chapter 1 homework; fall 2010 c 4wardrop/courses/bhomework14.pdf · chapter 2 homework; fall 2010...

Chapter 1 Homework; FALL 2010

1. Consider the following CM: A 10-sided dieis tossed, with faces marked 1, 2, . . . , 10.The outcome is the number on the face thatlands up.

(a) Determine the sample space.

(b) List the elements of the followingevent: A = the outcome is an oddnumber.

(c) List the elements of the followingevent: B = the outcome is an evennumber larger than 6.

(d) Describe the following event inwords:C = {7, 8, 9, 10}.

2. Refer to the previous exercise. Assume theELC. Calculate the probability of each ofthe events,A, B andC.

3. A CM has a sample space that consists offive outcomes: 1, 2, 3, 4 and 5. For each ofthe following assignments, decide whetherit is a mathematically valid way to assignprobabilities for this situation. If not, ex-plain why not.

(a) P (1) = 0.30, P (2) = 0.15, P (3) =0.25, P (4) = 0.20, P (5) = 0.10.

(b) P (1) = 0.30, P (2) = 0.15, P (3) =0.20, P (4) = 0.20, P (5) = 0.10.

(c) P (1) = 0.30, P (2) = 0.15, P (3) =0.25, P (4) = 0.20, P (5) = 0.20.

(d) P (1) = 0.30, P (2) = 0.15, P (3) =0.25, P (4) = 0.40, P (5) = −0.10.

4. Refer to the previous exercise. Use assign-ment (a) to calculate the probability of eachof the following events.

(a) A = {1, 3, 5}.

(b) B = {2, 4}.

(c) C = {4}.

(d) D = {3, 4, 5}.

(e) E = {1, 3}.

5. Refer to the previous exercise.

(a) Verify that:

P (B or D) 6= P (B) + P (D).

(b) Verify that Rule 6 is true for eventsBandD.

(c) Given thatP (B) = 0.15 + 0.20 =0.35, explain why you know thatP (A) = 0.65 without adding theprobabilities of outcomes 1, 3 and 5.

(d) Of the five events listed in Exercise 4,find all pairs that illustrate Rule 5.

6. You are given the following information:the eventsA andB are disjoint;P (A) =0.30; andP (B) = 0.55. Calculate the fol-lowing probabilities.

(a) P (A or B).

(b) P (Ac).

(c) P (Bc).

7. You are given the following information:P (A) = 0.65; P (B) = 0.45; P (AB) =0.30. CalculateP (A or B).

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Chapter 1 Homework Continued

8. Consider a sample space with five mem-bers: 0, 1, 2, 3 and 4. Assume the ELC andi.i.d. trials. DefineX = X1 + X2, the to-tal of the numbers obtained in the first twotrials. Find the sampling distribution ofX.

9. Consider a sample space with three mem-bers: 1, 2 and 3. Assume the ELC and i.i.d.trials.

DefineX = X1 + X2 + X3, the total ofthe numbers obtained in the first three tri-als. Find the sampling distribution ofX.

10. Consider a sample space with four mem-bers: 1, 2, 3 and 4. Do not assume the ELC.Instead assume the following:

P (1) = 0.1, P (2) = 0.2, P (3) = 0.3

andP (4) = 0.4.

Assume i.i.d. trials. DefineX = X1 + X2,the total of the numbers obtained in the firsttwo trials. Find the sampling distribution ofX.

11. Refer to the previous question. DefineX =X1 + X2 + X3, the total of the numbersobtained in the first three trials. Find thesampling distribution ofX.

12. Refer to Table 1.4 in the Course Notes, onpage 16.

(a) What is our approximation toP (X = 20)? What is our pretty cer-tain interval for the exactP (X = 20)?

(b) What is the exact value ofP (X = 8)? What is our approxi-mation toP (X = 8)? What is ourpretty certain interval for the exactP (X = 8)? Is the pretty certain in-terval correct?

(c) What is the exact value ofP (X = 28)? What is our approxima-tion toP (X = 28)?

13. Refer to the sampling distribution and com-puter simulation on page 17 of the CourseNotes.

(a) For each value ofx = 0, 1, . . . , 6, cal-culate the pretty certain interval forP (X = x). Which, if any, of theseintervals is incorrect?

(b) Calculate the pretty certain intervalfor P (X ≤ 3). Is it correct?

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1. Brian likes to throw darts. He particularlyenjoys hitting a ‘20’ and for the purposeof this exercise we will not distinguish be-tween single, double and triple values. As-sume that Brian’s tosses are BT withp =0.58, where a success is hitting a ‘20.’

(a) Brian throws five darts. Calculate theprobability that he obtains S, F, S, S,F in that order.

(b) Brian will continue throwing darts un-til he misses a ‘20.’ Calculate theprobability that he will throw exactlythree darts.

(c) Brian will continue throwing darts un-til he misses a ‘20.’ Calculate theprobability that he will throw morethan three darts.

(d) If Brian throws six darts, calculate, byhand, the probability that he obtainsexactly three successes.

(e) Next week Brian will throw exactlyfour darts on each of five days, Mon-day thru Friday. If Brian obtains threeor more successes on a day, we saythat the event Angelina has occurredon that day.

i. Calculate the probability that anAngelina occurs on any givenday.

ii. Calculate the probability that theevent Angelina occurs on exactlythree of the days next week.

(f) Next Saturday morning, Brian willthrow exactly five darts. He willthrow darts again on Saturday after-noon. The number of darts he throwsin the afternoon will equal the numberof successes he obtains on Saturday

morning. LetY denote the total num-ber of successes that Brian obtains onSaturday morning and afternoon com-bined.

i. CalculateP (Y = 7).

ii. CalculateP (Y = 3).

(g) Next Sunday Brian will throw eightdarts. LetA be the event that he ob-tains exactly four successes. LetB bethe event that he obtains failures onboth his first and last throws. Calcu-lateP (AB).

2. Let X ∼ Bin(1024, 0.50). Calculate themean, variance and standard deviation ofX. What is the formula forZ?

3. Let X ∼ Bin(192, 0.25). Use the normalcurve website to approximate the followingprobabilities.

(a) P (X ≥ 55).

(b) P (X = 48).

(c) P (X ≤ 48).

(d) P (42 ≤ X ≤ 60).

(e) P (45 ≤ X < 57).

4. Refer to the previous question. Use the bi-nomial website to obtain the exact proba-bilities.

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5. I went to the binomial website and entereda value forn andp and obtained the follow-ing probabilities:

P (X ≤ 100) = 0.5268 and

P (X ≤ 93) = 0.2277.

Using just these two given probabilities, wecan determine the probabilities of a numberof other events. For example,

P (X > 100) = 1 − P (X ≤ 100) =

1 − 0.5268 = 0.4732.

List all events whose probabilities can beobtained and obtain them. Do not list anyevent more than once; for example, theevent(X > 100) is the same as(X ≥ 101).

6. In the 2008 presidential election inNorth Carolina, Barack Obama received2,142,651 votes and John McCain received2,128,474 votes. I will ignore votes cast forany other candidates. The finite populationsize isN = 4,271,125. I will designatea vote for Obama as a success, givingp = 0.50166 andq = 0.49834.

We plan to selectn = 7 persons at ran-dom with replacement from this popula-tion. Calculate, by hand,P (X ≥ 4), whereX is the total number of votes for Obama.

7. Refer to the previous problem. Replacen = 7 by n = 801, and calculateP (X ≥401), using both the normal curve websitefor an approximate answer and the bino-mial website to for the exact answer.

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1. Givenn = 400 BT with a total ofx = 220successes, use the snc approximation to ob-tain 90%, 95% and 99% CIs forp.

2. Repeat the previous question withn =1600 andx = 880. Comment.

3. Use the website to calculate the both the ex-act 90% and 95% two-sided CIs forp foreach of the following situations.

(a) n = 32; x = 8.

(b) n = 64; x = 16.

(c) n = 51; x = 31.

4. Use the website to obtain the exact one-sided upper CI forp for the following sit-uations.

(a) n = 23; x = 1; 95%.

(b) n = 43; x = 4; 80%.

(c) n = 18; x = 2; 90%.

5. Givenp ± 0.03925 is the 80% snc approx-imate CI forp. Calculate the snc approxi-mate 90%, 95% and 98% CIs forp.

6. Refer to the previous question. Given thatp = 0.25, determine the value ofn.

7. Suppose thatn = 60 andp = 0.45. An-swer the following questions for the ap-proximate 95% two-sided CI forp.

(a) What values ofx will make the CIcorrect? I recommend using trial-and-error to find the values. Basically, youwant to find the interval ofx valueswith the property that eachx valuegenerates a CI that containsp = 0.45.

(b) What is the actual probability that theapproximate 95% two-sided CI forpis correct?

8. Suppose thatn = 40 andp = 0.30. Answerthe following questions for the exact 95%two-sided CI forp.

(a) What values ofx will make the CIcorrect? As discussed in 7(a) above,I recommend using trial-and-error.

(b) What is the actual probability that theexact 95% two-sided CI forp is cor-rect?

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1. Suppose thatX ∼ Poisson(49). Calculatethe mean, variance and standard deviationof X.

2. Suppose thatX ∼ Poisson(20). Use thewebsite to calculate the following probabil-ities.

(a) P (X = 20).

(b) P (X ≤ 20).

(c) P (X > 20).

(d) P (16 ≤ X ≤ 24).

3. Suppose thatX ∼ Poisson(64). Use the ap-propriate normal curve to approximate thefollowing probabilities. Also, use the web-site to obtain the exact probabilities.

(a) P (X > 75).

(b) P (60 < X < 70).

4. Suppose thatX ∼ Poisson(θ), with θ un-known. Given thatX = 72,

(a) Use the snc to obtain the approximate98% CI forθ.

(b) Use the website to obtain the exact98% CI forθ.

5. Tim is observing a Poisson Process with arate of 5 occurrences per hour. LetX de-note the number of successes he will ob-serve in a 4.5 hour period. Use the websiteto findP (X = 26).

6. Refer to the previous question. The nextday Tim observes his process for 105 min-utes. LetY denote the total number of suc-cesses observed by Tim on his two daysof data collection. Use the website to findP (Y ≤ 30).

7. Refer to the previous question. Supposethat Tim observes a total of 40 successeson his two days of collecting data. Now,pretending that we don’t know that the rateis 5 per hour, use the website to calculatethe 95% CI for the rate. Is your CI correct?

8. The following data appeared in the Wiscon-sin State Journal on July 13, 2010, for acci-dents involving autos in Madison, Wiscon-sin.

2005 2006 2007Average weekdayarterial volume 26,271 25,754 25,760Total crashes 4,577 4,605 4,779Bike crashes 97 95 118Pedestrian crashes 84 87 80Fatal crashes 9 12 13

2008 2009 TotalAverage weekdayarterial volume 24,416 24,222 126,423Total crashes 4,578 4,753 23,292Bike crashes 95 115 520Pedestrian crashes 76 77 404Fatal crashes 6 14 54

In each of the following, assume that thenumber of crashes of interest follows aPoisson Process with unknown rateλ peryear. Use all five years of data to obtain theapproximate 95% CI forλ.

(a) Total crashes.

(b) Bike crashes.

(c) Pedestrian crashes.

(d) Fatal crashes. Also obtain the exactCI.

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1. Use the calculator linked to the course pageto find the area to the right of the numberχ2 under theχ2 curve with degrees of free-dom denoted bydf , for each combinationof values ofχ2 anddf given below.

(a) χ2 = 12, df = 5.

(b) χ2 = 16.721, df = 8.

(c) χ2 = 21.12, df = 10.

2. Use the calculator linked to the course pageto find χ2

α(df) for each combination ofα

anddf given below.

(a) α = 0.01, df = 10.

(b) α = 0.05, df = 11.

(c) α = 0.10, df = 9.

3. (Optional.) Do you know a genetic modelthat could be tested with the Goodness ofFit Test? If yes, briefly describe it.

4. I discussed and tested my blue round-cornered die in the class notes. I also havea white round-cornered die that I cast 1000times. Below are myO’s:

Outcome 1 2 3 4 5 6Oi 181 139 135 160 141 244

Test the null hypothesis that the die is bal-anced. Useα = 0.01 and find the P-value.

5. (Hypothetical data.) A certain breed of ger-bils were crossed and gave progeny of thefollowing colors: 40 black, 59 brown and42 white. Are these data consistent with a1:2:1 ratio (browns most likely) predictedby a genetic model? Useα = 0.05 and findthe P-value.

6. During the two NBA seasons, 1980–1982,Larry Bird of the Boston Celtics shot a pairof free throwsn = 338 times during theregular season. For these 338 pairs, Birdmade both free throws 251 times, missedboth free throws 5 times, and made exactlyone free throw the remaining 82 times. Dothese data appear to be the results of BT?Useα = 0.10 and find the P-value.

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1. I simulatedn = 200 dichotomous trials onmy computer. Below are the results.

• I obtained a total of 101 successes.

• Dividing the data into four segmentsof 50 trials each, I found that the num-ber of successes were: 35 in the first50 trials; 29 in the second 50 trials; 21in the third 50 trials; and 16 in the last50 trials.

• The first trial was anS and the lasttrial was anF . The pair ‘FF’ occurred51 times.

(a) Investigate the issue of the constancyof p by using the four segments ofdata. Use both descriptive and infer-ential (testing) methods.

(b) Repeat (a) with two segments ofdata—the first and second halves.

(c) Create the memory table and discusswhether the trials appear to be inde-pendent. Use both descriptive and in-ferential methods.

(d) Now, Nature (well, me) announcesthat, in fact, the trials were not BT.They were simulated as follows. Eachsegment of 50 trials satisfied the as-sumptions of BT, but the values ofp were 0.7, 0.6, 0.5 and 0.4 for thesegments one thru four, respectively.With this new knowledge, commenton your above analyses.

2. I simulatedn = 200 dichotomous trials onmy computer. Below are the results.

• I obtained a total of 113 successes.

• Dividing the data into four segmentsof 50 trials each, I found that the num-ber of successes were: 34 in the first

50 trials; 24 in the second 50 trials; 30in the third 50 trials; and 25 in the last50 trials.

• The first and last trials were both fail-ures. The pair ‘FF’ occurred 37 times.

(a) Investigate the issue of the constancyof p by using the four segments ofdata. Use both descriptive and infer-ential (testing) methods.

(b) Repeat (a) with two segments ofdata—the first and second halves.

(c) Create the memory table and discusswhether the trials appear to be inde-pendent. Use both descriptive and in-ferential methods.

(d) Now, Nature (well, me) announcesthat, in fact, the trials were not BT.They were simulated as follows. Eachsegment of 50 trials satisfied the as-sumptions of BT, but the values ofpwere 0.55, 0.52, 0.49 and 0.46 for thesegments one thru four, respectively.With this new knowledge, commenton your above analyses. Also, com-pare the results for this question toyour results for question 1.

3. View each of the Milwaukee Brewers’ 162games during the 2009 season as a trial,with a victory a success. You are given thefollowing information.

• Dividing the season into six segmentsof 27 games each, the Brewers wonthe following number of games ineach segment: 15, 16, 12, 11, 11 and15, respectively.

• The Brewers lost game one and wongame 162. The pair ‘SS’ occurred 39times.

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(a) Investigate the issue of the constancyof p by using the six segments of data.Use both descriptive and inferential(testing) methods.

(b) Repeat (a) with three segments ofdata—the first, second and last thirds.

(c) Repeat (a) with two segments ofdata—the first and second halves.

(d) Create the memory table and discusswhether the trials appear to be inde-pendent. Use both descriptive and in-ferential methods.

4. Two of my students, Burime and Farhad,performed a project on shooting freethrows. Below are the results of their 100shots.

1 1 1 0 0 1 1 0 1 10 0 1 1 1 1 1 1 0 11 1 1 1 1 0 0 0 1 00 1 0 0 1 1 1 0 0 11 0 0 0 0 0 1 1 0 0

1 1 0 0 0 1 1 0 1 00 0 1 1 1 0 0 0 1 11 0 0 0 0 0 1 1 0 11 0 0 0 0 1 0 0 0 01 0 0 0 0 1 0 0 0 1

(a) Divide the data into four segments of25 shots and investigate whetherp isconstant.

(b) Repeat (a) for two segments of 50shots.

(c) Create the memory table and investi-gate independence.

5. Below are three tables, labeled 1, 2 and 3.Use these tables to answer the questions be-low them.

Table 1Current

Prev. S F TotalS 90 210 300F 210 490 700

Total 300 700 1000

Table 2Current

Prev. S F TotalS 100 101 201F 100 200 300

Total 200 301 501

Table 3Current

Prev. S F TotalS 150 99 249F 100 25 125

Total 250 124 374

(a) Which table is from a sequence inwhich the majority of trials yielded anS?

(b) Which table is from a sequence inwhich the first and last trials were thesame outcome?

(c) Which table is from a sequence inwhich the first trial gave anS and thelast trial gave anF?

(d) Which table is from a sequence inwhich the first trial gave anF and thelast trial gave anS?

(e) Which table shows no evidence ofmemory?

(f) Which table shows evidence of posi-tive memory; i.e. that the success rateafter anS is larger than the successrate after anF?

(g) Which table shows evidence of nega-tive memory; i.e. that the success rateafter anS is smaller than the successrate after anF?

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1. Recall from a lecture example that a sum-mer squash resulting from a certain crosshas probability3/16 of being yellow.

Assuming BT, find the point prediction and90% prediction interval for the number ofyellow squash in a sample of 800 squash.

2. Recall from a lecture example that in a cer-tain setting the probability that a chickenwill have white feathers and a small combis 9/16.

Assuming BT and this particular setting,find the point prediction and 95% predic-tion interval for the number of chickens thatwill have white feathers and a small combin a sample of 320 such chickens.

3. During the 1993–94 NBA season, KarlMalone attempted 736 free throws andmade 511 of them.

(a) Given that he attempted 695 freethrows during the next season, use the1993–94 data to find the point predic-tion and 80% prediction interval forthe number of free throws he wouldmake in 1994–95. Write down the as-sumptions that you need to make toobtain your answers.

(b) Given that he actually made 516 freethrows in 1994–95, comment on youranswers to part (a).

4. During the 2000 baseball season, BarryBonds hit 49 home runs in 480 official at-bats.

(a) Assuming that these 480 at-bats areBT, with a home run a success, cal-culate the 95% CI forp.

(b) During the 2001 baseball season,Barry Bonds hit 73 home runs in 476official at-bats. Assuming that these476 at-bats are BT, with a home run asuccess, calculate the 95% CI forp.

(c) Refer to (b). Use the 2000 data to findthe point prediction and 98% predic-tion interval for the number of homeruns that Barry Bonds would hit in2001. Comment on your answers.

5. Refer to the data on bike crashes, by year,reported in Question 8 of the Chapter 4homework. In each of the following, as-sume that the number of bike crashes fol-lows a Poisson Process with unknown rateλ per year.

(a) Use the data from 2005 to obtain an80% prediction interval for the num-ber of bike crashes in 2006. Is yourPI correct?

(b) Use the total number of bike accidentsin 2005 and 2006 combined to ob-tain the 90% prediction interval forthe number of bike crashes in 2007.Is your PI correct?

(c) Use the total number of bike accidentsin the three years 2005–2007 to ob-tain the 95% prediction interval forthe number of bike crashes in 2008.Is your PI correct?

(d) Use the total number of bike accidentsin the four years 2005–2008 to ob-tain the 98% prediction interval forthe number of bike crashes in 2009.Is your PI correct?

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1. In 1982, the Wisconsin Legislature passeda comprehensive law on drinking alcoholand driving.

Question 7 on the Wisconsin Driver Sur-veys measured the respondent’s knowledgeof the legal limit of blood alcohol concen-tration in Wisconsin (0.10% at that time).In 1982, 50.1% of 1589 respondents gavethe correct answer compared to 56.2% of1072 in 1983.

(a) Construct the 95% CI for the increasein knowledge from 1982 to 1983.

(b) Test the null hypothesis of no changein knowledge from 1982 to 1983 ver-sus the alternative that knowledge in-creased. Use the Fisher’s website toobtain the exact P-value and use thenormal curve approximation to obtainthe approximate P-value. Would youreject the null hypothesis forα =0.10?

2. Question 12 on the 1983 and 1984 Wiscon-sin Driver Surveys asked, “Have you heardanything about Wisconsin’s recent drunkendriving law?”. In 1983, 64.8% of the 1072persons questioned responded ‘Yes’ and,in 1984, 59.0% of the 2632 persons ques-tioned responded ‘Yes’.

(a) Construct a 95% CI for the decreasein ‘awareness’ in the population from1983 to 1984. Briefly comment onyour findings.

(b) Test the null hypothesis of no changein awareness from 1983 to 1984versus the alternative that awarenesschanged. Use the Fisher’s website toobtain the exact P-value and use the

normal curve approximation to obtainthe approximate P-value. Would youreject the null hypothesis forα =0.05?

3. I collected data on the first fourteen weeksof the 1990 major league baseball season.In the American League, the team that wasleading at the end of the sixth inning wenton to win 458 out of 527 games. In theNational League, the team that was leadingat the end of the sixth inning went on towin 390 out of 450 games. Assuming thesedata are independent random samples fromtwo populations, construct a 95% confi-dence interval forp1−p2 (let the AmericanLeague be the first population).

Questions 4–8 belowexplore the situationwhen a basketball player shoots a pair offree throws during an NBA game. I want tosee whether the outcome of the first shothas any influence on the outcome of thesecond shot.

I have data on two players, Larry Birdand Rick Robey, who both played for theBoston Celtics. The data below for eachman are for the 1980–81 and 1981–82 sea-sons combined.

In these problems, the two populations aredetermined by the outcome of the first shot;i.e. the populations are ‘first shot success’(1) and ‘first shot failure’ (2). The responsewill be the outcome of the second shot.

4. Below are the data for Larry Bird.

Second ShotFirst Shot S F TotalSuccess 251 34 285Failure 48 5 53Total 299 39 338

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(a) Calculate the 95% CI forp1 − p2.What assumptions must you make?

(b) Test the null thatp1 = p2 versus thealternativep1 6= p2. Calculate boththe exact and approximate P-value.

5. Below are the data for Rick Robey.

Second ShotFirst Shot S F TotalSuccess 54 37 91Failure 49 31 80Total 103 68 171

(a) Calculate the 95% CI forp1 − p2.

(b) Test the null thatp1 = p2 versus thealternativep1 6= p2. Calculate boththe exact and approximate P-value.

6. Write a few sentences summarizing whatyou have learned about Bird and Robeyfrom these data.

7. Collapse the data for Bird and Robey intoone table. I will help by getting you started.

Second ShotFirst Shot S F TotalSuccess 376Failure 133Total 402 107 509

Calculate the 95% CI forp1 − p2. What as-sumptions must you make? Do these makesense?

8. Refer to the data in the previous four ques-tions. Show that Simpson’s Paradox is oc-curring. Discuss what is happening.

9. An observational study yields the following“collapsed table.”

Group S F Total1 99 231 3302 86 244 330

Total 185 475 660

Below are two component tables for thesedata.

Subgp A Subgp BGp S F Tot Gp S F Tot1 24 96 120 1 75 135 2102 225 2 105

Tot 345 Tot 315

Complete these tables so that Simpson’sParadox is occurringor explain why Simp-son’s Paradoxcannot occur for these data.You must present computations to justifyyour answer.

10. An observational study yields the following“collapsed table.”

Group S F Total1 180 320 5002 117 183 300

Total 297 503 800

Below are two component tables for thesedata.

Subgp A Subgp BGp S F Tot Gp S F Tot1 130 130 260 1 50 190 2402 180 2 120

Tot 440 Tot 360

Complete these tables so that Simpson’sParadox is occurringor explain why Simp-son’s Paradoxcannot occur for these data.You must present computations to justifyyour answer.

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1. I have drawn a histogram for 200 obser-vations. One of the rectangles has end-points of 0.50 and 0.60, and a height of 4.For each of the three situations below, de-terminehow many observations are in thisclass interval [0.50 to 0.60), with the usualendpoint convention.

(a) If it is a frequency histogram.

(b) If it is a relative frequency histogram.

(c) If it is a density scale histogram.

2. Below are 50 sorted observations.

0.05 0.07 0.08 0.09 0.130.22 0.33 0.42 0.43 0.460.49 0.71 0.93 0.99 1.361.50 1.58 1.59 2.01 2.502.64 2.67 2.88 2.91 3.013.18 3.34 3.53 3.57 3.713.76 3.96 4.00 4.04 4.284.35 4.41 4.45 4.56 4.574.63 4.65 4.75 4.78 4.834.85 4.92 4.93 4.94 4.96

(a) Calculate the median of these data.

(b) Draw a frequency histogram of thesedata. Use 0.00–1.00, 1.00–2.00, andso on, as your class intervals. Clearlylabel the height and endpoints of eachof the five rectangles.

(c) Draw a density scale histogram ofthese data. Use 0.00–0.50, 0.50–2.50,2.50–4.50 and 4.50–5.00 as your classintervals. Clearly label the height andendpoints of each of the four rectan-gles.

3. Refer to the data in problem 2 above.

(a) Calculate the first and third quartilesof these data.

(b) Given the mean and standard devia-tion of these data are 2.76 and 1.78,respectively, determine thepropor-tion of observations that are withinone standard deviation of the mean.How does your proportion compare tovalue predicted by the empirical rule?Are you are surprised by the agree-ment/disagreement? Comment.

4. A sample of size 33 yields the followingsorted data. Note that I have x-ed outx(32)

(the second largest number). This fact willNOT prevent you from answering the ques-tions below.

2.86 5.72 6.22 6.33 6.37 6.486.56 7.25 7.57 8.27 8.77 9.74

11.19 11.20 11.97 12.60 12.64 13.5914.47 17.39 18.59 18.77 18.98 19.0721.50 23.36 25.71 28.83 30.91 37.4744.18 x 56.85

Hint: The mean of these data is (exactly)17.56 and the standard deviation is 13.23.

(a) What proportion of these data lies inthe interval[x − s, x + 2s]?

(b) Calculate the median and quartiles ofthese data.

(c) Suppose we discover that the obser-vation 5.72 above is really 15.72. Af-ter making this change to the data set,calculate the first quartile, median andmean.

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1. A student in my Stat 301 class, SaraLamers, performed 40 golf trials. Each trialconsisted of hitting a golf ball with a 3-wood. Her response was the distance theball traveled, in yards. Assume that her tri-als are i.i.d. from a population with meanµ. Her sorted data are below.

22 32 38 56 58 77 8193 99 101 101 101 104 107

107 108 109 109 110 111 113114 115 116 118 122 122 127127 128 128 128 129 131 131137 139 139 140 147

Hint: The mean and standard deviation ofthese data are 106.87 and 29.87.

(a) Calculate Gosset’s 95% CI forµ.

(b) Calculate Gosset’s 99% CI forµ.

(c) Find the P-value for testing the nullhypothesis thatµ = 100 against thealternative thatµ > 100.

(d) Find the P-value for testing the nullhypothesis thatµ = 120 against thealternative thatµ < 120.

(e) Calculate the approximate 95% CI forthe median,ν, of the population.

2. Students in my Stat 301 class, Mei LanChan, Sin Fai Cheung, and Todd Willer,performed 15 trials. Each trial consisted ofaccelerating an automobile from 40 mph to65 mph. Their response was the time, inseconds, required to achieve the speed of65 mph. Assume that their trials are i.i.d.from a popoulation with meanµ and me-dianν. The sorted data are below.

6.67 6.78 6.79 6.86 6.886.92 6.94 7.05 7.11 7.127.23 7.34 7.36 7.56 7.86

Hint: The mean and standard deviation ofthese data are 7.10 and 0.3253.

(a) Calculate Gosset’s 90% CI forµ.

(b) Calculate Gosset’s 98% CI forµ.

(c) Find the P-value for testing the nullhypothesis thatµ = 7.00 against thealternative thatµ > 7.00.

(d) Find the P-value for testing the nullhypothesis thatµ = 7.20 against thealternative thatµ 6= 7.20.

(e) Calculate an exact CI for the median.Select a confidence level as close aspossible to 90%.

3. Recall that a confidence interval istoosmall if the number being estimated islarger than every number in the confidenceinterval. Similarly, a confidence intervalis too large if the number being estimatedis smaller than every number in the confi-dence interval.

Each of four researchers selects a randomsample from the same population. Each re-searcher calculates a confidence interval forthe mean of the population. The intervalsare below.

[43, 82], [49, 92], [57, 95] and[86, 112].

(a) Nature announces thatµ = 90. Howmany intervals are too large? Howmany are too small? How many arecorrect?

(b) Nature announces, “Two of the inter-vals are correct, one interval is toosmall and one interval is too large.”Given this information, determine allpossible values of the mean.

14

Chapter 11 Homework Cont.

(c) Nature announces, “Exactly one CI iscorrect.” Determine all possible val-ues of the mean.

4. Eighteen researchers select random sam-ples from the same population. Each re-searcher computes a confidence interval forthe mean of the population. Thus, eachresearcher is estimating the same number.Unfortunately, the lower and upper boundsof the 18 confidence intervals became dis-connected. The 18 lower bounds, sorted,are below.

153 163 188 198 245 246260 284 288 307 308 322334 344 360 375 393 409

The 18 upper bounds, sorted, are below.

294 326 331 345 351 364372 373 381 394 396 401403 408 411 439 445 460

(a) Given thatµ = 300, how many ofthe confidence intervals are too large?How many are too small? How manyare correct?

(b) Suppose we are told that exactly twoof the intervals are too large. Deter-mine all possible values ofµ.

(c) Suppose we are told that at least fiveof the intervals are too small. Deter-mine all possible values ofµ.

(d) Nature announces that exactly 11 ofthe CI are correct. Determine all pos-sible values for the mean.

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1. The following is taken from the abstract ofa Dutch medical study that was publishedin 1989.

Previous reports have indicatedthat coffee consumption may in-crease serum cholesterol levels.We studied the effects of cof-fee prepared by two commonbrewing methods (filtering andboiling) on serum lipid levelsin a twelve week randomizedtrial . . . . After a three-weekrun-in period during which theyall consumed filtered coffee, theparticipants were randomly as-signed to one of three groups re-ceiving four to six cups of boiledcoffee a day, four to six cups offiltered coffee a day, or no cof-fee, for a period of nine weeks.

For the purpose of this exercise, the nocoffee group will be disregarded (the re-searchers found no statistically signifi-cant differences between the filtered coffeegroup and the no coffee group). The sub-jects were selected from a group of 596 par-ticipants in another study. These 596 per-sons were believed to be a random samplefrom a population. Each of the 596 peoplewho were aged 18 or older and who habitu-ally drank coffee were invited to participatein the present study. One hundred sevensubjects volunteered to be in this study and101 completed the entire 12 week program.Before the study all subjects drank filteredcoffee.

(a) This can be viewed as a “Beforeand after” study within each treat-ment group. For example, for the 33

subjects who drank boiled coffee themean increase (from end of the run-into the end of the study) in total choles-terol was 0.52 mmol per liter with astandard deviation of 0.733. On theassumption of a random sample, con-struct Gosset’s 95% CI for the meanincrease in total cholesterol.

(b) Refer to the previous question. The34 subjects who drank filtered coffeehad a mean increase of 0.04 with astandard deviation of 0.744. On theassumption of a random sample, con-struct Gosset’s 95% CI for the meanincrease in total cholesterol.

(c) This study can be viewed as “Com-paring two treatments” with the treat-ments being boiled and filtered coffeeand the response being the change intotal cholesterol. Obtain the P-valuefor the test of the null hypothesis ofno treatment effect versus the third al-ternative.

(d) Refer to the previous question. Onthe assumption of independent ran-dom samples, construct the 95% CIfor the difference between the twomean increases in total cholesterol.Use Case 3.

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2. Recall the study on home runs and WrigleyField in the Lecture Examples. I have simi-lar data from the Astrodome in Houston forthe same 20 seasons, 1967–1987, exclud-ing 1981. The Astrodome wasnot a goodplace for hitting home runs, so I calculatedmy differences as Away minus Home, toavoid negatives. The summary statistics ared = 45.7 home runs andsd = 22.74.

(a) Calculate Gosset’s 98% CI for thepopulation mean of the differences.

(b) Find the P-value for the test of the nullhypothesis that the population meandifference is 40.5 versus the alterna-tive that it is larger than 40.5.

3. For this problem, refer to the t-curve calcu-lator linked to our course website. Remem-ber that there are three boxes and a choice.The top box is for thedf . The left box has achoice, either ‘Area left of’ and ‘Area rightof.’

For each of the situations below, first tellme what to enter in each box and whichchoice to select to obtain the desired an-swer. Next, do it and tell me the answer.

(a) n1 = 6, n2 = 12 and we want thetneeded for the 80% CI forµ1 − µ2.

(b) n1 = 14, n2 = 19 and we want thetneeded for the 95% CI forµ1 − µ2.

(c) n1 = 5, n2 = 15, t = 2.105 and wewant the P-value for the alternative>.

(d) n1 = 18, n2 = 16, t = 3.011 and wewant the P-value for the alternative6=.

(e) n1 = 10, n2 = 25, t = −1.875 andwe want the P-value for the alterna-tive <.

4. In each of the following situations,

• Calculate the 95% CI forµ1−µ2; and

• Obtain the P-value for testing the nullhypothesis thatµ1 = µ2 for each ofthe three possible alternatives.

(a) n1 = 7, n2 = 12, s1 = 6.50, s2 =4.00, x = 22.75 andy = 18.25.

(b) n1 = 17, n2 = 6, s1 = 11.50, s2 =14.00, x = 32.40 andy = 31.20.

(c) n1 = 19, n2 = 10, s1 = 21.50, s2 =18.20, x = 52.50 andy = 61.40.

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1. Below is the table of population counts fora disease and its screening test. (RecallthatA means the disease is present andBmeans the screening test is positive.)

B Bc TotalA 254 35 289Ac 199 3126 3325Total 453 3161 3614

(a) What proportion of the populationwould test positive?

(b) What proportion of the population isdisease free?

(c) What proportion of the population isfree of the disease and would test neg-ative?

(d) What proportion of the population hasthe disease and would test positive?

(e) Of those who would test negative,what proportion has the disease?

(f) Of those who are free of the disease,what proportion would test positive?

(g) What proportion of the populationwould receive a correct screening testresult?

(h) Of those who would receive an incor-rect screening test result, what pro-portion would receive a false posi-tive?

(i) What proportion of the populationdoes not have the disease or wouldtest negative?

2. A company has 900 employees. You aregiven the following information:

• Fifty-six percent of the employees arefemale.

• Forty percent of the employees driveto work.

• Of those who do not drive to work,seventy percent are female.

(a) Create the table of population countsfor these two variables. Be sure to la-bel the rows and columns of your ta-ble.

(b) Create the table of population propor-tions for these two variables.

(c) An employee is selected at randomfrom the company. Given that the em-ployee drives to work, calculate theprobability that the employee is male.

(d) An employee is selected at randomfrom the company. Given that the em-ployee is female, calculate the proba-bility that the employee does not driveto work.

3. A random sample of sizen = 400 from adichotomous population yields the follow-ing data.


(a) Test the null hypothesis thatpA = pB.For each of the three possible alterna-tives find the P-value.

(b) Construct the 90% confidence inter-val for pA − pB.

18


4. A random sample of sizen = 450 from adichotomous population yields the follow-ing data.


(a) Test the null hypothesis thatpA = pB.For each of the three possible alterna-tives find the P-value.

(b) Construct the 98% confidence inter-val for pA − pB.

5. This problem is about relative risks andodds ratios. Below is a table of hypothet-ical population counts, in thousands.

Group B Bc TotalA 12 188 200Ac 12 788 800

Total 24 976 1000

A case control study with 800 subjectsfrom this population yielded the data be-low.


Total 400 400 800

(a) Calculate the relative risk and odds ra-tio for the population; comment.

(b) Calculate the point estimate of thepopulation odds ratio.

(c) Obtain the 95% CI for the populationodds ratio.

6. This problem is about relative risks andodds ratios. Below is a table of hypothet-ical population counts, in thousands.


Total 45 955 1000

A case control study with 700 subjectsfrom this population yielded the data be-low.


Total 350 350 700

(a) Calculate the relative risk and odds ra-tio for the population; comment.

(b) Calculate the point estimate of thepopulation odds ratio.

(c) Obtain the 90% CI for the populationodds ratio.

19

Chapter 14 Homework; ; FALL 2010: DoNOT Submit for Grading, Solutions are be-low

The regression equation isWeight = - 304 + 6.57 Height

Predictor Coef SE CoefHeight 6.57 0.9256

S = 12.36 R-Sq = 86.3%

Analysis of Variance

Source DF SSRegression A DResidual Error B CTotal 9 E

Ht Wt Fit SE Fit75.0 184.00 188.47 4.5969.0 162.00 149.08 5.0274.0 162.00 181.91 4.1878.0 200.00 208.17 6.4980.0 240.00 221.30 8.05

Above is edited output from a study on theheights and weights ofn members of the WNBAChicago Sky team. Note that five numbers inthe ANOVA table have been replaced by A–E.For the purpose of the following questions, wewill view thesen women as a random samplefrom the population of all female professionalbasketball players.

Also, you are given the following table ofGosset’s values oft for confidence and predic-tion intervals.

df 80% 90% 95% 98% 99%7 1.415 1.895 2.365 2.998 3.4998 1.397 1.860 2.306 2.896 3.3559 1.383 1.833 2.262 2.821 3.250

1. (a) What are the values ofn, A and B?

(b) Determine the value of C in theANOVA table.

(c) Determine the values of D and E inthe ANOVA table.

2. Calculate the 98% CI for the slope of thepopulation regression line.

3. Calculate the 95% CI for the mean weightof players who are 78 inches tall.

4. Tamera Young is a WNBA player and is 74inches tall. Calculate the 90% PI for herweight.

5. (a) Calculate the predicted weight,y fora player who is 72 inches tall.

(b) Calculate the residual for the playerwho is 75 inches tall and weighs 184pounds.

(c) There is a player in the WNBA whoweighs 130 pounds and has a residualof −5.94. How tall is she?

20

Solutions for Chapter 14 Homework

1. (a) The ‘Total DF’ always equalsn − 1.Thus,

n − 1 = 9 or n = 10.

Next,A always equals 1. By subtrac-tion, B = 8.

(b) We know that

C/B = C/8 = s2 = (12.36)2.

Thus,C = 1222.

(c) We are given thatR2 = 0.863. Also,

R2 = D/E andD + C = E.

From (b), we get

0.863 = D/(D + 1222).

This gives

0.863(1222) = 0.137D, or

D = 7698.

Also, E = C + D = 1222 + 7698 =8920.

2. For questions 2–4, thedf is 8 b/c it is equalto thedf for error. For 98%, from the giventable,t = 2.896. Thus, the CI is

6.57 ± 2.896(0.9256) = 6.57 ± 2.68 =

[3.89, 9.25].

3. For 95%, from the given table,t = 2.306.Thus, the CI is

208.17 ± 2.306(6.49) = 208.17 ± 14.97 =

[193.20, 223.14].

4. For 90%, from the given table,t = 1.860.Thus, the PI is

181.91 ± 1.860√

(12.36)2 + (4.18)2 =

181.91±1.860(13.0477) = 181.91±24.27 =

[157.64, 206.18].

5. (a) We substitutex = 72 into the regres-sion line:

y = −304+6.57(72) = −304+473 =

169.

(b) From the computer output we see thaty = 188.47. Thus,e = y− y = 184−188.47 = −4.47.

(c) For this playery = 130 and

y = y − e = 130 + 5.94 = 135.94.

Thus, for this player,

135.94 = −304 + 6.57x.

Solving forx we get

x = (135.94+304)/6.57 = 67 inches.

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