Having Probability - Probablity - Exam | Exams Probability and Statistics

ST 311 ReilandFINAL EXAM PRACTICE PROBLEMS

See class web page for topics covered on the final exam.

Solutions are at end of file.

1. Determine which of the following functions is in fact a probability distribution function.

a. , 3, 4, 5, 6. p(x) xœ1

4œ

b. , 0, 1, 2, 3, 4. p(x) xœœ

c. , 0, 1, 2, 3. p(x) xœœ

x#

2. In a population of students the number of calculators owned is a random variable with (0) .2,xpœ

pp(1) .6, (2) .2. Find the expected value and standard deviation of this probability distribution.œœ

3. An oil firm is to drill three wells, with each well having probability 0.2 of successfully producing oil.

It costs the firm $20,000 to drill each well. A successful well will bring in oil worth $750,000. Let the

random variable X be the oil firm's from well , . The wells are in different geographic

3gain 33œ"ß#ß$

areas and so the drilling outcome at any well has no affect on the drilling outcomes at the other wells.

a. Find the firm's expected gain G from the three wells.

b. Find the standard deviation of the firm's gain.

4. Let the random variable denote the displacement in cubic inches ) of the engine in a particular\Ð38

model of automobile. The size of the engine (that is, the cubic inch displacement) varies depending on

the options chosen by the buyer of the automobile. It is known that in and 22 in .IÐ\Ñ œ "(( œ

If denotes the engine diplacement in cubic centimeters ), determine and . Note that\Ð-7IÐ\Ñ

‡$‡

5‡

1 in 16.4 cm .

5. Let be the number of accidents per week at a hazardous intersection; varies with mean 2.2 and\\

standard deviation 1.4. Let , and be the number of accidents in each of 3 different weeks at\ß\ \

"# $

this intersection. The number of accidents in a week is not affected by the number of accidents in any

other week. What is , the standard deviation of the sum ?5Ð\ \ \ Ñ "#$

"#$ \\\

6. The probability distribution below describes the number of repair calls that an appliance repair shop

may receive during an hour.

0123

0.1 0.3 0.4 0.2

Repair calls

Probability

a. How many calls should the shop to receive per hour? What is the standard deviation?expect

b. Find the expected value and standard deviation of the number of repair calls the appliance shop

should expect during an 8-hour day.

7. A college student on a meal plan reports that the amount of money he spends daily on food varies with

a mean (expected value) of $13.50 and a standard deviation of $7.

a. Find the expected value and standard deviation of the amount he spends on 2 consecutive days (the

amounts he spends on different days are independent).

b. Find the expected value and standard deviation of the amount he spends during a semester that spans

120 days.

8. Which of the following best describes the origin of in the binomial probability function?:Ð":Ñ

B8B

a) It is the probability of each path with exactly successes and failures.BÐ8BÑ

b) It is the probability that the first success occurs in the th trial among the trials.B8

c) It is the probability of or more successes among the trials.B8

d) It is the probability of or less successes among the trials.B8

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ST 311 FINAL EXAM PRACTICE PROBLEMS Reiland

See class web page for topics covered on the final exam. Solutions are at end of file.

Determine which of the following functions is in fact a probability distribution function.

a. p(x) œ 14 , x œ3, 4, 5, 6.

b. p(x) œ x , x œ0, 1, 2, 3, 4.

c. p(x) œ 5  6 x , x œ0, 1, 2, 3.

In a population of students the number of calculators owned is a random variable x with p (0) œ.2, p (1) œ .6, p (2) œ.2. Find the expected value and standard deviation of this probability distribution.
An oil firm is to drill three wells, with each well having probability 0.2 of successfully producing oil. It costs the firm $20,000 to drill each well. A successful well will bring in oil worth $750,000. Let the random variable X be the oil firm's 3 gain^ from well ,3 3 œ "ß #ß $. The wells are in different geographic areas and so the drilling outcome at any well has no affect on the drilling outcomes at the other wells. a. Find the firm's expected gain G from the three wells. b. Find the standard deviation of the firm's gain.
Let the random variable \ denote the displacement in cubic inches Ð38$) of the engine in a particular model of automobile. The size of the engine (that is, the cubic inch displacement) varies depending on the options chosen by the buyer of the automobile. It is known that IÐ\Ñ œ "(( in $^ and (^5) \œ22 in .$ If \ ‡denotes the engine diplacement in cubic centimeters^ Ð-7 $), determine^ IÐ\ Ñ‡^ and (^5) \ ‡. Note that 1 in $^ œ16.4 cm .$
Let \ be the number of accidents per week at a hazardous intersection; \varies with mean 2.2 and standard deviation 1.4. Let \ ß " #, and $be the number of accidents in each of 3 different weeks at this intersection. The number of accidents in a week is not affected by the number of accidents in any other week. What is (^5) Ð\ \ \ Ñ" # $ , the standard deviation of the sum "  #  $?
The probability distribution below describes the number of repair calls that an appliance repair shop may receive during an hour. 0 1 2 3 0.1 0.3 0.4 0.

Repair calls Probability a. How many calls should the shop expectto receive per hour? What is the standard deviation? b. Find the expected value and standard deviation of the number of repair calls the appliance shop should expect during an 8-hour day.

A college student on a meal plan reports that the amount of money he spends daily on food varies with a mean (expected value) of $13.50 and a standard deviation of $7. a. Find the expected value and standard deviation of the amount he spends on 2 consecutive days (the amounts he spends on different days are independent). b. Find the expected value and standard deviation of the amount he spends during a semester that spans 120 days.
Which of the following best describes the origin of : Ð"  :ÑB^ 8Bin the binomial probability function? a) It is the probability of each path with exactly B successes and Ð8  BÑfailures. b) It is the probability that the first success occurs in the Bth trial among the 8 trials. c) It is the probability of B or more successes among the 8 trials. d) It is the probability of B or less successes among the 8 trials.

Which of the following best describes the origin of 8 GBin the binomial probability function? a) 8xBx b) It is the number of successes among the 8 trials. c) It is the number of paths with B successes and Ð8  BÑfailures. d) It is the number of ways that the first success can occur on the Bth trial.
Let B be a binomial random variable with 8 œ #! and : œ !Þ%. Which of the following is(are) correct? a) the mean of Bis 12. b) the standard deviation of Bis 4.8. c)  Bœ!

#! :ÐBÑ œ "

d) all of the above are correct e) none of the above are correct

During the lunch-hour rush at a McDougalds restaurant orders for the Big Mac hamburger follow a Poisson distribution and occur at a rate of 4 per minute. What is the probability that 5 or more Big Macs will be ordered in one minute?
As the sample size n increases, the width of a 98% confidence interval for the population mean tends to: a. increase b. decrease c. stay the same
In a study to establish the absolute threshold of hearing, 71 male college freshmen 19-21 years of age are asked to participate. Each subject is seated in a soundproof room and a 150 Hz tone is presented at a large number of stimulus levels in a randomized order. The subject is instructed to press a button if he detects the tone. The sample mean for the group was C œ21.6 decibels and the sample standard deviation was s = 2.1 decibels (which we can use as an estimate of the unknown population standard deviation 5 ). Estimate the mean absolute threshold of all men 19-21 years of age with a 99% confidence interval.
In a Risk Management Study on fires in compartmented fire-resistant buildings, the data below was generated. The data in the table give the number of victims who died trying to evacuate for a sample of 14 recent fires.

NUMBER OF

FIRE VICTIMS

Las Vegas Hilton (Las Vegas) 5 Inn on the Park (Toronto) 5 Westchase Hilton (Houston) 8 Holiday Inn (Cambridge, Ohio) 10 Conrad Hilton (Chicago) 4 Providence College (Providence) 8 Baptist Towers (Atlanta) 7 Howard Johnson (New Orleans) 5 Cornell University (Ithaca) 9 Westport Central Apartments (Kansias City, MO) 4 Orrington Hotel (Evanston IL) 0 Hartford Hospital (Hartford, CT) 16 Milford Plaza (New York) 0 MGM Grand (Las Vegas) 36

To investigate the possible link between fluoride content of drinking water and cancer, the cancer death rates (number of deaths per 100,000 population) from 1982-1999 in 20 selected U. S. cities - the ten largest fluoridated cities and the ten largest cities not fluoridated by 1969 - were recorded. These data were used to calculate for each city the annual rate of increase in cancer death rate over this 18 year period. The data are given below:

FLUORIDATED NONFLUORIDATED

Annual Increase in Annual Increase in City Cancer Death Rate City Cancer Death Rate Chicago 1.0640 Los Angeles. Philadelphia 1.4118 Boston 1. Baltimore 2.1115 New Orleans 1. Cleveland 1.9401 Seattle. Washington 3.8772 Cincinnati 4. Milwaukee -.4561 Atlanta -1. St. Louis 4.8359 Kansas City 2. San Francisco 1.8875 Columbus 1. Pittsburgh 4.4964 Newark -. Buffalo 1.4045 Portland 2.

a. Construct a 95% confidence interval for the difference between the mean annual increases in cancer death rates for fluoridated and nonfluoridated cities.

b. Let. (^) " be the mean annual increase in the cancer death rate for fluoridated cities and let .#be the mean annual increase in the cancer death rate for nonfluoridated cities. Perform a hypothesis test to investigate if the mean annual increase in the cancer death rate for fluoridated cities is greater than the mean annual increase in the cancer death rate for nonfluoridated cities.

Perform the hypothesis test shown below, L (^) !À : œ Þ!& L (^) +À :  Þ!& given that a random sample of size 1000 revealed that the number of successes was 40. Compute the P-value and use it to make a conclusion concerning the hypothesis test.
Recently there have been campaigns encouraging people to save energy by carpooling to work. Some cities have created “carpool only" traffic lanes (i. e. only cars with 2 or more passengers can use these lanes). In order to evaluate the effectiveness of carpool only lanes, toll booth personnel in one city monitor 2,000 randomly selected cars in 2005 before the carpool lanes were established and 1500 cars in 2008 after the lanes were established. The results are shown below, where x (^) " ( x #) is the number of cars with 2 or more passengers in the data for 2008 (2005). Use a 95% confidence interval to determine whether the data indicate that the proportion of cars with carpool riders has increased over this period. year 2008 : n (^) " œ 1,500, x (^) " œ 576; year 2005 : n (^) # œ 2,000, x #œ652.
The president of a large university has been studying the relationship between male/female supervisory structures in his institution and the level of employees' job satisfaction. The results of a recent survey are shown in the table below. Conduct a test at the 5% significance level to determine whether the level of job satisfaction depends on the boss/employee gender relationship.

Male/Female Female/Male Male/Male Female/Female Satisfied 60 15 50 15 Neutral 27 45 48 50 Dissatsfied 13 32 12

Boss/Employee Level of Satisfaction

When two competing teams are equally matched, the probability that each team wins any game is 0.5. The National Basketball Association (NBA) championship goes to the team that wins four games in a best-of-seven series. If the 2 teams are evenly matched, the probability that the series ends with one of the teams winning the first four games would be 2(0.5) %= 0.125 [team A wins in 4 games with probability (0.5) %^ àteam B can also win in 4 games with the same probability, so the probability the series ends in 4 games is 2(0.5) ].%

Similarly team A can win in 5 games if team A wins 3 of the first 4 games and then wins game 5. So team A wins in 5 games with probability (^) % G Ð!Þ&Ñ Ð!Þ&Ñ‡Ð!Þ&Ñ œ !Þ"#&$ $. But team B can also win in 5 games with the same probability, so the probability that the series ends in 5 games is !Þ#&. Similar probability calculations show that the probability is !Þ$"#&that the series lasts six games, and the probability is !Þ$"#&that the series lasts the full seven games. The table below shows the number of games it took to decide each of the last 57 NBA champs. Do you think the teams are usually equally matched? Give statistical evidence to support your conclusion.

Length of series 4 games 5 games 6 games 7 games NBA finals 7 13 22 15

A professor claims that 70% of College of Business graduates earn more than $45,000 per year. In a random sample of 300 graduates, 195 earn more than $45,000. Perform a hypothesis test to evaluate the professor's claim.

21. (^) s p (^) (^) #!!) œ 1500576 œ .384; p s#!!&œ 2000652 œ.326;

(.384  .326) „ 1.96 ^ (.384)(.616) 1500  (.326)(.674) 2000 œ .058 „ .032 Ê(.026, .09) Conclusion: Since the interval is entirely positive, conclude that the proportion of all cars with carpool riders has increased over the time period from 2005 to 2008.

22. Expected cell counts are in parentheses:

Male/Female Female/Male Male/Male Female/Female Satisfied 60 (33.175) 15 (30.521) 50 (36.493) 15

Boss/Employee Level of Satisfaction (39.81) Neutral 27 (40.284) 45 (37.062) 48 (44.313 50 (48.341) Dissatsfied 13 (26.54) 32 (24.417) 12 (29.194) 55 (31.848) L (^) !ÀBoss/employee relationship and job satisfaction are independent L (^) +À Boss/employee relationship and job satisfaction are dependent Test statistic: ;# ; degrees of freedom 3 4 "

"# Ð9,=/<@/./B:/->/.Ñ œ (^) /B:/->/. œ #Þ(! œ Ð  "Ñ‡Ð  "Ñ œ '

Cutoff value from chi-square table is 12.592. Conclusion: reject H! and conclude that boss/employee relationship and job satisfaction are related.

23. L (^) !Àteams are evenly matched L (^) +À teams are not evenly matched

The table below shows the observed values in each cell and the expected cell values in parentheses if the teams are evenly matched.

Length of series 4 games 5 games 6 games 7 games NBA finals 7 (7.125) 13 (14.25) 22 (17.8125) 15 (17.8125)

Test statistic: \ #^ œ Ð((Þ"#&Ñ(Þ"#&  Ð"$"%Þ#&Ñ"%Þ#&  Ð##"(Þ)"#&Ñ"(Þ)"#&  Ð"&"(Þ)"#&Ñ"(Þ)"#& œ "Þ&%;

# #

degrees of freedom = (4 1) = 3; cutoff value from chi-square table is 7.815.

Conclusion : do not reject L (^) !. There is no evidence that the NBA championship series are inconsistent with the conjecture that the teams are evenly matched. 24. L (^)! À : œ !Þ(!ß L (^) + À : Á !Þ(!à : œs "&$!!^ œ Þ'&; test statistic D œ Þ'&Þ(! œ  "Þ)àP-  ÐÞ(!ÑÐÞ$!Ñ$!! value œ T ÐD   "Þ)Ñ  T ÐD  "Þ)Ñ œ Þ!&)). Since P-value > .05, do not reject the null hypothesis; there is no evidence that : differs significantly from !Þ(!.

Having Probability - Probablity - Exam, Exams of Probability and Statistics

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