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### 409Hw09

Course: STATISTICS stat 410, Spring 2011
School: University of Illinois,...
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409 Fall STAT 2009 Homework #9 (due Friday, October 30, by 4:00 p.m.) From the textbook: 8.2-16 9.1-4 9.1-6 9.1-10 9.1-7 9.1-12 7. 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 be tested. a) If the alternative hypothesis is H 1 : &gt; 100, compute the power of the appropriate test at = 101 and at = 102. Use = 0.05....

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409 Fall STAT 2009 Homework #9 (due Friday, October 30, by 4:00 p.m.) From the textbook: 8.2-16 9.1-4 9.1-6 9.1-10 9.1-7 9.1-12 7. 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 be tested. a) If the alternative hypothesis is H 1 : > 100, compute the power of the appropriate test at = 101 and at = 102. Use = 0.05. b) If the alternative hypothesis is H 1 : 100, compute the power of the appropriate test at = 101 and at = 102. Use = 0.05. c) For the test in (a) compute the p-value associated with X = 103.5. d) For the test in (b) compute the p-value associated with X = 103.5. 8. Let X 1 , X 2 , , X 12 be a random sample of size n =12 with mean . That is, k e , k! P( X1 = k ) = from a Poisson distribution k = 0, 1, 2, 3, . Consider the test H 0 : = 1 2 vs. H 1 : < 1 2 . / / The rejection region is given by { ( x 1 , x 2 , , x 12 ) : 12 i =1 x i 2 }. If K ( ) is the power function of the test, find the powers K ( 1 2 ), K ( 1 3 ), K ( 1 4 ), K ( 1 6 ), and K ( 1 12 ). Sketch the graph of K ( ). What is the / / / significance level of the test? / / 9. A certain automobile manufacturer claims that at least 80% of its cars meet the tough new standards of the Environmental Protection Agency (EPA). The EPA tests a random sample of 25 its cars and to wishes test H 0 : p 0.80 vs. H 1 : p < 0.80, where p is the proportion of the cars that meet the new EPA standards. a) Let X denote the number of cars in the sample that meet the new EPA standards. Find the best Rejection Region with the significance level closest to 0.05. b) What is the power of the Rejection Rule obtained in part (a) if p = 0.70 ? If p = 0.60 ? c) Suppose that 18 out of 25 cars in our sample meet the new EPA standards. Compute the p-value. d) Find the best randomized Rejection Region with the significance level 0.07. e) What is the power of the Rejection Rule obtained in part (d) if p = 0.70 ? If p = 0.60 ? 10. Alex wants to test whether a coin is fair or not. Suppose he decides to use 25 tosses. Let p denote the probability of obtaining heads. H 0 : p = 0.50 vs. H 1 : p 0.50. a) Find the Rejection Rule with the significance level closest to 0.05. b) What is the actual value of the significance level associated with the Rejection Rule obtained in part (a)? c) Suppose Alex got 9 heads and 16 tails. Find the p-value of the test. d) Suppose Alex got 20 heads and 5 tails. Find the p-value of the test. e) Suppose Alex got 20 heads and 5 tails. State your decision ( Reject H 0 or Do NOT Reject H 0 ) at a 5% level of significance. f) What is the power of this test ( using the Rejection Rule obtained in part (a) ) if p = 0.60 ?
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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
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Name _Version BQuiz 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 BQuiz 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
University of Illinois, Urbana Champaign - STATISTICS - stat 410
Cumulative Binomial Probabilities p n2Cumulative Binomial Probabilities p0.60 0.160 0.640 0.064 0.352 0.784 0.026 0.179 0.525 0.870 0.010 0.087 0.317 0.663 0.922 0.004 0.041 0.179 0.456 0.767 0.953 0.002 0.019 0.096 0.290 0.580 0.841 0.972 0.001 0.009
University of Illinois, Urbana Champaign - STATISTICS - stat 410
Left tailed testH 0 : p = p0vs.H 1 : p &lt; p0If H 0 is TRUE :Use p 0 .Reject H 0Do NOT Reject H 0Type I ErrorCorrect decisiona0a+1nRejection Rule for a Left tailed test:Find a such that P( Y a ) = CDF @ a .( using Binomial ( n , p 0 ) tables
University of Illinois, Urbana Champaign - STATISTICS - stat 410
Chapter 6Estimation6.1Sample Characteristics6.12 (a) x =(b) s2 =4= 1.333;388= 1.275.696.14 (a) x = 1.711, s = 0.486;(b) and (d) graphs.0.80.70.60.50.40.30.20.10.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7123456Figure 6.14: (b) Female un
University of Illinois, Urbana Champaign - STATISTICS - stat 410
Chapter 8Tests of Statistical Hypotheses8.1Tests about Proportions8.12 (a) C = cfw_x : x = 0, 1, 2;(b) = P (X = 0, 1, 2; p = 0.6)= (0.4)4 + 4(0.6)(0.4)3 + 6(0.6)2 (0.4)2 = 0.5248;= P (X = 3, 4; p = 0.4)= 4(0.4)3 (0.6) + (0.4)4 = 0.1792.OR(a ) C
University of Illinois, Urbana Champaign - STATISTICS - stat 410
Consider a coin being tossed 20 times.There are 2 20 = 1,048,576 possible outcomes (samples).Suppose we wish to test whether the coin is fair or not.We can rank all these outcomes (samples) from 20 C 10 = 184,756 best outcomes(samples) with 10 Hs and
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409 / MATH 409Monday, Wednesday, FridayInstructor:Office:E-mail:Office hours:Text:Fall 20092:00 2:50 p.m.103 Talbot LaboratoryAlexey Stepanov101-A Illini Hallstepanov@illinois.eduph.:265-6550Monday 3:30 4:30 p.m., Thursday 1:30 3:00 p.
University of Illinois, Urbana Champaign - STATISTICS - stat 410
William Gosset(1876-1937)Thet DistributionEXCEL: = TINV ( , v ) = TDIST ( t , v , 1 ) = TDIST ( t , v , 2 ) gives gives givest2for t distribution with v degrees of freedom the upper tail probability for t distribution with v degrees of freedom, P (
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