Practice Problems for Final Exam (pdf)

ST 101

FINAL EXAM PRACTICE PROBLEMS

Reiland

1.

What is the probability of getting a license plate that has a repeated letter or digit if you live in a state where the license plate scheme is four letters followed by two numerals?

2.

Four radar systems are arranged so that they work independently of each other. Each system has a 0.9 chance of detecting an approaching airborne object. Find the probability that at least one radar system will fail to detect an approaching object.

3.

How many four-digit serial numbers can be formed if no digit is to be repeated within any number? (The first digit may be a zero).

4.

A doctor notes that 20% of the patients he tests actually have mononucleosis. When a patient has mono, the test shows a positive result (indicating disease presence) 90% of the time. When a patient does not have the disease, the test shows a positive result 10% of the time. If a patient's test result is positive, what is the probability that the patient actually has the disease? a. .18 b. .26 c. .69 d. .77 e. .90

5.

A manufacturer of hand soap has introduced a new product. An extensive survey indicates that 40% of the people have seen advertising for the new product. It also showed that 20% of the people in the survey had tried the new product. In addition, 15% of those in the survey had seen it advertised and had tried the product. What is the probability that a randomly chosen person would have seen the advertising for the new product or have tried the product?

6.

Mathcounts is a national mathematics competition for seventh- and eighth-graders. In 1992 the contest was won by Andrei Gnepp of Orange, Ohio by answering the following question: The Bulls lead the Pistons 3 games to 2 in a seven game playoff. Assuming the probability that the Bulls win any particular game against the Pistons is 35 , what is the probability that the Pistons will win the playoff?

7.

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

8.

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 œ 21.6 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.

9.

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.

ST 101

Final Exam Practice Problems

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

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NUMBER OF VICTIMS 5 5 8 10 4 8 7 5 9 4 0 16 0 36

Source: Macdonald, J. N. “Is Evacuation a Fatal Flaw in Fire-Fighting Philosophy?” Risk Management, Vol. 33, No. 2, Feb. 1986, p. 37

a.

Construct a 98% confidence interval for the true mean number of victims per fire . who die attempting to evacuate compartmented fire-resistant buildings.

b.

Interpret the interval constructed in part b.

10. A 95% confidence interval for the mean mileage of the 2000 Mazda 626s is (23.8, 29.6). Choose all of the statements below that are correct. a. We are 95% confident that the interval (23.8, 29.6) contains the true but unknown mean mpg . of the 2009 Mazda 626s. b. If the same data were used to construct a 99% confidence interval, the resulting interval would be narrower than the 95% interval given above. c. If we repeatedly took samples of the same size, and from each sample computed a 95% confidence interval, then 95% of the resulting intervals would include the true mean mpg . of the 2009 Mazda 626s. d. 95% of the Mazdas in the sample had an mpg rating in the interval (23.8, 29.6). e. 95% of all 2009 Mazda 626s autos have an mpg rating in the interval (23.8, 29.6). f. If we selected many different samples of 2009 Mazda 626s autos, 95% of the resulting sample means would be in the interval (23.8, 29.6). 11. A random sample of size 49 was selected from a population with standard deviation 5 œ 8.2. Suppose the confidence interval formed for . was (63.27, 68.73). a. What is the sample mean, C ? b. What is the confidence level of this confidence interval? 12. A confidence coefficient of 0.99 for a confidence interval can correctly be interpreted to mean that a. 99% of the time in repeated sampling, intervals using an appropriate formula will contain the sample value. b. 99% of the time in repeated sampling, intervals using an appropriate formula will contain the relevant population parameter. c. 99% of the time in repeated sampling, intervals using an appropriate formula will contain the sample value as the midpoint of the interval. d. 99% of the time in repeated sampling, intervals using an appropriate formula will contain the sample mean as their midpoints.

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13. A random sample of 100 individuals was taken to determine the true percentage of people who smoke in a region of the eastern United States. Forty-six of them said “yes" when they were asked if they smoked. A 95% confidence interval for the true proportion of nonsmokers is: a. (.46, .54) b. (.36, .56) c. (.95, 1.0) d. (.44, .64) e. (.00, .95) 14. 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 City Chicago Philadelphia Baltimore Cleveland Washington Milwaukee St. Louis San Francisco Pittsburgh Buffalo

Annual Increase in Cancer Death Rate 1.0640 1.4118 2.1115 1.9401 3.8772 -.4561 4.8359 1.8875 4.4964 1.4045

NONFLUORIDATED City Los Angeles Boston New Orleans Seattle Cincinnati Atlanta Kansas City Columbus Newark Portland

Annual Increase in Cancer Death Rate .8875 1.7358 1.0165 .4923 4.0155 -1.1744 2.8132 1.7451 -.5676 2.4471

Construct a 95% confidence interval for the difference between the mean annual increases in cancer death rates for fluoridated and nonfluoridated cities. 15.

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# œ 652.

16. 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.

Level of Satisfaction Satisfied Neutral Dissatsfied

Male/Female 60 27 13

Boss/Employee Female/Male 15 45 32

Male/Male 50 48 12

Female/Female 15 50 55

17. 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)% à team B can also win in 4 games with the same probability, so the probability the series ends in 4 games is 2(0.5)% ].

ST 101


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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 NBA finals

4 games 7

5 games 13

6 games 22

7 games 15

ST 101


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SOLUTIONS 1. To solve this problem use the law of complements. The event of interest is A = {license plate has a repeated letter or digit}. The complement is A' = {there are NO repeated letters or digits}. The total number of license plate configurations is #' ‚ #' ‚ #' ‚ #' ‚ "! ‚ "! œ %&ß '*(ß '!!. the number of configurations with NO repeated letter or digit is #' ‚ #& ‚ #% ‚ #$ ‚ "! ‚ * œ $#ß #*#ß !!!. So T ÐEw Ñ, the probability of a license plate with NO repeated letter or digit is w T ÐEw Ñ œ $#ß#*#ß!!! %&ß'*(ß'!! œ Þ(!''&. So T ÐEÑ œ " œ T ÐE Ñ œ "  Þ(!''& œ Þ#*$$&. 2. The probability that no radar system fails to detect an airborne object (i. e. all four radar systems work) is (.9)% , therefore the probability that at least one fails is 1  (.9)% . "!x 3. "! T% œ Ð"!%Ñx œ 10(9)(8)(7) œ 5040. 4. c.

Þ") P(patient actually has the disease if test result positive) œ Þ")Þ!) œ Þ") Þ#' œ .69 5. P(seen advertising or tried the product) œ .4  .2  .15 œ .45. 6. The winner of the playoff is the first team to win 4 games. Therefore, the Pistons must win 2 games in a row. P(Pistons win any particular game against Bulls) œ #& , so P(Pistons win playoff) # % œ ˆ #& ‰ œ #& .

8. 21.6 „ 2.58Š È#Þ"(" ‹ œ 21.6 „ .643 œ (20.957, 22.243).

7. decrease 9.

a. c.

From the 14 observations, B œ 8.36, s œ 8.94 (use to estimate 5 ); )Þ*% x „ D‡Š È58 ‹ œ 8.36 „ 2.33Š È ‹ œ 8.36 „ 5.57 œ (2.79, 13.93) "%

“We are 98% confident that the interval (2.79, 13.93) encloses the true mean number of victims who die attempting to evacuate compartmented fire-resistant buildings. 10. a and c 11. a. the midpoint of the interval Ð'$Þ#(ß ')Þ($Ñ is the mean C; C œ '$Þ#(')Þ($ œ ''Þ # b. ')Þ($  '' œ #Þ($ß so #Þ($ œ D ‡ È)Þ#%* ; solving for D ‡ gives D ‡ œ #Þ($‡( œ #Þ$$ ; therefore the )Þ#

confidence level is 98%. 12. b. 13. d 14. B" œ 2.2573, s21 œ 2.753, n" œ 10; B# œ 1.3411, s22 œ 2.429, n# œ 10, use =#" and =## to estimate 5"# and 5## ; (2.2573  1.3411) „ 1.96É 2.753 10 

15. sp#!!) œ

576 1500

œ .384; sp#!!& œ

652 2000

2.429 10

œ .9162 „ 1.4109 œ (  .4947, 2.3271).

œ .326;

(.384  .326) „ 1.96É (.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. 16. Expected cell counts are in parentheses:

ST 101


Boss/Employee Level of Satisfaction Male/Female Female/Male Male/Male Satisfied 60 (33.175) 15 (30.521) 50 (36.493) Neutral 27 (40.284) 45 (37.062) 48 (44.313 Dissatsfied 13 (26.54) 32 (24.417) 12 (29.194) L! À Boss/employee relationship and job satisfaction are independent L+ À Boss/employee relationship and job satisfaction are dependent

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Female/Female 15 (39.81) 50 (48.341) 55 (31.848)

Test statistic: ;# œ ! Ð9,=//.Ñ œ *#Þ(!*; degrees of freedom œ Ð3  "Ñ‡Ð4  "Ñ œ ' /B:/->/. "#

#

"

Cutoff value from chi-square table is 12.592. Conclusion: reject H! and conclude that boss/employee relationship and job satisfaction are related. 17. L! À teams 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 NBA finals Test statistic: # \ # œ Ð((Þ"#&Ñ  (Þ"#&

4 games 7 (7.125)

Ð"$"%Þ#&Ñ# "%Þ#&



5 games 13 (14.25)

Ð##"(Þ)"#&Ñ# "(Þ)"#&

6 games 22 (17.8125)



Ð"&"(Þ)"#&Ñ# "(Þ)"#&

7 games 15 (17.8125)

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.