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Homework Assignment #3 - Solutions 1. Social Network Friends The following is a data set from the DVD attached to the book, supposedly corresponding to the “Try It Yourself” question on page 304, even though the data is completely different. Consider this a simple random sample of 30 members of a social network, listing their friends in the network: 249 227 215 214 201 196 179 172 162 156 146 145 137 137 135 131 130 127 125 124 114 108 1.1. Descriptive Statistics Compute the following for the sample: • Sample Mean • Population and Sample Variance • Population and Sample Standard Deviation • Median • Suggest your choice for 1st and 3rd quartiles Solutions Mean Median 1 quartile 3 quartile Pop. S.D. Pop. Variance Standard Deviation Sample Variance 138.5 133 102 169.5 50.1881 2518.85 51.0461 2605.71 100 95 87 80 74 71 63 55 1.2. Confidence Intervals Compute confidence intervals, at a level of your choice, for • The “true” mean (note that the “true” variance is unknown) • The “true” variance (note that the “true” mean is unknown) Note 1: The two estimates are not independent, since they share the same data. ¡It is incorrect to ¢ combine them in some simple “2-dimensional confidence interval” for the pair ¹; ¾ 2 Note 2: Both estimates require the assumption that the data is coming from a normal distribution. This is a bit of a tall order, however it happens that applying a typical nonparametric test for normality to this data set does not rule out that possibility (the cautious language should become more familiar once we go into the chapter on statistical tests) Note 3: You could probably work out the required statistics from the raw data, given that you have time, but, just in case, here are our customary summaries: Count 30 Sum 4,155 Sum of Squares 651,033 Solutions For the mean: 90% CI for the Mean from to 122.665 154.335 95% CI for the Mean from to 119.439 157.561 99% CI for the Mean from to 112.811 164.189 For the variance: levels 0.9 0.95 0.99 left right 1775.632 4267.220 1652.706 4708.99 1443.864 5759.061 The numerator in the chi-square formula, 75565.5 X 2 2 (xi ¡ x¹ ) = nS = (n ¡ 1)s2 , is 2. Body Temperatures This is a reduced sample of readings of body temperatures. Consider it a simple random sample of size 32: 99.6 99.4 99.2 98.9 98.9 98.7 98.7 98.6 2.1. 98.6 98.6 98.6 98.6 98.5 98.4 98.4 98.4 98.4 98.3 98 98 98 98 97.9 97.8 Descriptive Statistics Compute the following for the sample: • Sample Mean • Population and Sample Variance • Population and Sample Standard Deviation • Median • Suggest your choice for 1st and 3rd quartiles Solutions Mean Median 1 quartile 3 quartile Pop. S.D. Pop. Variance Standard Deviation Sample Variance 98.2875 98.4 97.8 98.6 0.56555 0.31984 0.5746 0.33016 97.8 97.7 97.6 97.6 97.6 97.6 97.4 97.4 2.2. Confidence Intervals Compute confidence intervals, at a level of your choice, for • The “true” mean (note that the “true” variance is unknown) • The “true” variance (note that the “true” mean is unknown) Note 1: The two estimates are not independent, since they share the same data. ¡It is incorrect to ¢ combine them in some simple “2-dimensional confidence interval” for the pair ¹; ¾ 2 Note 2: Both estimates require the assumption that the data is coming from a normal distribution. This is a bit of a tall order, however it happens that applying a typical nonparametric test for normality to this data set does not rule out that possibility (the cautious language should become more familiar once we go into the chapter on statistical tests) Note 3: You could probably work out the required statistics from the raw data, given that you have time, but, just in case, here are our customary summaries: Count Sum Sum of Squares 32 3,145.2 309,144.08 Solutions 90% CI for the Mean from to 98.1153 98.4597 95% CI for the Mean from to 98.0803 98.4947 99% CI for the Mean from 98.0088 to 98.5662 For the variance, the numerator (n ¡ 1)s2 turns out to be 10.235 so that we have the confidence intervals levels 0.9 0.95 0.99 left right 0.186082 0.707924 0.227519 0.530845 0.212204 0.583565