The distribution of heights of 18-year-old men in the United States is approximately normal, with mean 68 inches and standard deviation 3 inches (U.S. Census Bureau). In Minitab, we can simulate the drawing of random samples of size 20 from this population (Calc ž¤ Random Data ž¤ Normal, with 20 rows from a distribution with mean 68 and standard deviation 3). Then we can have Minitab compute a 95% confidence interval and draw a boxplot of the data (Stat ž¤ Basic Statistics ž¤ 1€”Sample t, with boxplot selected in the graphs). The boxplots and confidence intervals for four different samples are shown in the accompanying figures. The four confidence intervals are

(a) Examine the figure [parts (a)€“(d)]. How do the boxplots for the four samples differ? Why should you expect the boxplots to differ?

(b) Examine the 95% confidence intervals for the four samples shown in the printout. Do the intervals differ in length? Do the intervals all contain the expected population mean of 68 inches? If we draw more samples, do you expect all of the resulting 95% confidence intervals to contain μ = 68? Why or why not?

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VARIABLE Sample l Sample 2 Sample 3 Sample4 STDEV 20 68.050 2.901 3.137 20 67.976 2.639 2.440 MEAN SEMEAN 0.649 0.702 0.590 0.546 95.0 % CI (66.692 69.407) (66.49069.426) (66.741,69.211) (65.766 68.050) 20 67.958 20 66.908 (a) Boxplot of Sample 1 (with 95%-confidence interval for the mean) (b) Boxplot of Sample 2 (with 95% t-confidence interval for the mean) 64 69 74 60 62 64 66 68 70 72 74 (c) Boxplot of Sample 3 (with 95% t-confidence interval for the mean) (d) Boxplot of Sample 4 (with 95% t-confidence interval for the mean) 63 64 65 66 67 68 69 70 71 72 73 62 63 64 65 66 67 68 69 70 71 72