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11_02_09

Course: STATISTICS stat 410, Spring 2011
School: University of Illinois,...
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409 Fall STAT 2009 Examples for 11/02/2009 Example 1: 1 2 3 4 5 p0(x ) 0.25 0.15 0.20 0.05 0.35 p1(x ) 0.20 0.20 0.20 0.20 0.20 H 0 : X has p.m.f. p 0 ( x ) vs. H 1 : X has p.m.f. p 1 ( x ). 1 (x) 2 3 4 5 1.25 0.75 1 0.25 1.75 Reject H 0 if x = 4 = 0.05, Power = 0.20. Reject H 0 if x = 4 or 2 = 0.20, Power = 0.40. = 0.10, Power = Reject H 0 if x = 4 Reject H 0 with...

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409 Fall STAT 2009 Examples for 11/02/2009 Example 1: 1 2 3 4 5 p0(x ) 0.25 0.15 0.20 0.05 0.35 p1(x ) 0.20 0.20 0.20 0.20 0.20 H 0 : X has p.m.f. p 0 ( x ) vs. H 1 : X has p.m.f. p 1 ( x ). 1 (x) 2 3 4 5 1.25 0.75 1 0.25 1.75 Reject H 0 if x = 4 = 0.05, Power = 0.20. Reject H 0 if x = 4 or 2 = 0.20, Power = 0.40. = 0.10, Power = Reject H 0 if x = 4 Reject H 0 with probability 1 if x = 2 3 8 . 30 Example 2: 1 2 3 4 5 p0(x ) 0.25 0.15 0.20 0.05 0.35 p1(x ) 0.05 0.30 0.40 0.20 0.05 H 0 : X has p.m.f. p 0 ( x ) vs. H 1 : X has p.m.f. p 1 ( x ). 1 (x) 2 3 4 5 5 0.5 0.5 0.25 7 = 0.05, Power = 0.20. 1 if x = 2 3 = 0.10, Power = 0.30. 1 if x = 3 4 = 0.10, Power = 0.30. = 0.10, Power = 0.30. Reject H 0 if x = 4 Reject H 0 if x = 4 Reject 0 H with probability Reject H 0 if x = 4 Reject H 0 with probability Reject H 0 if x = 4 1 if x = 2 6 1 Reject H 0 with probability if x = 3 8 Reject H 0 with probability H0 : 0 vs. 0 1 = . H1 : 1 max L( ) Reject H 0 = if 0 max L( ) max L( ) k 0 * = 1 max 0 1 L( ) k since * = min ( , 1 ) H0 : = 0 H1: 0 vs. Reject H 0 if = L( 0 ) k L () Example 3: Let X 1 , X 2 , , X n be a random sample of size distribution with mean 1 / . That is, n from an Exponential f X ( x ) = e x , H0 : = 0 L( ) = vs. n e H1: 0 xi i =1 n = n exp x i . i =1 = L( 0 ) = L () x > 0. n 1 = . X n 0 exp 0 x i () { } i =1 n n = e n 0 x exp n 0 x . n 1 n 1 exp x i x x i =1 k x exp{ 0 x } c Reject H 0 if x exp{ 0 x } c.
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University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 11/04/2009H0 : 0vs. 0 1 = .H1 : 1max L( )Reject H 0if=max L( ) 0kmax L( )* 1 0=max 0 1L( )ksince * = min ( , 1 )H0 : = 0vs.H1: 0Reject H 0=ifL( 0 )kL()Example 1:Let X 1 , X 2 , , X n be a ra
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 11/09/2009Pearsons 2 Test for Goodness of Fit ( Based on Large n )A random sample of size n is classified into k categories or cells.kYi = n.Let Y 1 , Y 2 , , Y k denote the respective cell frequencies.i =1Denote the
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 11/09/2009Alex buys a package of Sour Jelly Beans. On the package, it says that 50% of alljelly beans are lemon, 30% are cherry, and 20% are lime. When Alex opens thepackage, he finds 15 lemon, 9 cherry and 12 lime je
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Examples for 11/11/2009Fall 2009It is claimed that X has a Poisson distribution, and 120 observations of X are given in thefrequency distribution:xfrequency012345678121422242317521Use a chi-square goodness of fit test to v
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Examples for 11/13/2009The 2 Test of HomogeneityFall 2009( one margin fixed )Independent random samples from r populations.Each sample classified in c response categories.H 0 : In each response category, the probabilities are equal for all
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 11/13/2009A group of 500 children (250 boys and 250 girls) were asked to identify theirfavorite color. We obtain the following data:Favorite ColorSexYellowRedBlueBrownBoys107853325Girls93656725A toy ma
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 11/16/2009The following is a random sample from a certain population:2419171220112215222618We wish to test H 0 : m = 17 vs. H 1 : m &gt; 17, where m is the populationmedian.a)Use the sign test. What is the p
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 11/16/2009The following is a random sample from a certain population:2419171220112215222618We wish to test H 0 : m = 17 vs. H 1 : m &gt; 17, where m is the populationmedian.a)Use the sign test. What is the p
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 11/18/2009An investment analyst wants to test whether differences exist between the returnson two mutual funds. Paired data of annualized rates of return for the two mutualfunds during 15 randomly chosen months are as
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 11/18/2009An investment analyst wants to test whether differences exist between the returnson two mutual funds. Paired data of annualized rates of return for the two mutualfunds during 15 randomly chosen months are as
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Examples for 11/30/2009Fall 2009A statistics instructor teaches two statistics classes, one in the morning and onein the afternoon. Suppose the probability that the instructor gives a good morninglecture is 0.3. If the morning lecture is g
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 11/30/2009A statistics instructor teaches two statistics classes, one in the morning and onein the afternoon. Suppose the probability that the instructor gives a good morninglecture is 0.3. If the morning lecture is g
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Examples for 12/07/2009Fall 2009
University of Illinois, Urbana Champaign - STATISTICS - stat 410
University of Illinois, Urbana Champaign - STATISTICS - stat 410
University of Illinois, Urbana Champaign - STATISTICS - stat 410
University of Illinois, Urbana Champaign - STATISTICS - stat 410
University of Illinois, Urbana Champaign - STATISTICS - stat 410
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #1(due Friday, September 4, by 4:00 p.m.)1.Let &gt; 0 and let X be a random variable with the probability density functionf(x) =x +1,x &gt; 1,zero otherwise.Let W = ln ( X ). What is the probability distribution of W ?2.Le
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #1(due Friday, September 4, by 4:00 p.m.)1.Let &gt; 0 and let X be a random variable with the probability density functionf(x) =x +1x &gt; 1,,zero otherwise.Let W = ln ( X ). What is the probability distribution of W ?x&gt;1d
University of Illinois, Urbana Champaign - STATISTICS - stat 410
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #2(due Friday, September 11, by 4:00 p.m.)1.Let X 1 , X 2 , , X n be a random sample of sizeprobability density functionf X (x) = f X ( x ; ) = ( 1 ) 2 a)ln xx,nfrom the distribution withx &gt; 1, &gt; 1.We already know
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #2(due Friday, September 11, by 4:00 p.m.)1.Let X 1 , X 2 , , X n be a random sample of sizeprobability density functionf X (x) = f X ( x ; ) = ( 1 ) 2 a)ln xxn,from the distribution withx &gt; 1, &gt; 1.We already know
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #3(due Friday, September 18, by 4:00 p.m.)1.Let X 1 , X 2 , , X n be a random sample of sizeprobability density functionf X (x) = f X ( x ; ) = ( 1 ) 2 ln xxnfrom the distribution with &gt; 1.x &gt; 1,,a)Find the suffic
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #3(due Friday, September 18, by 4:00 p.m.)1.Let X 1 , X 2 , , X n be a random sample of sizeprobability density functionf X (x) = f X ( x ; ) = ( 1 ) 2 a)ln xxnfrom the distribution withx &gt; 1,, &gt; 1.Find the suffic
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2005Homework #4(due Friday, September 25, by 4:00 p.m.)From the textbook:6.4-86.4-106.8-26.5-26.4-126.8-46.4-186.8-66.5-4Hint: Equal variances10.The National Security Agency (NSA) hires you to compute a 95% confidenceinterval
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2005Homework #4(due Friday, September 25, by 4:00 p.m.)From the textbook:6.4-8a) x = 46.42;b) s 2 = 41.682, s 6.456, t 0.05 ( 4 ) = 2.132,46.42 2.132 6.4565or46.42 6.156or( 40.264, 52.576 ).6.4-10t 0.10 ( 27 ) = 1.314,( 21.45
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #5(due Thursday, October 1, by 4:00 p.m.)From the textbook:6.6-26.6-46.6-86.6-106.7-26.7-46.7-126.7-186.8-12 if we believe the seed distributors claim;b)10.a) if we ignore the seed distributors claim.Let X have
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Homework #5(due Thursday, October 1, by 4:00 p.m.)From the textbook:6.6-2For these 9 weights, x = 20.90, s = 1.858.a)A point estimate for is s = 1.858.b)8 1.858 2,17.548 1.858 2 = [ 1.255, 3.559 ].2.180 c)8 1.858 2,15.518 1.858
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #6(due Friday, October 9, by 4:00 p.m.)1.Let a &gt; 0, &gt; 0 and let X 1 , X 2 , , X n be a random sample of sizedistribution with probability density functionf X (x) = f X ( x ; ) =aa x a 1 ,nfrom anfrom a0 &lt; x &lt; .Sup
University of Illinois, Urbana Champaign - STATISTICS - stat 410
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #7 (Answers)8.1-6H 0 : p = 1/6 vs. H 1 : p &lt; 1/6orH 0 : p 1/6 vs. H 1 : p &lt; 1/6Left tailed.n = 8000.a)The test statistic isz= = 0.05.b)y py1n0=80006.p 0 (1 p 0 )1 1 166n8000(The critical (rejection) reg
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Homework #8Fall 2009(due Friday, October 23, by 4:00 p.m.)1.In a random sample of 100 Hawk &amp; Hummingbird Airline (HHA) direct flightsfrom New York to Boston, the average number of passengers was 56.3, withsample standard deviation 11.33.a)
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #8(due Friday, October 23, by 4:00 p.m.)1.In a random sample of 100 Hawk &amp; Hummingbird Airline (HHA) direct flightsfrom New York to Boston, the average number of passengers was 56.3, withsample standard deviation 11.33.X
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #9(due Friday, October 30, by 4:00 p.m.)From the textbook:8.2-169.1-49.1-69.1-109.1-79.1-127.Let X 1 , X 2 , , X 25 be a random sample from a N ( , 100 ) population,and suppose the null hypothesis H 0 : = 100 is to b
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #9(due Friday, October 30, by 4:00 p.m.)From the textbook:8.2-16H 0 : 2 = 30 vs. H 1 : 2 = 80.Right tailed.Recall: If X 1 , X 2 , , X n are i.i.d. N ( , 2 ), thena)Test Statistic:2 =2is 2 ( n 1 ).(n 1) s 2 = 18 s 2
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Homework #10Fall 2009(due Thursday, November 5, by 4:00 p.m.)1 2.Bert and Ernie noticed that thefollowing are satisfied whenCookie Monster eats cookies:(a)the number of cookies eaten duringnon-overlapping time intervals areindependent;
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Homework #10Fall 2009(due Thursday, November 5, by 4:00 p.m.)1 2.Bert and Ernie noticed that thefollowing are satisfied whenCookie Monster eats cookies:(a)the number of cookies eaten duringnon-overlapping time intervals areindependent;
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Homework #11Fall 2009(due Wednesday, November 13, by 4:00 p.m.)1.Let &gt; 0 and let X 1 , X 2 , , X n be independent random variables, each withthe probability density function +1f(x) = x0x 1.x &lt;1We wish to test H 0 : = 1 vs. H 1 : &gt; 1.
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #11(due Friday, November 13, by 4:00 p.m.)1.Let &gt; 0 and let X 1 , X 2 , , X n be independent random variables, each withthe probability density function +1f(x) = x0x 1.x &lt;1We wish to test H 0 : = 1 vs. H 1 : &gt; 1.a)
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #12(due Friday, November 20, by 4:00 p.m.)8.6-195(O E )3237.268.423.446.2360.038710.09883264318322859.2E45532210.432883E7188.8O3624.845.615.630.8300240.246154 0.031169 0.4444442000.649324 0
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #13(due Friday, December 4, by 4:00 p.m.)1.Seventy percent of the light aircraft that disappear while in flight in Neverlandare subsequently discovered. Of the aircraft that are discovered, 60% have anemergency locator, wh
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Homework #13(due Friday, December 4, by 4:00 p.m.)1.Seventy percent of the light aircraft that disappear while in flight in Neverlandare subsequently discovered. Of the aircraft that are discovered, 60% have anemergency locator, wh
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Homework #14Fall 2009(due Wednesday, December 9, by 4:30 p.m.)1.When correctly adjusted, a machine that makes widgets operates with a 5% defectiverate. However, there is a 10% chance that a disgruntled employee kicks the machine,in which ca
University of Illinois, Urbana Champaign - STATISTICS - stat 410
University of Illinois, Urbana Champaign - STATISTICS - stat 410
1.Let X 1 , X 2 , , X n be a random sample from the distribution with probabilitydensity functionf (x ) = 4 x 3 e x4 &gt; 0.x&gt;0a)Find the sufficient statistic Y = u ( X 1 , X 2 , , X n ) for .b)Obtain the maximum likelihood estimator of ,c)Isa c
University of Illinois, Urbana Champaign - STATISTICS - stat 410
Practice Problems1.The label on 1-gallon can of paint states that the amount of paint in the can issufficient to paint at least 400 square feet (on average). Suppose the amount ofcoverage is approximately normally distributed, and the overall standard
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Name _Quiz 1(3 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.1.Metaltech Industries manufactures carbide drill tips used in drilling oil wells.The lif
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009NameANSWERS.Quiz 1(3 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.1.Metaltech Industries manufactures carbide drill tips used in drilling oil wells.
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Name _Quiz 2(3 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.1.At Anytown State University, the population standard deviation of the SAT scoresof ente
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009NameANSWERS.Quiz 2(3 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.1.At Anytown State University, the population standard deviation of the SAT scores
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Name _Version AQuiz 3(3 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.a) (1) Suppose we Do NOT Reject H 0 for = 0.05. Then for = 0.10, we _.( Circle o
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009NameANSWERS.Version AQuiz 3(3 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.a) (1) Suppose we Do NOT Reject H 0 for = 0.05. Then for = 0.10, we _.(
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Name _Version BQuiz 3(3 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.a) (1) Suppose we Do NOT Reject H 0 for = 0.05. Then for = 0.01, we _.( Circle o
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009NameANSWERS.Version BQuiz 3(3 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.a) (1) Suppose we Do NOT Reject H 0 for = 0.05. Then for = 0.01, we _.(
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Name _Version AQuiz 4(5 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.1.At Anytown College, the administration would like the students grade distribut
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009NameANSWERS.Version AQuiz 4(5 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.1.At Anytown College, the administration would like the students grade d
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Name _Version BQuiz 4(5 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.1.At Anytown College, the administration would like the students grade distribut
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009NameANSWERS.Version BQuiz 4(5 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.1.At Anytown College, the administration would like the students grade d
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Name _Version AQuiz 5(5 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.1.A travel agent randomly sampled individuals in her target market to determine
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009NameANSWERS.Version AQuiz 5(5 points)Be sure to show all your work, your partial credit might depend on it.No credit will be given without supporting work.1.A travel agent randomly sampled individuals in her target market to de