STAT 400 Lecture AL1 Answers for 5 6 Spring 2015 Dalpiaz The true weight of 10 pound sacks of potatoes processed at a certain packaging house follows a normal distribution with mean of 10 1 pounds and standard deviation of 0 2 pounds a What is the probability that a sack weighs at least 10 pounds 10 10 1 P X 10 P Z 0 2 P Z 0 50 1 0 50 1 0 3085 0 6915 b A random sample of 9 sacks is selected What is the probability that the average weight of these 9 is at least 10 pounds X Case 2 Z n 10 10 1 P X 10 P Z 0 2 9 P Z 1 50 1 1 50 1 0 0668 0 9332 1 A student commission wants to know the mean amount of money spent by college students for textbooks in one semester Suppose the population mean is 450 and the population standard deviation is 40 A random sample of 625 students is taken a What is the probability that the sample mean will be less than 452 450 40 n 625 large n 625 Need P X 452 Central Limit Theorem X Z n 452 450 P X 452 P Z P Z 1 25 0 8944 40 625 b What is the probability that the sample mean will be within 2 of 450 That is what is the probability that the sample mean will be between 448 and 452 448 450 452 450 Z P 448 X 452 P P 1 25 Z 1 25 0 7888 40 40 625 625 c What is the probability that the sample mean will be within 10 of 450 That is what is the probability that the sample mean will be between 440 and 460 440 450 460 450 Z P 440 X 460 P P 6 25 Z 6 25 1 00 40 40 625 625 2 The amount of sulfur in the daily emissions from a power plant has a normal distribution with mean of 134 pounds and a standard deviation of 22 pounds For a random sample of 5 days find the probability that the total amount of sulfur emissions will exceed 700 pounds Normal distribution 134 22 n 5 Need P X 1 X 2 X 3 X 4 X 5 700 P Total 700 Total has N n n 2 distribution Total n n Z 700 5 134 P Total 700 P Z P Z 0 61 1 0 7291 0 2709 5 22 OR Need P X 1 X 2 X 3 X 4 X 5 700 P X 140 2 distribution X has N n X Z n 140 134 P X 140 P Z P Z 0 61 1 0 7291 0 2709 22 5 3 An economist wishes to estimate the average family income in a certain population The population standard deviation is known to be 4 500 and the economist uses a random sample of 225 families What is the probability that the sample mean will fall within 600 of the population mean 4 500 Need P 600 X 600 n 225 n 225 large Central Limit Theorem X Z n 600 600 P 600 X 600 P Z 4 500 4 500 225 225 P 2 00 Z 2 00 0 9772 0 0228 0 9544 4 Forty eight measurements are recorded to several decimal places Each of these 48 numbers is rounded off to the nearest integer The sum of the original 48 numbers is approximated by the sum of these integers If we assume that the errors made by rounding off are i i d and have uniform distribution over the interval 12 12 compute approximately the probability that the sum of the integers is within 2 units of the true sum X 1 X 2 X 48 are i i d Uniform 1 1 2 2 n 48 large X2 X 0 Central Limit Theorem X 2 Need P X 1 X 2 X 48 2 P X P Z 48 P Z 1 00 0 8413 0 1587 0 6826 Z n 48 1 12 48 2 1 12
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