STAT 400 Lecture AL1 1 Spring 2015 Dalpiaz Practice Problems 5 A gas station sells three grades of gasoline regular unleaded extra unleaded and super unleaded These are priced at 1 55 1 70 and 1 85 per gallon respectively Let X 1 X 2 and X 3 denote the amounts of these grades purchased gallons on a particular day Suppose the X i s are independent with 1 1 000 2 500 3 300 1 100 2 80 and 3 50 If the X i s are normally distributed what is the probability that revenue exceeds a 2 2 600 b 3 000 Suppose that the actual weight of 10 pound sacks of potatoes varies from sack to sack and that the actual weight may be considered a random variable having a normal distribution with the mean of 10 2 pounds and the standard deviation of 0 6 pounds Similarly the actual weight of 3 pound bags of apples varies from bag to bag and that the actual weight may be considered a random variable having a normal distribution with the mean of 3 15 pounds and the standard deviation of 0 3 pounds A boy scout troop is planning a camping trip If the boy scouts buy 3 10 pound sacks of potatoes and 4 3 pound bags of apples selecting them at random what is the probability that the total weight would exceed 42 pounds 3 Let X 1 and X 2 be independent with normal distributions N 6 1 and N 7 1 respectively Find P X 1 X 2 Hint Write P X 1 X 2 P X 1 X 2 0 and determine the distribution of X 1 X 2 4 Compute P X 1 2 X 2 2 X 3 7 if X 1 X 2 X 3 are i i d with common distribution N 1 4 This problem was written long time ago 5 An instructor gives a test to a class containing several hundred students It is known that the standard deviation of the scores is 14 points A random sample of 49 scores is obtained a What is the probability that the average score of the students in the sample will differ from the overall average by more than 2 points b What is the probability that the average score of the students in the sample will be within 3 points of the overall average 6 The yield at a given coal mine in tons of ore per day is approximately normally distributed with mean 785 tons and unknown standard deviation a Find the standard deviation of the daily output if it is known that on 13 of the days the output falls below 717 2 tons b Find the 68th percentile of the daily outputs c Find the probability that at least 700 tons of ore will be mined on a given day 6 continued A random sample of 10 days is obtained Assume each day is independent of all other days d What is the probability that average yield for 10 days is at most 800 tons e What is the probability that at least 700 tons of ore will be mined on 8 out of 10 days Answers 1 Total 1 55 X 1 1 70 X 2 1 85 X 3 E Total 1 55 E X 1 1 70 E X 2 1 85 E X 3 1 55 1 000 1 70 500 1 85 300 2 955 Var Total 1 55 2 Var X 1 1 70 2 Var X 2 1 85 2 Var X 3 1 55 2 100 2 1 70 2 80 2 1 85 2 50 2 51 077 25 SD Total 51 077 25 226 Total has Normal distribution a 2 600 2 955 P Total 2 600 P Z P Z 1 57 1 0 0582 0 9418 226 b 3 000 2 955 P Total 3 000 P Z P Z 0 20 1 0 5793 0 4207 226 2 Since weights vary from sack to sack and from bag to bag Total P 1 P 2 P 3 A 1 A 2 A 3 A 4 E Total E P 1 E P 2 E P 3 E A 1 E A 2 E A 3 E A 4 10 2 10 2 10 2 3 15 3 15 3 15 3 15 43 2 Var Total Var P 1 Var P 2 Var P 3 Var A 1 Var A 2 Var A 3 Var A 4 0 6 2 0 6 2 0 6 2 0 3 2 0 3 2 0 3 2 0 3 2 1 44 SD Total 1 44 1 2 Total has Normal distribution 42 43 2 P Total 42 P Z P Z 1 00 0 8413 1 2 3 E X 1 X 2 E X 1 E X 2 1 Var X 1 X 2 Var X 1 Var X 2 2 X 1 X 2 has Normal distribution 0 1 P X 1 X 2 P X 1 X 2 0 P Z P Z 0 707 0 24 2 4 E X 1 2 X 2 2 X 3 E X 1 2 E X 2 2 E X 3 1 Var X 1 2 X 2 2 X 3 Var X 1 4 Var X 2 4 Var X 3 36 SD X 1 2 X 2 2 X 3 6 X 1 2 X 2 2 X 3 has Normal distribution 7 1 P X 1 2 X 2 2 X 3 7 P Z P Z 1 00 0 1587 6 5 14 a Need 1 P 2 X 2 n 49 large n 49 Central Limit Theorem X Z n 2 2 1 P 2 X 2 1 P Z 14 14 49 49 1 P 1 00 Z 1 00 0 3174 b Need P 3 X 3 3 3 P 3 X 3 P Z 14 14 49 49 P 1 50 Z 1 50 0 8664 6 785 a Find z such that P Z z 0 13 z 0 13 Using the standard normal table z 1 13 x z 717 2 785 1 13 67 8 1 13 60 tons b Find z such that P Z z 0 68 z 0 68 Using the standard normal table z 0 47 c x z x 785 60 0 47 813 2 tons 700 785 P X 700 P Z 60 P Z 1 42 1 1 42 1 0 0778 0 9222 d Need P X 800 Normal distribution Case 2 800 785 P X 800 P Z P Z 0 79 0 79 0 7852 60 10 e Let Y the number of days out of 10 with at least 700 tons of ore mined Then Y has Binomial Distribution n 10 p 0 9222 see part c 10 Need P Y 8 0 9222 8 0 0778 2 0 1425 8
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