TPS2e_IM_ch10_161_172

The Practice of Statistics: TI-83/89 Graphing Calculator Enhanced

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161 10 10.1 (a) 44% to 50%. (b) We do not have information about the whole population; we only know about a small sam- ple. We expect our sample to give us a good estimate of the population value, but it will not be exactly correct. (c) The procedure used gives an estimate within 3 percentage points of the true value in 95% of all samples. 10.2 (a) The sampling distribution of is normal with mean m 5 280 and standard deviation 5 2.1. (b) Below. (c) 2 standard deviations— m 4.2. (d) Below; the confi- dence intervals drawn may vary, of course. (e) 95% (by the 68-95-99.7 rule). 60 / 1 840 s / 1 n x 273.7 275.8 277.9 280 282.1 284.2 286.3 10.3 This is a statement about the mean score for all young men, not about individual scores. We are attempting only to estimate the center of the population distribution; the scores for individu- als are much more variable. Also, “95%” is not a probability or a proportion; it is a confidence level. 10.4 (a) The sampling distribution of is normal with mean m and standard deviation 5 5 0.05657. (b) See the sketch on the next page. For this problem, the “numbers” below the axis would be m 2 0.16971, m 2 0.11314, m 2 0.05657, m , etc. (c) m 5 0.11314 (2 standard deviations). (d) 95%. (e) See the next page. The actual confidence intervals drawn may vary. 0.4 / 1 50 s / 1 n x 6851F_ch10_161_172 17/9/02 19:34 Page 161
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162 Chapter 10 10.11 7.91—which is half the margin of error with n 5 20. 10.5 .84 6 0.10, or .830 to .851 grams per liter. 10.6 11.78 6 0.77, or 11.01 to 12.55 years. 10.7 (a) The distribution is slightly skewed to the right. (b) 224.002 6 0.029, or 223.973 to 224.031. 2239 01 2239 66788889 2240 01 2240 589 2241 2 10.8 5 123.8 bu/acre, and 5 bu/acre. (a)–(c) See the table below; the intervals are 6 z * . (d) The margin of error increases with the confidence level. Conf. Level z * Interval 90% 1.645 119.6 to 128.0 bu/acre 95% 1.960 118.7 to 128.9 bu/acre 99% 2.576 117.1 to 130.5 bu/acre 10.9 With n 5 60, 5 bu/acre. (a) 95% confidence interval: 6 1.960 5 121.3 to 126.3 bu/acre. (b) Smaller: with a larger sample comes more information, which in turns gives less uncertainty (“noise”) about the value of m . (c) They will be narrower. 10.10 (a) 294 to 318.6. (b) 274.7 to 337.9. (c) Below—increasing confidence makes the interval wider. s x x 10 / 1 60 8 1.291 s x s x x 10 / 1 15 8 2.582 s x x m 2 .17 m 2 .11 m 2 .06 mm 1 .06 m 1 .11 m 1 .17 275 300 325 350 99.9% 99% 90% 80% 6851F_ch10_161_172 17/9/02 19:34 Page 162
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Introduction to Inference 163 10.12 (a) 10.00209 to 10.00251. (b) 22 (21.64). 10.13 68 (67.95). 10.14 35 (34.57). 10.15 (a) The computations are correct. (b) Since the numbers are based on a voluntary response, rather than an SRS, the methods of this section cannot be used—the interval does not apply to the whole population. 10.16 (a) The interval was based on a method that gives correct results 95% of the time. (b) Since the margin of error was 2%, the true value of p could be as low as 49%. The con- fidence interval thus contains some values of p , which give the election to Bush. (c) The proportion of voters that favor Gore is not random—either a majority favors Gore, or they don’t. Discussing probabilities about this proportion has little meaning: the “proba- bility” the politician asked about is either 1 or 0 (respectively).
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