After rounding to four decimal places (one more than the original data), we can express the confidence interval as 8.5901 g 6 m 6 9.0213 g. StatCrunch Manual Calculation First find the critical value t0.025 = 2.571 from Table A-3 using n - 1 = 5 degrees of freedom (as shown in Example 1). Find the standard deviation s = 0.2054 g from the original sample values. Now find the margin of error E: E = ta>2 s2 n = 2.571# 0.2054 2 6 = 0.215589 Captured Tank Serial Numbers Reveal Population Size During World War II, Allied intelligence specialists wanted to determine the number of tanks Germany was producing. Traditional spy techniques provided unreliable results, but statisticians obtained accurate estimates by analyzing serial numbers on captured tanks. As one example, records show that Germany actually produced 271 tanks in June 1941. The estimate based on serial numbers was 244, but traditional intelligence methods resulted in the extreme estimate of 1550. (See “An Empirical Approach to Economic Intelligence in World War II,” by Ruggles and Brodie, Journal of the American Statistical Association, Vol. 42.)
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