251dscr_D 9/25/02 (Open this document in 'Outline' view!)Appendix D: Review5. Reviewa. Grouped Data.Solution:b. Ungrouped Data.Solution:251dscr_D 9/25/02 (Open this document in 'Outline' view!)Appendix D: Review 5. Reviewa. Grouped Data. Consider the following sample: Class f 0.5 - 1.5 1 1.5 - 2.5 0 2.5 - 3.5 1 3.5 - 4.5 2 a. Calculate the meanb. Calculate the variance and standard deviationc. Calculate and interpret the skewness and relative skewnessd. Calculate and interpret Pearson's measure of skewnesse. Calculate the medianf. Calculate the interquartile rangeSolution:You need the following: Class x(midpoint) f F fx 2fx 3fx 0.5 - 1.5 1 1 1 1 1 1 1.5 - 2.5 2 0 1 0 0 0 2.5 - 3.5 3 1 2 3 9 27 3.5 - 4.5 4 2 4 8 32 128 4 12 42 156Note: ,4nf12fx, ,422fx1563fx.a. Calculate the mean 3412nfxxb. Calculate the variance and standard deviation 2363344212222nxnfxs414.12variance s 471.032xsCc. Calculate and interpret* the skewness and relative skewness 323323)2)(1(xnfxxfxnnnk 33424233156)2)(3(4 6324 . 414.1243331skgd. Calculate and interpret* Pearson's measure of skewness The mode is 4, since that occurs most. 121.22433.3deviationstdmodemeanSK2e. Calculate the median First use 1 npposition , then usewfFpNLxppp1 to find the value. Here 5.p. So 5.255.1 np. This location is above 2 and below 4, so use the class 3.5 to 4.5. Then 5.312245.5.35.x.f. Calculate the interquartile range For the first quartile 25.1525.1 npposition. This location is above 1 and below2 , so use the class 2.5 to 3.5. Then, using the second formula, we find 5.2111425.5.2175.Qx For the third quartile 75.3575.1 np. This location is above 2 and below 4, so use the class 3.5 to 4.5. Then, we find 0.4122475.5.3325.Qx. 5.15.20.413 QQIQRb. Ungrouped Data. Consider the following sample: 1, 3, 4, 4 -which can also be written as x 1x 1 2x 3 3x 4 4x 4 a. Calculate the meanb. Calculate the variance, standard deviation and coefficient of variation.c. Calculate and interpret the skewness and relative skewnessd. Calculate and interpret Pearson's measure of skewnesse. Calculate the medianf. Calculate the interquartile rangeSolution:You need the following: x 2x 3x 1x 1 1 1 2x 3 9 27 3x 4 16 64 4x 4 16 64 12 42 156Note: ,4n12x, ,422x1563x.a. Calculate the mean 3412nxx3b. Calculate the variance and standard deviation 2363344212222nxnxs414.12variance s 471.032xsCc. Calculate and interpret* the skewness and relative skewness 323323)2)(1(xnxxxnnnk 33424233156)2)(3(4 6324 . 414.1243331skgd. Calculate and interpret* Pearson's measure of skewness The mode is 4, since that occurs most. 121.22433.3deviationstdmodemeanSKe. Calculate the median First use banpposition .1 , then use aaapxxbxx 11. to find the value. Here5.p. So 5.255.1 np. Thus 2a and 5.0. b. So, using the second formula, .5.3345.035.05.023221225.xxxxxxxf. Calculate the interquartile range For the first quartile 25.1525.1 npposition. Thus 1a and 25.0. b. So .5.11325.0125.025.01121111175.xxxxxxQx For the third quartile 75.3575.1 np. Thus 3a and 75.0. b. So .44475.0475.075.03343313325.xxxxxxQx 5.25.10.413 QQIQR* These negative numbers all indicate skewness to the
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