Statistics - Champlain College St. Lawrenceweb2.slc.qc.ca/sh/Math_337/E17) Non-Standard Normal...

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Statistics Math 201-337-77 Filename: E17) Non-Standard Normal Distributions.Doc 1. If the assembly time of an “easy to assemble” toy is a random variable having the normal distribution with µ = 12.8 minutes and σ = 4.0 minutes, what are the probabilities that this kind of toy can be assembled in a) less than 10.0 minutes; b) anywhere from 11.0 to 14.6 minutes ? 2. The reduction of a person’s oxygen consumption during periods of transcendental meditation may be looked upon as random variable having the normal distribution with µ = 38.6 cc per minutes and σ = 6.5 cc per minute. Find the probabilities that during a period of transcendental meditation a person’s oxygen consumption will be reduced by a) at least 33.4 cc per minute; b) at most 34.7 cc per minute. 3. The sardines processed by a cannery have a mean length of 4.54 inches with a standard deviation of 0.25 inch. If the distribution of the lengths of the sardines can be approximated closely with a normal distribution, what percentage of the sardines are a) shorter than 4.00 inches; b) between 4.40 and 4.60 inches long ? 4. In a photographic process, the developing time of prints may be looked upon as a random variable having the normal distribution with µ = 15.40 seconds and σ = 0.48 second. Find the probabilities that the time it takes to develop one of the prints will be a) at least 16.00 seconds; b) at most 14.20 seconds; c) anywhere from 15.0 to 15.8 seconds. 5. In a very large class in European history, the final examination grades have a mean of 71.6 and a standard deviation of 12.6. If it is reasonable to approximate the distribution of these grades with a normal distribution, what percentage of the grades should exceed 79 ? 6. If the amount of time a tourist spends in a famous museum is a random variable having the normal distribution with µ = 43.4 minutes and σ = 6.8 minutes, find the probabilities that a tourist will spend a) at most 36.0 minutes in the museum; b) anywhere from 40.0 to 50.0 minutes in the museum. 7. If a random variables has the normal distribution with µ = 102.4 and σ = 3.6, find the probabilities that it will take on a value a) less than 107.6; b) greater than 99.7; c) between 106.9 and 110.5; d) between 96.1 and 104.2. 8. The average time required to perform job A is 78.5 minutes with a standard deviation of 16.2 minutes, and the average time required to perform job B is 103.2 minutes with a standard deviation of 11.3 minutes with a standard deviation of 11.3 minutes. Assuming normal distributions, what proportion of the time will job A take longer than the average job B, and what proportion of time will job B take less time than the average job A ? 9. The Beanstalk Club, a social organization for tall people, has a requirement that women must be at least 70 in. (or 5 ft 10 in.) tall. Suppose you are trying to decide whether to open a branch of the Beanstalk Club at your college with 500 female students where the mean height is 66 in. and the standard deviation is 1.5 in., then: a) Find the percentage of female students who are eligible for membership because they meet the minimum height requirement of 70 in. b) Among the 500 female students in your college, how many would be eligible for Beanstalk Club membership? 10. Replacement times for TV sets are normally distributed with a mean of 8.2 years and a standard deviation of 1. 1 years (based on data from " Getting Things Fixed," Consumer Reports). Find the probability that a randomly selected TV set will have a replacement time less than 7.0 years.

Transcript of Statistics - Champlain College St. Lawrenceweb2.slc.qc.ca/sh/Math_337/E17) Non-Standard Normal...

Page 1: Statistics - Champlain College St. Lawrenceweb2.slc.qc.ca/sh/Math_337/E17) Non-Standard Normal Distributions.pdf · Statistics Math 201-337-77 1RQ 6WDQGDUG 1RUPDO ... The reduction

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Filename: E17) Non-Standard Normal Distributions.Doc

1. If the assembly time of an “easy to assemble” toy is a random variable having the normal distribution with µ = 12.8minutes and σ = 4.0 minutes, what are the probabilities that this kind of toy can be assembled ina) less than 10.0 minutes; b) anywhere from 11.0 to 14.6 minutes ?

2. The reduction of a person’s oxygen consumption during periods of transcendental meditation may be looked upon asrandom variable having the normal distribution with µ = 38.6 cc per minutes and σ = 6.5 cc per minute. Find theprobabilities that during a period of transcendental meditation a person’s oxygen consumption will be reduced bya) at least 33.4 cc per minute; b) at most 34.7 cc per minute.

3. The sardines processed by a cannery have a mean length of 4.54 inches with a standard deviation of 0.25 inch. If thedistribution of the lengths of the sardines can be approximated closely with a normal distribution, what percentage of thesardines area) shorter than 4.00 inches; b) between 4.40 and 4.60 inches long ?

4. In a photographic process, the developing time of prints may be looked upon as a random variable having the normaldistribution with µ = 15.40 seconds and σ = 0.48 second. Find the probabilities that the time it takes to develop one ofthe prints will bea) at least 16.00 seconds;b) at most 14.20 seconds;

c) anywhere from 15.0 to 15.8 seconds.

5. In a very large class in European history, the final examination grades have a mean of 71.6 and a standard deviation of12.6. If it is reasonable to approximate the distribution of these grades with a normal distribution, what percentage of thegrades should exceed 79 ?

6. If the amount of time a tourist spends in a famous museum is a random variable having the normal distribution withµ = 43.4 minutes and σ = 6.8 minutes, find the probabilities that a tourist will spenda) at most 36.0 minutes in the museum; b) anywhere from 40.0 to 50.0 minutes in the

museum.

7. If a random variables has the normal distribution with µ = 102.4 and σ = 3.6, find the probabilities that it will take on avaluea) less than 107.6;b) greater than 99.7;

c) between 106.9 and 110.5;d) between 96.1 and 104.2.

8. The average time required to perform job A is 78.5 minutes with a standard deviation of 16.2 minutes, and the averagetime required to perform job B is 103.2 minutes with a standard deviation of 11.3 minutes with a standard deviation of11.3 minutes. Assuming normal distributions, what proportion of the time will job A take longer than the average job B,and what proportion of time will job B take less time than the average job A ?

9. The Beanstalk Club, a social organization for tall people, has a requirement that women must be at least 70 in. (or 5 ft 10in.) tall. Suppose you are trying to decide whether to open a branch of the Beanstalk Club at your college with 500female students where the mean height is 66 in. and the standard deviation is 1.5 in., then:a) Find the percentage of female students who are eligible for membership because they meet the minimum height

requirement of 70 in.b) Among the 500 female students in your college, how many would be eligible for Beanstalk Club membership?

10. Replacement times for TV sets are normally distributed with a mean of 8.2 years and a standard deviation of 1. 1 years(based on data from " Getting Things Fixed," Consumer Reports). Find the probability that a randomly selected TV setwill have a replacement time less than 7.0 years.

Page 2: Statistics - Champlain College St. Lawrenceweb2.slc.qc.ca/sh/Math_337/E17) Non-Standard Normal Distributions.pdf · Statistics Math 201-337-77 1RQ 6WDQGDUG 1RUPDO ... The reduction

11. Replacement times for CD players are normally distributed with a mean of 7.1 years and a standard deviation of 1.4years (based on data from "Getting Things Fixed," Consumer Reports). Find the probability that a randomly selected CDplayer will have a replacement time less than 8.0 years.

12. One classic use of the normal distribution is inspired by a letter to Dear Abby in which a wife claimed to have givenbirth 308 days after a brief visit from her husband, who was serving in the navy. The lengths of pregnancies are normallydistributed with a mean of 268 days and a standard deviation of 15 days. Given this information, find the probability of apregnancy lasting 308 days or longer. What does the result suggest?

13. The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. If westipulate that a baby is premature if born at least three weeks early, what percentage of babies are born prematurely?(Such information is important to hospital administrators, who need to ensure that correct equipment is available tohandle premature babies' special needs.)

14. According to the Opinion Research Corporation, men spend an average of 11.4 min in the shower. Assume that the timesare normally distributed with a standard deviation of 1.8 min. If a man is randomly selected, find the probability that hespends at least 10.0 min in the shower.

15. Measurements of human skulls from different epochs are analysed to determine whether they change over time. Themaximum breadth is measured for skulls from Egyptian males who lived around 3300 B.C. Results show that thosebreadths are normally distributed with a mean of 132.6 mm and a standard deviation of 5.4 mm (based on data fromAncient Races of the Tbebaid by Thomson and Randall-Maciver). An archeologist discovers a male Egyptian skull and afield measurement reveals a maximum breadth of 119 mm. Find the probability of getting a value of 119 or less if a skullis randomly selected from the period around 3300 B.C. Is the newly found skull likely to come from that era?

16. Some vending machines are designed so that their owners can adjust the weights of the quarters that are accepted. Ifmany counterfeit coins are found, adjustments are made to reject more coins, with the effect that most of the counterfeitcoins are rejected along with many legal coins. Assume that quarters have weights that are normally distributed with amean of 5.67 g and a standard deviation of 0.070 g. If a vending machine is adjusted to reject quarters weighing less than5.50 g or more than 5.80 g, what is the percentage of legal quarters that are rejected?

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ANSWERS:

1. a) 0.2420 b) 0.3472 2. a) 0.7881 b) 0.2743

3. a) 0.0154 b) 0.3071

4. a) 0.1056 b) 0.0062 c) 0.5934 5. 0.2643

6. a) 0.1379 b) 0.5255

7. a) 0.9332 b) 0.7734 c) 0.0934 d) 0.6514

8. 0.0643 0.0143 (0.4933)

9. a) 0.0038 b) 2 (i.e. 1.94) 10. 0.1379 11. 0.7389

12. 0.0038; either a very rare event has occurred or the husband is not the father.

13. 0.0808 14. 0.7823 15. 0.0059 16. 0.0389