Exercise Set 1 The normal distribution and sampling distributions

Transcription

1 Eercise Set 1 The normal distribution and sampling distributions 1). An orange juice producer buys all his oranges from a large orange grove. The amount of juice squeezed from each of these oranges is approimately normally distributed with a mean of 4.70 ounces and a standard deviation of 0.40 ounce. a. What is the probability that a randomly selected orange will contain between 4.70 and 5.00 ounces? a. X - the amount of juice squeezed from an orange P(4.7 < X < 5) = b. 77% of the oranges will contain at least how many ounces of juice? 0.77 b. X = ounces 1

2 c. 80% of the oranges are between what two values (in ounces) symmetrically distributed around the population mean? 0.8 X1 X2 c. (X1 < P < X2) = 0.8 X1 = , X2 = ounces 2) A random variable (X) is normally distributed with a mean of 25 and a standard deviation of 5. a) Find the probability that X is less than 27. b) What value will 10% of the observations be below? c) Determine 2 values of which the smallest has 25% of the values below it and the largest has 25% of the values above it. 3) A designer of maternity wear sells dresses and pants priced around $ 150 each for an average total sale of $ The total sale has a normal distribution with a standard deviation of $ 350. a) Calculate the probability that a random selected customer will have a total sale of more than $ b) Compute the probability that that the total sale will be within 2 standard deviations of the mean total sales. c) Determine the median total sale. 2

3 4) The time to get an oil change at a certain car dealership averages 42.3 minutes with a standard deviation of 8.6 minutes. a) There are 45 cars booked for oil changes today. What is the probability that the jobs can be done in an average of 38 minutes or less? b) Suppose 90 oil changes were done in one particular week. There is a 95% chance that the mean time was more than minutes. 5) The manager of a cereal-filling process in a large factory claims that, on average, its machines deviate from a perfect filling by an average of 30g in a month, with a standard deviation of 10g. A quality controller eamined 40 cereal bags and found that the average deviation from a perfect filling was 35g after one month. a) If the manufacturer s claim is correct, what is the probability that the average deviation from a perfect filling for 40 cereal bags would be 35g or more? b) If the average bag weight deviates by 33g from a perfect filling in one month, what is the probability that the average deviation from a perfect filling for the 40 bags would be 35g or more? 6) The lifetime of a certain brand of tire is normally distributed with a mean of km and standard deviation of km. The tire carries a warranty for km. a) What is the probability that the tire you recently purchased will last more than km? b) What percent of this brand of tire will fail before the warranty epires? c) What should the mileage warranty be so that only 4% of the tires need to be replaced under warranty? 7) At a certain university, the cumulative grade point average (CGPA) of first year students usually averages 2.73 with a standard deviation It has been found that the marks are usually approimately normally distributed. 3

4 a) What is the probability that a student will have a CGPA that is between 2.00 and 3.00? b) What percent of students will be on probation, i. e. their CGPA is less than 2.00? c) Academic scholarships are awarded to the top 1% of first year students. What minimum CGPA is needed to receive a scholarship? 8) The breaking strength of plastic bags used for packaging produce is normally distributed with a mean of 5 pounds per square inch and a standard deviation of 1.5 pounds per square inch. What proportion of the bags has a breaking strength of a) less than 3.17 pounds per square inch? b) at least 3.6 pounds per square inch? c) between 5 and 5.5 pounds per square inch? d) Between what 2 values symmetrically distributed around the mean will 95% of the breaking strengths fall? 9) A statistical analysis of long-distance telephone calls made from the headquarters of the Bricks and Clicks Computer Corporation indicates that the length of these calls is normally distributed with seconds. µ = 240 seconds and σ = 40 a) What is the probability that a particular call lasted less than 180 seconds? b) What is the probability that a particular call lasted between 180 and 300 seconds? c) What is the probability that a call lasted between 110 and 180 seconds? d) What is the length of a particular call if only 1% of all calls are shorter? Sampling Distributions 10) The time,, a student spends learning a computer software package is normally distributed with a mean of 8 hours and a standard deviation of 1.5 hours. The approimate probability that the average learning time for 5 students 4

5 eceeds 8.5 hours is: A B C D ) The distribution of scores for the final eam in a Statistics course has a mean of 74 and a standard deviation of 15. A random sample of 36 eam papers is selected. What is the probability that the average score is higher than 77? A B C D ) The mean lifetime of a fluorescent light bulb is 1570 hours with a standard deviation of 200 hours. Suppose we take 100 bulbs at random, what is the probability that the average lifetime eceeds 1560 hours? A B C D ) Eample A machine targets to produce its ball bearings with a diameter of 0.75 inch. The lower and upper specification limits are 0.74 and 0.76 inches. The actual diameter of the ball bearings is approimately normally distributed with a mean of inch and a standard deviation of inch. If you select a random sample of 25 ball bearings, what is the probability that the sample mean is a. between the target and the population mean? b. between the lower specification limit and the target? c. greater than the upper specification limit? 5

6 d. less than the lower specification limit? e. greater than what value when this probability is 93%? σ σ = = = a. µ = 0.75, n 25 P (0.75 < µ < 0.753) b. c. d. e. = P (0.74 < µ < 0.75) = 0.5 P ( µ > 0.76) P ( µ < 0.74) = almost 0 = almost 0 P( µ > X ) = 93%, X = ) A tire company claims that one of their tires lasts an average of km. It is known that the tire life is normally distributed with a standard deviation of 2800 km. Ryerson Inc. has just purchased all new tires for its fleet of 10 cars. Assuming that the tire company s claim is true: a. what is the probability that a tire will last longer than km? b. what is the probability that the tires of a Ryerson Inc. company car will last an average of at least km? c. what is the probability that the average life of all the tires purchased by Ryerson Inc. will be at least km? d. what is the probability that all the tires of a Ryerson Inc. car will last more than km? 15) A political pollster is conducting an analysis of sample results in order to make predictions on election night. Assuming a two-candidate election, if a specific candidate receives at least 55% of the votes in the sample, then that candidate will be forecasted as the winner of the election. If you select a random sample of 100 voters, what is the probability that a candidate will be forecasted as the winner when a. The true percentage of her vote is 50.1%? 6

7 b. The true percentage of her vote is 60%? c. The true percentage of her vote is 49%? (she is a loser) d. If the sample size is increased to 400, what are your answers to (a) through (c)? Discuss. Answers 2) a) b) c) (-21.63, 21.63) 3) a) b) c) $ ) a) b) ) a) b) ) a) b) 45% c) km 7) a) b) 2% c) ) a) b) c) d) 2.06 and ) a) b) c) d) 147 minutes 10) A 11) A 12) C 14) a) (not a sampling distribution) b) c) d) = ) a) b) c) d) , , , increasing the sample size decreases the standard error of the sampling distribution and the probability estimates become more reliable. 7