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Unformatted text preview: STAT 310/MATH 363 Ditlev Monrad Final Examination December 16, 1995 OPEN BOOK—SHOW ALL WORK Problem 1. (25 points) There are four ABO blood types. 41% of all Americans are type A, 9% are type B, 4%
are type AB, and 46% are type 0. There are two Rh blood types. 85% of Americans are type Rh+ and 15% are type Rh—.
The three characteristics, ABO blood type, Rh factor, and gender are independent. (Each
genetic trait is located on a different chromosome.) The distribution of the blood types is
the same among females as among males. What percent of American females are (a) either type A or type 0,
(b) both type A and Rh+,
(c) both type AB and Rh—. (d) In a random sample of 400 female blood donors, the number of donors of type AB will be around , give or take , or so. Problem 2. (20 points)
865% of all newborn US babies have Rh+ blood type. 15% haVe Rh— blood type. Find the chance that out of 25 randomly chosen babies, (a) At least 20 are Rh+,
(b) Exactly 20 are Rh+,
(c) At most 20 are Rh+. (Use calculator or statistical tables.) Problem 3. (20 points) The literacy rate of a population represents the percent of the population over 15 years
old that can read and write. According to the Reader’s digest Almanac and Yearbook, the literacy rate in Russia is 99.8%. (a) If 2000 residents of Russia are selected at random what is the expected number of
illiterate persons in the sample? Find the probability that in a random sample of 2000 Russians, the number who are illiterate is (b) at most 3, (c) exactly 4, (d) at least 8. (Use calculator or statistical tables for the last 3 questions.) Broblem 4. (30 points) Let a be an unknown parameter satisfying —1 g a g 1. Let X1,X2, . .. ,Xn be 71 inde— pendent random variables, each with the p.d.f. 1 3
f(x;a)=% , 1<x<1. (a) Compute the mean u of this distribution. (b) Given the observations $1,232, . . . ,ccn, determine the likelihood function
L(a) = L(a; $1,302, . .. ,xn). (c) Find the likelihood equation that has the maximum likelihood estimator as its solution.
Can you solve it? (d) Find the methodof—moments estimator d of a. Compute d if i = ~0.15. (f) Compute o? if 5: = 0.23. Problem 5. (50 points) Let (9 > 0 be an unknown parameter and let X1,X2, . .. ,Xn be n independent random variables, each with the p.d.f. f(a:;6) = %$(2_9)/0 , 0 < 3: <1. (a) Given the observations x1, :52, . . . ,xn determine the likelihood function
L(9) = L(0;$1,x2, . .. ,mn). (b) Find the maximum likelihood estimator d of 0. (c) Show that é is unbiased. (d) Compute V'ar(é). (e) Show that d is a consistent estimator of 6. (f) Find the standard error of (g) Assume that n = 25 and 63 = 7.5. Give an approximate 95% conﬁdence interval for (9. Problem 6. (25 points) Johns Hopkins researchers conducted a study of pregnant IBM employees. Among 30 who
worked with glycol ethers, 10 (or 33.3%) had miscarriages, but among 750 who were not
exposed to glycol ethers, 120 (or 16.0%) had miscarriages. Can this difference be explained
away as chance variation, or can we conclude that women exposed to glycol ethers have an increased risk of miscarriage? (a) State the null hypothesis. (b) State the alternative hypothesis. (c) Compute the value of the test statistic. ((1) Estimate the p—value. { e) Can we conclude that women exposed to glycol ethers have an increased risk of mis— carriage? Problem 7. (25 points) The effectiveness of a test preparation course was studied for a random sample of 75
subjects who took the SAT before and after coaching. The differences between the scores
resulted in a mean increase of 0.6 and a standard deviation of 3.8. (See “An Analysis
of the Impact of Commercial Test Preparation Courses on SAT Scores,” by Sesnowitz,
Bernhardt, and Kwain, American Education Research Journal, Vol. 19, No. 3.) Can this increase be explained away as chance variation or can we conclude that the course is effective in raising scores? (a) State the null hypothesis. (b) State the alternative hypothesis. (c) Compute the value of the test statistic.
((1) Estimate p—value. (e) Does the course raise scores? Problem 8. (25 points) A study of seat belt use involved children who were hospitalized as a result of motor vehicle
crashes. For a group of 290 children who were not wearing seat belts, the number of days
in Intensive Care Units (ICU) has a mean of 1.39 and a standard deviaiton of 3.06. For a
group of 123 children who were wearing seat belts, the number of days in ICU has a mean
of 0.83 and a standard deviation of 1.77 (based on data from “Morbidity Among Pediatric
Motor Vehicle Crash Victims: The Effectiveness of seat Belts,” by Osberg and Di Scala,
American Journal of Public Health, Vol. 82, No. 3). Can this difference be explained away as chance variation, or is there signiﬁcant evidence in favor of seat belt use for children? (a) State the null hypothesis. (b) State the alternative hypotheis. (c) Compute the value of the test statistic.
(d) Estimate the p—value. (e) Does the useof seat belts protect children against serious injuries? Problem 9. (30 points) Many people believe that fatal DWI crashes occur because of casual drinkers who tend to
binge on Friday and Saturday nights, whereas others believe that fatal DWI crashes are
caused by alcoholics who drink every day of the week. In a study of fatal car crashes, 216
cases are randomly selected from the pool in which the driver was found to have a blood
alcohol content over 0.10. These cases are broken down according to the day of the week,
with the results listed in the accompanying table. At the 0.05 significance level, test the claim that such fatal crashes occur on the different days of the week with equal frequency. Day Sun Mon Tues Wed Thurs Fri Sat
Number 40 24 25 28 29 32 38 Based on data from the Dutchess County STOP—DWI Program. (a) State the null hypothesis. (b) Compute the expected frequencies under the null hypothesis. (c) Compute the value of the test statistic. (d) How many degrees of freedom does the test statistic have? (e) Estimate the p—value. f ) Does the evidence support the theory that fatal DWI car crashes are due to casual drinkers or those Who drink daily? STAT 3 lO/MATH 363 Ditlev Monrad
Final Examination August 7, 1998 OPEN BOOK. SHOW ALL WORK. Problem 1. (10 points) A fair die is tossed 180 times. The number of sixes will be around , give or take , or so. Problem 2. (20 points) A multiple choice test has 72 questions. To each questions there are 3 possible answers,
one of which is correct. Imagine that you don't know the answer to any of the questions
and blindly guess all the answers. Then your number of correct answers will be around , give or take , or so. Your chance of getting at least 25 correct answers is closest to
35% 40% 45% 50% 55% (Use the statistical tables or a calculator.) Problem 3. (35 points)
Let 6 > 0 be an unknown parameter and let X I,X 2,...,)( n be n independent random variables, each with the p.d.f.
. x ~X/{7
f(x,0)=(9—Ze forx>0 (a) Given the observations x1,x2,... x determine the likelihood function a n 1 L09) 2 L(6;xl,x2,...,xn).
(b) Find the maximum likelihood estimator of (9.
(c) Show that is unbiased.
(d) Compute Va/‘(Oll (e) Show that is a consistent estimator of 6.
(0 Find the standard error of (g) Assume that n = 32 and é : 7.2. Give an approximate 95% conﬁdence interval for 6. Problem 4. (25 points)
In a study of the relationship between birth order and college success, an investigator
found that 126 in a sample of 180 college graduates were ﬁrst—born or only children; in a
sample of 100 non—graduates of comparable age and socioeconomic background the
number of ﬁrst—bom or only children was 54.
(a) Estimate the proportion of ﬁrst—bom or only children in the two populations from
which these samples were drawn.
(b) Estimate the difference between the proportion of ﬁrst—born or only children in the
two populations. (c) Compute the standard error for the estimate of the difference. Problem 5. (10 points)
A researcher discovered a new drug for baldness, and perfomied an experiment to see if
the new drug worked better than the old drug, Rogaine. (Assume the experiment was
double~blind and used randomized controls.) The new drug perfomied better in the
experiment. The p—value was 18%. Which of the following statements most accurately
describes the implication of that p—value: (i) The test proves that the new drug is better than Rogaine. (ii) We can reject the null hypothesis.
(iii) It is quite possible that the superior performance of the new drug was due to chance. (iv) The test proves that the new drug is not better than Rogainc. Choose one option. Problem 6. (25 points) In a study to assess various effects of using a female model in automobile advertising,
each of 100 male subjects was shown photographs of two automobiles matched for price,
color, and size, but of different makes. One of the automobiles was shown with a female
model to 50 of the subjects (group A), and both automobiles were shown without the
model to the other 50 subjects (group B). In group A, the automobile shown with the
model was judged to be more expensive by 37 subjects. In group B, the same automobile
was judged as more expensive by 23 subjects. Do these results indicate that using a
female model increases the perceived value of an automobile? Or can the different responses from the males in groups A and B be explained away as chance variation? (a) State the null hypothesis H 0 (which we want to disprove). (b) State the alternative hypothesis H A (which we want to prove). (c) What is the test statistic?
(d) Find the pvalue.
(e) Can we conclude that the use of a female model increases the perceived value of a car? Problem 7. (25 points) An agronomist wants to determine whether a larger corn crop can be obtained if sterilized
males of an insect pest are introduced to control the pest population instead of using an
insecticide. Eighty l—acre plots are randomly divided into two groups of forty l—acre
plots. The insecticide is used on each l—acre plot in the ﬁrst group and the sterilized male
insects on each l—acre plot in the second group. The average yield for the plots treated with insecticide was 100.15 bushels with a standard deviation of 5.73 bushels. The average yield for the plots treated with sterilized male insects was 109.53 bushels with a standard deviation of6.06 bushels. (a) What is the null hypothesis?
(b) What is the alternative hypothesis?
(e) Compute the value of the test statistic. (d) Estimate the p—value. (e) Can we conclude that the use of sterilized male insects gives more effective pest control than the use of insecticide? Problem 8. (25 points) The table below is a cross—tab for a simple random sample of 103 adults aged 25—29, living in Wyoming. Men ‘ Women
Never Married 21 9
Married 20 3 9 Widowed/Divorced/ Separated 7L 7 Is the distribution the same for men as for women? (a) What is the null hypothesis H 0 ? (b) What is the test statistic? (c) How many degrees of freedom does the test statistic have? ((1) Estimate the p—value. (e) How do you interpret the results? How can the percent of married men be so much smaller than the percent of married women? Name: UIN: STAT 310/MATH 363 Ditlev Monrad
Final Examination December 15, 2003
OPEN BOOK SHOW ALL WORK Problem 1. (25 points) 1n the United States the annual death rate due to stroke is 60 per 100,000 population.
(a) What is the expected number of deaths due to stroke over a 3 month period
in a random sample of12,000 Americans?
Estimate the probability that over a 3 month period a random sample of 12,000
Americans has (b) no deaths due to stroke (0) at most 4 deaths due to stroke
(d) at least 4 deaths due to stroke
(e) exactly 4 deaths due to stroke (Use a calculator or statistical tables to compute the probabilities.) Problem 2. (10 points)
A survey organization takes a simple random sample of 1,500 persons from the residents
ofa large city. Among these sample persons, 1,035 were renters. (a) Estimate the fraction of all residents of the city who are renters.
(b) Find a 95% conﬁdence interval for the fraction of the population who are renters. Problem 3. (15 points)
(a) Compute the mean u of the continuous distribution with p.d.f.
foge) : ext‘H”, x>l, where the constant 9 is greater than 1 . (b) (liVen n independent obscrmtions from this distribution. x‘i'n X3. X”, find the method—ot—nioinents estimator () ol‘the unkntmn parameter ('3. (c) Compute ifx : 1.03 Problem 4 (25 points) Let (9 > 0 be an unknown parameter and let Y1, Y3. Y” be 11 independent random variables, each with the p.d.f f0; 9) = 3y2/93, 05 y < (9, (a) Compute the corresponding distribution function F<r>=t€oo f0»; em» ,w<r< oo.
(b) Determine the p.d.f. onmax = max {Y1, Y2, Yn}.
(c) Compute E(Ymax).
((1) Compute Var(Ymax). (6) Explain why Ymax is a consistent estimator oft). Problem 5. (35 points) Let G > 0 be an unknown parameter and let X1, X2, X” be )1 independent random variables, each with the p.d.f. 2
x 203 73/0
9 ‘ for x > 0 ftxﬂ) = (a) Given the obsen'ations x], X; “Urn, determine the likelihood function [4(0) : [4((9; x1, Xg. uni”).
(b) Find the maximum likelihood estimator 6 of(}. (c) Show that 63 is unbiased.
(d) Compute Var(éi).
(e) Show that (3 is a consistent estimator of 6. (l) Find the standard error ofé , assuming that n: 27 and (5 z 7.2. (g) Assume that n = 27 and 6’ = 7.2. Give an approximate 95% conﬁdence interval for 6. Problem 6. (25 points) In a random sample of 4200 Puerto Ricanborn women living in New York City during
1980—1986 there were 87 cancer cases. The corresponding cancer rate for female Puerto
Rican residents was only 0.013 according to the National Cancer Institute. Can this
difference be explained away as a chance variation, or can we conclude that female
Puerto Ricans increase their cancer risk by moving to New York? (Such studies of migrants can give insight into the role that the environment plays in cancer rates.) (a) State the null hypothesis Hy. (b) State the alternative hypothesis // I.
(c) Compute the value ofthe test statistic.
((1) Compute the p—value. (e) Can we conclude that Puerto Rican—born women who move to New York have a greater risk of cancer than those who remain in Puerto Rico? Problem 7. (25 points)
A motorist is stopped for drunk driving. Four independent measurements are made of the
driver’s blood alcohol level and average 0.1 12 percent. It is know that the instrument
used is unbiased and produces measurements that follow a normal curve with an SD of
0.012 percent. The driver is legally drunk ifhis alcohol level exceeds 0.100 percent. Can
he claim innocence? (a) Formulate the null hypothesis. (b) Formulate the alternative hypothesis. (e) What is the test statistic? ((1) Compute the p—Value. (e) Can we conclude that the motorist was legally drunk when he was stopped? Problem R (20 pninm) Ruxrior has; it that there is a lot of drinking;r 120111;: on in fraternities and sororities. VVC
Want to (LUIIIpELI‘C thC beer drinking habits (if the stlniﬁnts iivillg 111 1.1 "ALHI 11.11.1145 2111(1 snrtn'iLit‘N with the drinking habits“ of the students living in dermis and private housing. Independent
rundoni SILITIDICS of 100 and 144 students were taken frein the above two groups. The I’llllllljtif ()f 5t11k1‘711t5 V‘vllk) (llif1lrlk 111()1(‘ {1117(1‘II G 1)(Jiriil(",N (()I (iHJIN) (11'11t4t‘1‘ lHJI' VVf"!“1\' i‘llrllf‘ll (“IL
11111:) :11111 SH. 1'1N111w'1ivvly in 11117» twlx «unplug. Test the 11111111yp0t1101<is that on average
students: in fraternities and Hororitim; drink no 111<>1‘(‘ than HtudClltij 111 (101mm 01‘ 1)L‘l\'(’\17(‘
housing.
(11‘) [i‘lll'f H” 1111 1.1(4(1”.HIHII(',P1)(51,V\/H(‘ll HIP l‘wr) ln‘nlnn‘finiis ix lest‘t 111: (i) 0.040 (ii) 0.050 (iii) 0.06/1 (iv) 0.090
(b) The observed value of the test statistic is closest to: (:1) 2.0 (:11) 2.0 (111) 3.7 (iv) 4.6
(c) The p value is closest to: (/i) 1 in 10,000 (ii) 2 in 1.000 (iii) 29/“ (iv) 52% (v) 35‘}?
(d) We conlcude that students in fraternities and sororities do drink more than students in (11111115 (11 private, housing), FPTHP (ii) False ...
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This note was uploaded on 02/10/2009 for the course STAT 400 taught by Professor Tba during the Spring '05 term at University of Illinois at Urbana–Champaign.
 Spring '05
 TBA
 Statistics, Probability

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