Exercises

Prerequisites

• All material presented in the Normal Distributions chapter

1. If scores are normally distributed with a mean of 35 and a standard deviation of 10, what percent of the scores is:

a. greater than 34?

b. smaller than 42?

c. between 28 and 34?

2. What are the mean and standard deviation of the standard normal distribution? (b) What would be the mean and standard deviation of a distribution created by multiplying the standard normal distribution by 8 and then adding 75?

3. The normal distribution is defined by two parameters. What are they?

4. What proportion of a normal distribution is within one standard deviation of the mean? (b) What proportion is more than 2.0 standard deviations from the mean? (c) What proportion is between 1.25 and 2.1 standard deviations above the mean?

5. A test is normally distributed with a mean of 70 and a standard deviation of 8. (a) What score would be needed to be in the 85th percentile? (b) What score would be needed to be in the 22nd percentile?

6. Assume a normal distribution with a mean of 70 and a standard deviation of 12. What limits would include the middle 65% of the cases?

7. A normal distribution has a mean of 20 and a standard deviation of 4. Find the Z scores for the following numbers: (a) 28 (b) 18 (c) 10 (d) 23

@Assume the speed of vehicles along a stretch of 1-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph.

a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit?

b. What proportion of the vehicles would be going less than 50 mph?

c. A new speed limit will be initiated such that approximately 10% of vehicles will be over the speed limit. What is the new speed limit based on this criterion?

d. In what way do you think the actual distribution of speeds differs from a normal distribution?

9. A variable is normally distributed with a mean of 120 and a standard deviation of 5. One score is randomly sampled. What is the probability it is above 127?

10. You want to use the normal distribution to approximate the binomial distribution. Explain what you need to do to find the probability of obtaining exactly 7 heads out of 12 flips.

@A group of students at a school takes a history test. The distribution is normal with a mean of 25, and a standard deviation of 4. (a) Everyone who scores in the top 30% of the distribution gets a certificate. What is the lowest score someone can get and still earn a certificate? (b) The top 5% of the scores get to compete in a statewide history contest. What is the lowest score someone can get and still go onto compete with the rest of the state?

@use the normal distribution to approximate the binomial distribution and find the probability of getting 15 to 18 heads out of 25 flips. Compare this to what you get when you calculate the probability using the binomial distribution. Write your answers out to four decimal places.

13. True/false: For any normal distribution, the mean, median, and mode will be equal.

14. True/false: In a normal distribution, 11.5% of scores are greater than Z = 1.2.

15. True/false: The percentile rank for the mean is 50% for any normal distribution.

16. True/false: The larger then, the better the normal distribution approximates the binomial distribution.

17. True/false: AZ-score represents the number of standard deviations above or below the mean.

Figure 6.17 b. P <x< ____ _,

59. Find the 70th percentile of the distribution for the time a CD player lasts. a. Sketch the situation. Label and scale the axes. Shade the region corresponding to the lower 70%.

Figure 6.18 b. P(x < k) = ____ Therefore, k = ___ _

HOMEWORK

6.1 The Standard Normal Distribution

Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days.

~at is the median recovery time? a. 2.7 b. 5.3 c. 7.4 d. 2.1

61. What is the z-score for a patient who takes ten days to recover? a. 1.5 b. 0.2 c. 2.2 d. 7.3

62. The length of time to find it takes to find a parking space at 9 A.M. follows a normal distribution with a mean of five minutes and a standard deviation of two minutes. If the mean is significantly greater than the standard deviation, which of the following statements is true?

I. The data cannot follow the uniform distribution. II. The data cannot follow the exponential distribution ..

Ill. The data cannot follow the normal distribution.

a. I only b. II only c. III only d. I, II, and Ill

362 CHAPTER 6 I THE NORMAL DISTRIBUTION

63. The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005-2006 season. The heights -0f basketball players have an approximate normal distribution with mean, µ = 79 inches and a standard deviation, a= 3.89 inches. For each of the following heights, calculate the z-score and interpret it using complete sentences.

a. 77 inches b. 85 inches c. If an NBA player reported his height had a z-score of 3.5, would you believe him? Explain your answer.

64. The systolic blood pressure (given in millimeters) of males has an approximately normal distribution with mean µ = 125 and standard deviation a = 14. Systolic blood pressure for males follows a normal disttibution.

a. Calculate the z-scores for the male systolic blood pressures 100 and 150 millimeters. b. If a male friend of yours said he thought his systolic blood pressure was 2.5 standard deviations below the mean,

but that he believed his blood pressure was between 100 and 150 millimeters, what would you say to him?

65. Kyle's doctor told him that the z-score for his systolic blood pressure is 1.75. Which of the following is the best interpretation of this standardized score? The systolic blood pressure (given in millimeters) of males has an approximately normal distribution with meanµ = 125 and standard deviation a= 14. If X = a systolic blood pressure score then X ~ N (125, 14).

a. Which answer(s) is/are correct? · i. Kyle's systolic blood pressure is 175. ii. Kyle's systolic blood pressure is 1.75 times the average blood pressure of men his age. iii. Kyle's systolic blood pressure is 1.75 above the average systolic blood pressure of men his age. iv. Kyles's systolic blood pressure is 1.75 standard deviations above the average systolic blood pressure for

men. b. Calculate Kyle's blood pressure.

1'66)Height and weight are two measurements used to track a child's development. The World Health Organization measures ~d development by comparing the weights of children who are the same height and the same gender. In 2009, weights for

all 80 cm girls in the reference population had a mean µ = 10.2 kg and standard deviation a= 0.8 kg. Weights are normally distributed. X ~ N(l0.2, 0.8). Calculate the z-scores that correspond to the following weights and interpret them.

a. 11 kg b. 7.9 kg c. 12.2 kg

67. In 2005, 1,475,623 students heading to college took the SAT. The distribution of scores in the math section of the SAT follows a normal distribution with meanµ = 520 and standard deviation a = 115.

a. Calculate the z-score for an SAT score of 720. Interpret it using a complete sentence. b. What math SAT score is 1.5 standard deviations above the mean? What can you say about this SAT score? c. For 2012, the SAT math test had a mean of 514 and standard deviation 117. The ACT math test is an alternate to

the SAT and is approximately normally disttibuted with mean 21 and standard deviation 5.3. If one person took the SAT math test and scored 700 and a second person took the ACT math test and scored 30, who did better with respect to the test they took?

6.2 Using the Normal Distribution

Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days.

68. What is the probability of spending more than two days in recovery? a. 0.0580 b. 0.8447 c. 0.0553 d. 0.9420

69. The goth percentile for recovery times is? a. 8.89 b. 7.07 c. 7.99 d. 4.32

Use the following information to answer the next three exercises: The length of time it takes to find a parking space at 9 A.M. follows a normal distribution with a mean of five minutes and a standard deviation of two minutes.

70. Based upon the given information and numerically justified, would you be surprised if it took less than one minute to find a parking space?

a. Yes b. No

c. Unable to determine

71. Find the probability that it takes at least eight minutes to find a parking space. a. 0.0001 b. 0.9270 c. 0.1862 d. 0.0668

72. Seventy percent of the time, it takes more than how many minutes to find a parking space? a. 1.24 b. 2.41 c. 3.95 d. 6.05

73. According to a study done by De Anza students, the height for Asian adult males is normally distributed with an average of 66 inches and a standard deviation of 2.5 inches. Suppose one Asian adult male is randomly chosen. Let X = height of the individual.

a. x ~ __ L__,__)

b. Find the probability that the person is between 65 and 69 inches. Include a sketch of the graph, and write a probability statement.

c. Would you expect to meet many Asian adult males over 72 inches? Explain why or why not, and justify your answer numerically.

d. The middle 40% of heights fall between what two values? Sketch the graph, and write the probability statement.

74. IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let X = IQ of an individual.

a. x ~ __ L__,__)

b. Find the probability that the person has an IQ greater than 120. Include a sketch of the graph, and write a probability statement.

c. MENSA is an organization whose members have the top 2% of all IQs. Find the minimum IQ needed to qualify for the MENSA organization. Sketch the graph, and write the probability statement.

d. The middle 50% of IQs fall between what two values? Sketch the graph and write the probability statement.

75. The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of 10. Suppose that one individual is randomly chosen. Let X = percent of fat calories.

a. x ~ __ L__,__) b. Find the probability that the percent of fat calories a person consumes is more than 40. Graph the situation. Shade

in the area to be determined. c. Find the maximum number for the lower quarter of percent of fat calories. Sketch the graph and write the

probability statement.

~Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and ~andard deviation of 50 feet.

a. If X = distance in feet for a fly ball, then X ~ __ L__,__) b. If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than

220 feet? Sketch the graph. Scale the horizontal axis X. Shade the region corresponding to the probability. Find the probability.

c. Find the 80th percentile of the distribution of fly balls. Sketch the graph, and write the probability statement.

77. In China, four-year-olds average three hours a day unsupervised. Most of the unsupervised children live in rural areas, considered safe. Suppose that the standard deviation is 1.5 hours and the amount of time spent alone is normally distributed. We randomly select one Chinese four-year-old living in a rural area. We are interested in the amount of time the child spends alone per day.

a. In words, define the random variable X. b. x ~ __ L__,__) c. Find the probability that the child spends less than one hour per day unsupervised. Sketch the graph, and write the

probability statement. d. What percent of the children spend over ten hours per day unsupervised? e. Seventy percent of the children spend at least how long per day unsupervised?

78. In the 1992 presidential election, Alaska's 40 election districts averaged 1,956.8 votes per district for President Clinton. The standard deviation was 572.3. (There are only 40 election districts in Alaska.) The distribution of the votes per district for President Clinton was bell-shaped. Let X = number of votes for President Clinton for an election district.

a. State the approximate distribution of X. b. Is 1,956.8 a population mean or a sample mean? How do you know? c. Find the probability that a randomly selected district had fewer than 1,600 votes for President Clinton. Sketch the

graph and write the probability statement.

40,000 40,000 45,050 45,500 46,249 48,134

49,133 50,071 50,096 50,466 50,832 51,100

51,500 51,900 52,000 52,132 52,200 52,530

52,692 53,864 54,000 55,000 55,000 55,000

55,000 55,000 55,000 55,082 57,000 58,008

59,680 60,000 60,000 60,492 60,580 62,380

62,872 64,035 65,000 65,050 65,647 66,000

66,161 67,428 68,349 68,976 69,372 70,107

70,585 71,594 72,000 72,922 73,379 74,500

75,025 76,212 78,000 80,000 80,000 82,300

Table 6.4

a. Calculate the sample mean and the sample standard deviation for the maximum capacity of sports stadiums (the data).

b. Construct a histogram. c. Draw a smooth curve through the midpoints of the tops of the bars of the histogram. d. In words, describe the shape of your histogram and smooth curve. e. Let the sample mean approximate µ and the sample standard deviation approximate a. The distribution of X can

then be approximated by X - __ (__,__J. f. Use the distribution in part e to calculate the probability that the maximum capacity of sports stadiums is less than

67,000 spectators. g. Determine the cumulative relative frequency that the maximum capacity of sports stadiums is less than 67,000

spectators .. Hint: Order the data and count the sports stadiums that have a maximum capacity less than 67,000. Divide by the total number of sports stadiums in the sample.

h. Why aren't the answers to part f and part g exactly the same?

84. An expert witness for a paternity lawsuit testifies that the length of a pregnancy is normally distributed with a mean of 280 days and a standard deviation of 13 days. An alleged father was out of the country from 240 to 306 days before the birth of the child, so the pregnancy would have been less than 240 days or more than 306 days long if he was the father. The birth was uncomplicated, and the child needed no medical intervention. What is the probability that he was NOT the father? What is the probability that he could be the father? Calculate the z-scores first, and then use those to calculate the probability.

85. A NUMMI assembly line, which has been operating since 1984, has built an average of 6,000 cars and trucks a week. Generally, 10% of the cars were defective coming off the assembly line. Suppose we draw a random sample of n = 100 cars. Let X represent the number of defective cars in the sample. What can we say about X in regard to the 68-95-99. 7 empirical rule (one standard deviation, two standard deviations and three standard deviations from the mean are being referred to)? Assume a normal distribution for the defective cars in the sample.

86. We flip a coin 100 times (n = 100) and note that it only comes up heads 20% (p = 0.20) of the time. The mean and standard deviation for the number of times the coin lands on heads is µ = 20 and a = 4 (verify the mean and standard deviation). Solve the following:

a. There is about a 68% chance that the number of heads will be somewhere between _ and _ . b. There is about a __ chance that the number of heads will be somewhere between 12 and 28. c. There is about a __ chance that the number of heads will be somewhere between eight and 32.

87. A $1 scratch off lotto ticket will be a winner one out of five times. Out of a shipment of n = 190 lotto tickets, find the probability for the lotto tickets that there are

a. somewhere between 34 and 54 prizes. b. somewhere between 54 and 64 prizes. c. more than 64 prizes.

@Facebook provides a variety of statistics on its Web site that detail the growth and popularity of the site.

On average, 28 percent of 18 to 34 year olds check their Facebook profiles before getting out of bed in the morning. Suppose this percentage follows a normal distribution with a standard deviation of five percent.

Q Find the probability that the percent of 18 to 34-year-olds who check Facebook before getting out of bed in the morning is at least 30.

~ Find the 95th percentile, and express it in a sentence.

REFERENCES

6.1 The Standard Normal Distribution

"Blood Pressure of Males and Females." StatCruch, 2013. Available online at http://www.statcrunch.com/5.0/ viewreport.php?reportid=11960 (accessed May 14, 2013).

"The Use of Epidemiological Tools in Conflict-affected populations: Open-access educational resources for policy-makers: Calculation of z-scores." London School of Hygiene and Tropical Medicine, 2009. Available online at http://conflict.lshtm.ac.uk/page_125.htm (accessed May 14, 2013).

"2012 College-Bound Seniors Total Group Profile Report." CollegeBoard, 2012. Available online at http://media.collegeboard.com/digitalServices/pdf/research/TotalGroup-2012.pdf (accessed May 14, 2013).

"Digest of Education Statistics: ACT score average and standard deviations by sex and race/ethnicity and percentage of ACT test takers, by selected composite score ranges and planned fields of study: Selected years, 1995 through 2009." National Center for Education Statistics. Available online at http://nces.ed.gov/programs/digest/d09/tables/dt09_147.asp (accessed May 14, 2013).

Data from the San Jose Mercury News.

Data from The World Almanac and Book of Facts.

"List of stadiums by capacity." Wikipedia. Available online at https://en.wikipedia.org/wiki/List_of_stadiums_by_capacity (accessed May 14, 2013).

Data from the National Basketball Association. Available online at www.nba.com (accessed May 14, 2013).

6.2 Using the Normal Distribution

"Naegele's rule." Wikipedia. Available online at http://en.wikipedia.org/wiki/Naegele's_rule (accessed May 14, 2013).

"403: NUMMI." Chicago Public Media & Ira Glass, 2013. Available online at http://www.thisamericanlife.org/radio­archives/episode/403/nummi (accessed May 14, 2013).

"Scratch-Off Lottery Ticket Playing Tips." WinAtTheLottery.com, 2013. Available online at http://www.winatthelottery.com/publiddepartment40.cfm (accessed May 14, 2013).

"Smart Phone Users, By The Numbers." VIsual.ly, 2013. Available online at http://visual.ly/smart-phone-users-numbers (accessed May 14, 2013).

"Facebook Statistics." Statistics Brain. Available online at http://www.statisticbrain.com/facebook-statistics/(accessed May 14, 2013).

SOLUTIONS

1 ounces of water in a bottle

3 2

5 -4

7 -2

9 The mean becomes zero.

11 z=2

13 z = 2.78

15 x = 20

17 x = 6.5

61. Previously, De Anza statistics students estimated that the amount of change daytime statistics students carry is exponentially distributed with a mean of $0.88. Suppose that we randomly pick 25 daytime statistics students.

a. In words, X = ____ _ b. x ~ __ L._,__j

c. In words, X = ____ _

d. x ~ __ (......_ _ _,. __J

e. Find the probability that an individual had between $0.80 and $1.00. Graph the situation, and shade in the area to be determined.

f. Find the probability that the average of the 25 students was between $0.80 and $1.00. Graph the situation, and shade in the area to be determined.

g. Explain why there is a difference in part e and part f.

@suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.

(ii) If X = average distance in feet for 49 fly balls, then ~ ~ ( , .

@ What is the probabi~ity that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the

horizontal axis for X . Shade the region corresponding to the probability. Find the probability.

0 Find the 80th percentile of the distribution of the average of 49 fly balls.

63. According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, learn, prepare, copy, assemble, and send) IRS Form 1040 is 10.53 hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is two hours. Suppose we randomly sample 36 taxpayers.

a. In words, X = _____ _

b. In words, X = _____ _

c. x ~ __ L._,__)

d. Would you be surprised if the 36 taxpayers finished their Form 1040s in an average of more than 12 hours? Explain why or why not in complete sentences.

e. Would you be surprised if one taxpayer finished his or her Form 1040 in more than 12 hours? In a complete sentence, explain why.

64. Suppose that a category of world-class runners are known to run a marathon (26 miles) in an average of 145 minutes

with a standard deviation of 14 minutes. Consider 49 of the races. Let X the average of the 49 races.

a. X ~ _ _ L._,__J

b. Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons.

c. Find the 80th percentile for the average of these 49 marathons. d. Find the median of the average running times.

65. The length of songs in a collector's iTunes album collection is uniformly distributed from two to 3.5 minutes. Suppose we randomly pick five albums from the collection. There are a total of 43 songs on the five albums.

a. In words, X = ___ _ b. x~ ____ _ c. In words, X = _____ _

d. x ~--~__J

e. Find the first quartile for the average song length. f. The IQR(interquartile range) for the average song length is from ___ __ _

66. In 1940 the average size of a U.S. farm was 174 acres. Let's say that the standard deviation was 55 acres. Suppose we randomly swvey 38 farmers from 1940.

a. In words, X = _____ _

b. In words, X = _____ _

c. x ~ __ L._,__)

d. The IQR for X is from ___ acres to ___ acres.

67. Determine which of the following are true and which are false. Then, in complete sentences, justify your answers.

a. When the sample size is large, the mean of X is approximately equal to the mean of X.

b. When the sample size is large, X is approximately normally distributed.

c. When the sample size is large, the standard deviation of X is approximately the same as the standard deviation

ofX.

68. The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about

36 and a standard deviation of about ten. Suppose that 16 individuals are randomly chosen. Let X =average percent of fat

calories.

a. X ~ .. ___ _,. _____)

b. For the group of 16, find the probability that the average percent of fat calories consumed is more than five. Graph the situation and shade in the area to be determined.

c. Find the first quartile for the average percent of fat calories.

69. The distribution of income in some Third World countries is considered wedge shaped (many very poor people, very few middle income people, and even fewer wealthy people). Suppose we pick a country with a wedge shaped distribution. Let the average salary be $2,000 per year with a standard deviation of $8,000. We randomly survey 1,000 residents of that country.

a. In words, X = _____ _

b. In words, X = _____ _

c. x ~ <"--~-__, d. How is it possible for the standard deviation to be greater than the average? e. Why is it more likely that the average of the 1,000 residents will be from $2,000 to $2,100 than from $2,100 to

$2,200?

~ch of the following is NOT TRUE about the distribution for averages? a. The mean, median, and mode are equal. b. The area under the curve is one. c. The curve never touches the x-axis. d. The curve is skewed to the right.

71. The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations. The distribution to use for the average cost of gasoline for the 16 gas stations is:

a. X ~ N(4.59, 0.10)

b. X ~ N( 4.59 0.10) ' v'I6

c. x ~ N (4.59, 0~~0)

d. x ~ ( 16 ) N 4.59, O.lO

7.2 The Central Limit Theorem for Sums

72. Which of the following is NOT TRUE about the theoretical distribution of sums? a. The mean, median and mode are equal. b. The area under the curve is one. c. The curve never touches the x-axis. d. The curve is skewed to the right.

73. Suppose that the duration of a particular type of criminal trial is known to have a mean of 21 days and a standard deviation of seven days. We randomly sample nine trials.

a. In words, I:X = _____ _ b. I:X ~ _ _ L-..,___j c. Find the probability that the total length of the nine trials is at least 225 days. d. Ninety percent of the total of nine of these types of trials will last at least how long?

7 4. Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from two to six pounds with a mean of four pounds and standard deviation of 1.1547. We randomly survey 64 homes with children.

a. In words, X = _____ _ b. The distribution is __ _ c. In words, I:X = _ _____ _

94. The Screw Right Company claims their t inch screws are within ±0.23 of the claimed mean diameter of 0. 750 inches

with a standard deviation of 0.115 inches. The following data were recorded.

0.757 0.723 0.754 0.737 0.757 0.741 0.722 0.741 0.743 0.742

0.740 0.758 0.724 0.739 0.736 0.735 0.760 0.750 0.759 0.754

0.744 0.758 0.765 0.756 0.738 0.742 0.758 0.757 0.724 0.757

0.744 0.738 0.763 0.756 0.760 0.768 0.761 0.742 0.734 0.754

0.758 0.735 0.740 0.743 0.737 0.737 0.725 0.761 0.758 0.756

Table 7.8

The screws were randomly selected from the local home repair store.

a. Find the mean diameter and standard deviation for the sample b. Find the probability that 50 randomly selected screws will be within the stated tolerance levels. ls the company's

diameter claim plausible?

95. Your company has a contract to perform preventive maintenance on thousands of air-conditioners in a large city. Based on service records from previous years, the time that a technician spends servicing a unit averages one hour with a standard deviation of one hour. In the coming week, your company will service a simple random sample of 70 units in the city. You plan to budget an average of 1.1 hours per technician to complete the work. Will this be enough time?

@typical adult has an average IQ score of 105 with a standard deviation of 20. If 20 randomly selected adults are given an IQ tesst, what is the probability that the sample mean scores will be between 85 and 125 points?

97. Certain coins have an average weight of 5.201 grams with a standard deviation of 0.065 g. If a vending machine is designed to accept coins whose weights range from 5.111 g to 5.291 g, what is the expected number of rejected coins when 280 randomly selected coins are inserted into the machine?

REFERENCES

7.1 The Central Limit Theorem for Sample Means (Averages)

Baran, Daya. "20 Percent of Americans Have Never Used Email."WebGuild, 2010. Available online at http://www.webguild.org/20080519/20-percent-of-americans-have-never-used-email (accessed May 17, 2013).

Data from The Flurry Blog, 2013. Available online at http://blog.flurry.com (accessed May 17, 2013).

Data from the United States Department of Agriculture.

7.2 The Central Limit Theorem for Sums

Farago, Peter. "The Truth About Cats and Dogs: Smartphone vs Tablet Usage Differences." The Flurry Blog, 2013. Posted October 29, 2012. Available online at http://blog.flurry.com (accessed May 17, 2013).

7.3 Using the Central Limit Theorem

Data from the Wall Street Journal.

"National Health and Nutrition Examination Survey." Center for Disease Control and Prevention. Available online at http://www.cdc.gov/nchs/nhanes.htm (accessed May 17, 2013).

SOLUTIONS

1 mean = 4 hours; standard deviation = 1.2 hours; sample size = 16

3 a. Check student's solution. b. 3.5, 4.25, 0.2441

5 The fact that the two distributions are different accounts for the different probabilities.

Discrete Probability Distribution ? ? ?

Imagine you are in a game show, where

Now, let us start the money give-away! There are 4 prizes hidden on a game board with 16 spaces. One prize is worth $4000, another is worth $1500, and two are worth $1000.

But, wait!!! You are also told that, in the rest of the spaces, there will be a bill of $50 that you have to pay to the host as a penalty for not making the "wise" choice.

OK, you are lucky that you only have to pay $50 for making a bad choice. Imagine that you failed to answer the question asked by

in the Monty Python and the Holy Grail!

But, of course, it is a much kinder and gentler world now than the time of King Arthur and his knights.

In this modem game show, you are actually given a choice, a real choice.

Choice # 1 : You are offered a sure prize of $400 cash, and you just take the money and walk away. Period. No question asked .... .

Choice #2: Take your chance and play the game ...... .

What would be your choice? Take the money and run, or play the game? Why??? Hmmmm .... .. .

You have to make a decision ...... quick ..... .

By the way, I am not that devil above. He is just a friend, a drinking pal, from hell.