Chapter 9

# Chapter 9 - 6851F_ch09_154_160 16/9/02 18:50 Page 154 9 9.1...

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9.1 m 5 2.5003 is a parameter; 5 2.5009 is a statistic. 9.2 5 7.2% is a statistic. 9.3 5 48% is a statistic; p 5 52% is a parameter. 9.4 Both 5 335 and 5 289 are statistics. 9.5 (a) Since the proportion of times the toast will land butter-side down is 0.5, the result of 20 coin flips will simulate the outcomes of 20 pieces of falling toast (landing butter-side up or but- ter-side down). (b) Answers will vary. (c) Answers will vary; however, it is more likely that the center of this distribution will be close to 0.5, and it is more likely that the shape will be close to normal. (d) Answers will vary. (e) We obtain a more accurate representation of a sampling distribution when many samples are taken. 9.6 (a) x 2 x 1 ˆ p ˆ p x The results appear to be quite variable. (b) The center is close to 0.5, and the shape is approximately normal. 9 154 6851F_ch09_154_160 16/9/02 18:50 Page 154

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Sampling Distributions 155 (c) The median and mean are extremely close. (d) The spread of the distribution did not seem to change. To decrease the spread, I would increase the number of trials, n . For example, use randBin (50, .5). 9.7 (a) The scores will vary depending on the starting row. Note that the smallest possible mean is 61.75 (from the sample 58, 62, 62, 65) and the largest is 77.25 (from 73, 74, 80, 82). (b)–(c) Answers will vary; shown below are two views of the sampling distribution. The first shows all possible values of the experiment (so the first rectangle is for 61.75, the next is for 62.00, etc.); the other shows values grouped from 61 to 61.75, 62 to 62.75, etc. (which makes the histogram less bumpy). The tallest rectangle in the first picture is 8 units; in the second, the tallest is 28 units. (d) There are (10 3 9)/2 5 45 possible samples of size 2 that can be drawn from the population. (e) The shapes and centers for the two distributions are roughly the same. However, the spread is a little larger for the distribution corresponding to n 5 2. This distribution is some- what more irregular, reflecting the fact that sample means based on samples of size 2 tend to be more variable than those based on samples of size 4. 9.8 (a) Table is on the next page; histogram not shown. (b) The histogram actually does not appear to have a normal shape. The sampling distribution is quite normal in appearance, but even a 61.75 69.4 77.25 61 69.4 77 6851F_ch09_154_160 16/9/02 18:50 Page 155
156 Chapter 9 sample of size 100 does not necessarily show it. (c) The mean of is 0.0981. The bias seems to be small. (d) The mean of the sampling distribution should be p 5 0.10. (e) The mean would still be 0.10, but the spread would be smaller. p Count p Count p Count 9 0.045 1 18 0.090 12 24 0.120 10 13 0.065 3 19 0.095 9 25 0.125 4 14 0.070 2 20 0.100 7 26 0.130 1 15 0.075 5 21 0.105 5 27 0.135 2 16 0.080 11 22 0.110 6 28 0.140 2 17 0.085 12 23 0.115 7 30 0.150 1 9.9 (a) Below, left. (b) For the 72 survival times, m 5 141.847 days. (c) Means will vary with sam- ples. (d) It would be unlikely (though not impossible) for all five values to fall on the same side of m . This is one implication of the unbiasedness of : Some values will be higher and some lower than m . (But note, it is not necessarily half and half.) (e) Shown (below, right) is a stemplot for one set of 100 sample means, which approximates the sampling distribution of . This set of means var- ied from 85 to 225 days and had mean 138.1 and standard deviation 25.9 days. The mean of the

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## This note was uploaded on 12/08/2011 for the course STAT 101 taught by Professor O during the Fall '08 term at Lake Land.

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Chapter 9 - 6851F_ch09_154_160 16/9/02 18:50 Page 154 9 9.1...

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