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Trial 3 880 880 880 910 880 850 860 870 720 840 720 840 Trial 4 890 910 810 920 810 890 820 860 800 880 770 720 Trial 5 890 870 840 870 780 810 810 740 760 810 810 940 790 950 810 800 820 810 850 870 760 840 740 850 750 850 760 780 620 850 860 840 970 840 950 840
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sistent with respect to the variability of the measurements Are all ve trials centered on the same value How does each group of trials compare to the true value Could there have been startup effects in the experiment that Michelson performed Could there have been bias in the measuring instrument 6-86. In 1789, Henry Cavendish estimated the density of the earth by using a torsion balance. His 29 measurements follow, expressed as a multiple of the density of water. 5.50 5.55 5.57 5.34 5.42 5.30 5.61 5.36 5.53 5.79 5.47 5.75 4.88 5.29 5.62 5.10 5.63 5.86 4.07 5.58 5.29 5.27 5.34 5.85 5.26 5.65 5.44 5.39 5.46
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The currently accepted true velocity of light in a vacuum is 299, 792.5 kilometers per second. Stigler (1977, The Annals of Statistics) reports that the true value for comparison to these measurements is 734.5. Construct comparative box plots of these measurements. Does it seem that all ve trials are con-
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(a) Calculate the sample mean, sample standard deviation, and median of the Cavendish density data. (b) Construct a normal probability plot of the data. Comment on the plot. Does there seem to be a low outlier in the data (c) Would the sample median be a better estimate of the density of the earth than the sample mean Why
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6-87. Consider the airfoil data in Exercise 6-12. Subtract 30 from each value and then multiply the resulting quantities by 10. Now compute s2 for the new data. How is this quantity related to s2 for the original data Explain why. n 6-88. Consider the quantity g i 1 1xi a2 2 . For what value of a is this quantity minimized 6-89 Using the results of Exercise 6-87, which of the n n two quantities g i 1 1xi x2 2 and g i 1 1xi 22 will be smaller, provided that x 6-90. Coding the Data. Let yi a bxi, i 1, 2, . . . , n, where a and b are nonzero constants. Find the relationship between x and y, and between sx and sy. 6-91. A sample of temperature measurements in a furnace yielded a sample average ( F ) of 835.00 and a sample standard deviation of 10.5. Using the results from Exercise 6-90, what are the sample average and sample standard deviations expressed in C 6-92. Consider the sample x1, x2, . . . , xn with sample mean x and sample standard deviation s. Let zi 1xi x 2 s, i 1, 2, . . . , n. What are the values of the sample mean and sample standard deviation of the zi 6-93. An experiment to investigate the survival time in hours of an electronic component consists of placing the parts in a test cell and running them for 100 hours under elevated temperature conditions. (This is called an accelerated life test.) Eight components were tested with the following resulting failure times: 75, 63, 100 , 36, 51, 45, 80, 90 The observation 100 indicates that the unit still functioned at 100 hours. Is there any meaningful measure of location that can be calculated for these data What is its numerical value 6-94. Suppose that we have a sample x1, x2, p , xn and we have calculated xn and s2 for the sample. Now an n (n 1)st observation becomes available. Let xn 1 and s2 1 be the sample mean and sample variance for the n sample using all n 1 observations. (a) Show how xn 1 can be computed using xn and xn 1. (b) Show that ns2
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n 1 (c) Use the results of parts (a) and (b) to calculate the new sample average and standard deviation for the data of Exercise 6-22, when the new observation is x38 64 . 6-95. The Trimmed Mean. Suppose that the data are arranged in increasing order, T% of the observations are removed from each end and the sample mean of the remaining numbers is calculated. The resulting quantity is called a trimmed mean. The trimmed mean generally lies between the sample mean x and the sample median x. Why (a) Calculate the 10% trimmed mean for the yield data in Exercise 6-17. (b) Calculate the 20% trimmed mean for the yield data in Exercise 6-17 and compare it with the quantity found in part (a). (c) Compare the values calculated in parts (a) and (b) with the sample mean and median for the yield data. Is there much difference in these quantities Why 6-96. The Trimmed Mean. Suppose that the sample size n is such that the quantity nT 100 is not an integer. Develop a procedure for obtaining a trimmed mean in this case.
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