IPS6e.ISM.Ch04

IPS6e.ISM.Ch04 - Chapter 4 Solutions 4.1 Eleven of the rst...

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Chapter 4 Solutions 4.1. Eleven of the Frst 20 digits on line 109 correspond to “heads” so the proportion of heads is 11 20 = 0 . 55. This is close to 0.5, but not exactly the same because of random variation. 36009 19365 15412 39638 HTHHT HTHTT HTHHH HTTHT 4.3. If you hear music (or talking) one time, you will almost certainly hear the same thing for several more checks after that. (±or example, if you tune in at the beginning of a 5-minute song and check back every 5 seconds, you’ll hear that same song over 30 times.) 4.4. To estimate the probability, count the number of times the dice show 7 or 11, then divide by 25. ±or “perfectly made” (fair) dice, the number of winning rolls will nearly always (99.4% of the time) be between 1 and 11 out of 25. 4.5. The table on the right shows information from www.mms.com as of this writing. The exercise speci- Fed M&M’s Milk Chocolate Candies, but based on these numbers, results will be similar for other popular varieties. Of course, answers will vary, but students who take reason- ably large samples should get results close to the numbers in this table. (±or example, samples of size 50 will almost always be within ± 12%, while size 75 should give results within ± 10%.) M&M’s variety Green % Milk Chocolate 16% Peanut 15% Dark Chocolate 16% Dark Choc. Peanut 15% Almond 20% Peanut Butter 20% 4.6. Out of a very large number of patients taking this medication, the fraction who experience this bad side effect is about 0.00001. Note: Student explanations will vary, but should make clear that 0.00001 is a long-run average rate of occurrence. Because a probability of 0.00001 is often stated as “1 in in 100,000,” it is tempting to interpret this probability as meaning “exactly 1 out of every 100,000.” While we expect about 1 occurrence of side effects out of 100,000 patients, the actual number of side effects patients is random; it might be 0, or 1, or 2, ... . 4.7. (a) Most answers will be between 35% and 65%. (b) Based on 10,000 simulated trials—more than students are expected to do—there is about an 80% chance of having a longest run of four or more (i.e., either making or missing four shots in a row), a 54% chance of getting Fve or more, a 31% chance of getting six or more, and a 16% chance of getting seven or more. The average (“expected”) longest run length is about six. 4.8. (a) (c) Results will vary, but after n tosses, the distribution of the proportion ˆ p is approximately Nor- mal with mean 0.5 and standard deviation 1 /( 2 n ) , while the distribution of the count of heads is ap- proximately Normal with mean 0 . 5 n and standard deviation n / 2, so using the 68–95–99.7 rule, we have the results shown in the table on the right. Note that the range for ˆ p gets narrower, while the range for the count gets wider. 99.7% Range 99.7% Range n for ˆ p for count 40 0 . 5 ± 0 . 237 20 ± 9 . 5 120 0 . 5 ± 0 . 137 60 ± 16 . 4 240 0 . 5 ± 0 . 097 120 ± 23 . 2 480 0 . 5 ± 0 . 068 240 ± 32 . 9 149
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150 Chapter 4 Probability: The Study of Randomness 4.9. The true probability (assuming perfectly fair dice) is 1 ³ 5 6 ´ 4 .
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This note was uploaded on 03/07/2010 for the course ECON 41 taught by Professor Guggenberger during the Spring '07 term at UCLA.

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IPS6e.ISM.Ch04 - Chapter 4 Solutions 4.1 Eleven of the rst...

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