Stat104_studyguide_part3_SOLUTIONSv2

Stat104_studyguide_part3_SOLUTIONSv2 - Stat 104:...

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Study Guide Solutions, part 3 Stat 104: Quantitative Methods for Economists
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1) A tire manufacturer claims that his tires will last 40,000 miles with a standard deviation of 5000 miles. i. Assuming that the claim is true, describe the sampling distribution of the mean lifetime of a random sample of 160 tires. Remember that “describe” means discuss center, spread, and shape. By the central limit theorem, the sample distribution of xbar will look bell shaped with mean 40000 and standard deviation 5000/sqrt(160)=395.29 ii. What is the probability that the mean life time of the sample of 160 tires will be less than 39,000 miles? Interpret the probability in terms of the truth of the manufacturer’s claim. P(xbar < 39000) = P(Z < (39000-40000)/395.29)=0.006 2) The probability of winning a bet on red in roulette is 0.474. The binomial probability of winning money if you play 10 games is 0.31 and drops to 0.27 if you play 100 games. Use a normal approximation to the binomial to estimate your probability of coming out ahead (that is, winning more than 1/2 of your bets) if you play 1000 times. p=0.474 mean = 0.474 std dev = sqrt(.474*(1-.474)/1000) = 0.0158 P(phat > 0.5) = P(Z > (.5-.474)/0.0158) = 0.488 3) Crabs off the coast of Northern California have a mean weight of 2 lbs. with a standard deviation of 5 oz. A large trap captures 35 crabs. Ignore this question (a) Describe the sampling distribution for the average weight of a random sample of 35 crabs taken from this population. (b) What would the mean weight of a sample of 35 crabs have to be in order to be in the top 10% of all such samples?
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4) A certain type of light bulb is advertised to have an average life of 1200 hours. If, in fact, light bulbs of this type only average 1185 hours with a standard deviation of 80 hours, what is the probability that a sample of 100 bulbs will have an average life of at least 1200 hours? P(xbar >1200) = P(Z > (1200-1185)/(80/sqrt(100)) = P(Z>15/8) 5) Your task is to explain to your friend Gretchen, who knows virtually nothing (and cares even less) about statistics, just what the sampling distribution of the mean is. Explain the idea of a sampling distribution in such a way that even Gretchen, if she pays attention, will understand. The distribution of Xbar appears bell-shaped as we collect more and more data. 6) The mean TOEFL score of international students at a certain university is normally distributed with a mean of 490 and a standard deviation of 80. Suppose that groups of 30 students are studied. The mean and the standard deviation for the distribution of sample means, respectively, will be (a) 490 and 8/3. (b) 16.33 and 80. (c) 490 and 14.61.
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This note was uploaded on 03/27/2012 for the course STATS 104 taught by Professor Michaelparzen during the Fall '11 term at Harvard.

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Stat104_studyguide_part3_SOLUTIONSv2 - Stat 104:...

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