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### Homework 0 (m124_125)

Course: MATH 124, Spring 2011
School: Arizona
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Arizona - MATH - 124
Arizona - MATH - 124
Arizona - MATH - 124
Arizona - MATH - 124
Arizona - MATH - 124
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409p.m.f. or p.d.f.1.Fall 2009Examples for 08/26/2009 .f ( x ; ),Suppose = cfw_ 1, 2, 3 and the p.m.f. parameter space.f ( x ; ) is = 1:f ( 1 ; 1 ) = 0.6,f ( 2 ; 1 ) = 0.1,f ( 3 ; 1 ) = 0.1,f ( 4 ; 1 ) = 0.2. = 2:f ( 1 ; 2 ) = 0.2,f
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409p.m.f. or p.d.f.1.Fall 2009Examples for 08/26/2009f ( x ; ), .Suppose = cfw_ 1, 2, 3 and the p.m.f. parameter space.f ( x ; ) is = 1:f ( 1 ; 1 ) = 0.6,f ( 2 ; 1 ) = 0.1,f ( 3 ; 1 ) = 0.1,f ( 4 ; 1 ) = 0.2. = 2:f ( 1 ; 2 ) = 0.2,f
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4095.Fall 2009Examples for 08/28/2009Let X 1 , X 2 , , X n be a random sample of size n from a uniformdistribution on the interval ( 0 , ).1f ( x; ) = 0E( X ) =20xF( x; ) = 10&lt; x&lt;otherwiseVar ( X ) =2x&lt;00&lt; x&lt;x&gt;12a)~Obtain the
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4095.Let X 1 , X 2 , , X n be a random sample of size n from a uniformdistribution on the interval ( 0 , ).1f ( x; ) = 0E( X ) =a)otherwiseVar ( X ) =2x&lt;00xF( x; ) = 10&lt; x&lt;20&lt; x&lt;x&gt;12~Obtain the method of moments estimator of ,
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 09/02/2009PDefAn estimator for is said to be consistent if , i.e.,for all &gt; 0, P 0 as n .The (Weak) Law of Large Numbers:Let X 1 , X 2 , be i.i.d. with mean and standard deviation . LetXn =PX 1 + . + X nnThen X n
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409DefExamples for 09/02/2009Fall 2009Let U 1 , U 2 , be an infinite sequence of random variables, and let W beanother random variable. Then the sequence cfw_ U n converges in probabilityto W, if for all &gt; 0,lim Pn ()Un W = 0 ,Pand writ
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 09/02/2009If the random variable Y denotes an individuals income, Paretos law claims thatP ( Y y ) = k , where k is the entire populations minimum income. It yfollows that +1 1 fY( y ) = k ,y k; 1. yAssume k
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Let X 1 , X 2 , , X n be a random sample of sizeprobability density function x 1nfrom the distribution with0 &lt; x &lt;1otherwisef X ( x; ) = a)Fall 2009Examples for 09/04/200900 &lt; &lt; .~Find the method of moments estimator of .1()0
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409DefExamples for 09/09/09Fall 2009Let X 1 , X 2 , , X n denote random variables with joint p.d.f. orp.m.f. f ( x 1 , x 2 , , x n ; ), which depends on the parameter .The statistic Y = u ( X 1 , X 2 , , X n ) is said to be sufficientfor if th
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 410Fal 2009Examples for 09/09/2009Let X 1 , X 2 , , X n be a random sample from a population with the populationmean and the population standard deviation .By CLT,X is approximately N ( 0 ,2n)for large n.If g ( x ) is differentiable at and
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 09/09/09Let X 1 , X 2 , , X n be a random sample of size n from a Poisson distributionwith mean . That is, k ef ( k; ) = P ( X 1 = k ) =,k = 0, 1, 2, 3, .k!a)Use Factorization Theorem to find Y = u ( X 1 , X 2 ,
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Examples for 09/14/2009Fall 20091.Suppose the lifetime of a particular brand of light bulbs is normally distributed withstandard deviation of 75 hours and unknown mean.a)What is the probability that in a random sample of n = 49 bulbs, the a
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Examples for 09/14/2009Fall 20091.Suppose the lifetime of a particular brand of light bulbs is normally distributed withstandard deviation of 75 hours and unknown mean.a)What is the probability that in a random sample of 49 bulbs, the avera
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 09/18/2009A (1 ) 100% confidence interval for the population mean x z2nx t2snn 1 degrees of freedom1.A manufacturer of TV sets wants to find the average selling price of a particularmodel. A random sample of 25
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 09/18/2009A manufacturer of TV sets wants to find the average selling price of a particularmodel. A random sample of 25 different stores gives the mean selling price as \$342with a standard deviation of \$14. Assume the
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 09/21/2009Two independent samples:X1, X2, , Xn1from population 1mean 1 , std. dev. 1Y1, Y2, , Yn2from population 2mean 2 , std. dev. 2If n 1 and n 2 are large, or population 1 and population 2 are approximately no
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 09/21/2009Two independent samples:X1, X2, , XnY1, Y2, , Yn1from population 12from population 2mean 1 , std. dev. 1mean 2 , std. dev. 2If n 1 and n 2 are large, or population 1 and population 2 are approximately no
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Examples for 09/23/2009Fall 2009EXCEL:= CHIINV ( , v )2 (v ) for 2 distribution with v degreesgivesof freedom= CHIDIST ( y , v )givesthe upper tail probability for 2 distributionwith v degrees of freedom, P ( Y &gt; y ).Recall:If X 1 ,
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 09/23/2009A machine makes -inch ball bearings. In a random sample of 41 bearings, thesample standard deviation of the diameters of the bearings was 0.02 inch. Assumethat the diameters of the bearings are approximately
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 09/25/2009If U and Vare independent 2 ( r 1 ) and2( r2 ) randomvariables, respectively, thenF=U r1V r2has an F distribution with r 1 and r 2 degrees of freedom.2X 1 , , X n ~ N X , X ,2SY2Y2SX2X=2Y 1 ,
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 09/25/2009If U and Vare independent 2 ( r 1 ) and2( r2 ) randomvariables, respectively, thenF=U r1V r2has an F distribution with r 1 and r 2 degrees of freedom.2X 1 , , X n ~ N X , X ,2SY2Y=2SX2X2Y 1 ,
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 09/28/2009The sample proportion:p= xnwhere x is the number of elements in the sample found to belongto the category of interest (the number of &quot;successes&quot;), and n isthe sample size.E( P ) = p,p (1 p ) ,nVar( P ) =
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 09/28/2009The sample proportion:p= xnwhere x is the number of elements in the sample found to belongto the category of interest (the number of &quot;successes&quot;), and n isthe sample size.E( P ) = p,p (1 p ) ,nVar( P ) =
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.A scientist wishes to test if a new treatment has a bettercure rate than the traditional treatment which cures only60% of the patients. In order to test whether the newtreatment is more effective or not, a test group of 20patients were giv
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009A scientist wishes to test if a new treatment has a better cure rate than the traditionaltreatment which cures only 60% of the patients. In order to test whether the newtreatment is more effective or not, a test group of 20 patient
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Examples for 10/07/2009Fall 2009Comparing Two Population ProportionsTest Statistic:p1 p 2z=p (1 p ) 1 n1.1+1n2 where, Y +Y n p + n pp= 1 2 = 1 1 2 2n1 + n 2n1 + n 2In a comparative study of two new drugs, A and B, 120 patients we
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 10/07/2009In a comparative study of two new drugs, A and B, 120 patients were treated withdrug A and 150 patients with drug B, and the following results were obtained.Drug ADrug BCuredNot cured54668763Total12
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 10/09/2009Testing Hypotheses about a Population ProportionNullAlternativeH 0 : p = p0vs.H1 : p &lt; p0Left tailed.H 0 : p = p0vs.H1 : p &gt; p0Right tailed.H 0 : p = p01.pvs.H1 : p p0Two tailed.Let X have a Bino
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4093.Fall 2009Examples for 10/09/2009Alex wants to test whether a coin is fair or not. Suppose he decides to use 15 tosses.Let p denote the probability of obtaining heads.H 0 : p = 0.50vs.H 1 : p 0.50.Two-tailed.n = 15.Cumulative Binomial
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 10/12/2009Hypotheses Testing for the population mean NullAlternativeH0 : 0vs.H1 : &lt; 0Left - tailed.H0 : 0vs.H1 : &gt; 0Right - tailed.H0 : = 0vs.H1 : 0Two - tailed.Z=Test Statistic:X 0ORnT=X 0sORnReje
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Examples for 10/12/2009Fall 20091.The overall standard deviation of the diameters of the ball bearings is = 0.005 mm.The overall mean diameter of the ball bearings must be 4.300 mm. A sample of 81 ballbearings had a sample mean diameter of 4
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 10/14/2009Matched Pair Comparison:PairDifference12X1X2Y1Y2D1 = X1 Y1D2 = X2 Y2nXnYnDn = Xn YnAssume that the differences D i = X i Y i are a random sample from normal distribution withmean and standard devi
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Examples for 10/14/2009 (Answers)Claim: &lt; 7H0: 7Test Statistic:T=D 0sDnRejection Region:=Fall 2009H1: &lt; 757= 1.4142.48Reject H 0 if T &lt; t 0.05 ( 8 1 = 7 d.f. ) = 1.895.The value of the test statistic does not fall into the Rej
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 10/16/2009A new method of storing snap beans is believed to retain more ascorbic acid thanthe old method. In an experiment, snap beans were harvested under uniformconditions and frozen in 250 equal-size packages. One
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 10/16/2009A new method of storing snap beans is believed to retain more ascorbic acid thanthe old method. In an experiment, snap beans were harvested under uniformconditions and frozen in 250 equal-size packages. One
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 10/19/2009H 0 trueH 0 is NOT trueDo Not Reject H 0Type II ErrorReject H 0Type I Error = significance level = P ( Type I Error ) = P ( Reject H 0 | H 0 is true ) = P ( Type II Error ) = P ( Do Not Reject H 0 | H 0 is
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 10/19/2009H 0 trueH 0 is NOT trueDo Not Reject H 0Type II ErrorReject H 0Type I Error = significance level = P ( Type I Error ) = P ( Reject H 0 | H 0 is true ) = P ( Type II Error ) = P ( Do Not Reject H 0 | H 0 is
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 10/21/2009An advertisement for a particular brand of automobile states that it accelerates from0 to 60 mph in an average of 5.0 seconds. Makers of a competing automobile feelthat the true average number of seconds it
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 4091.Fall 2009Examples for 10/21/2009An advertisement for a particular brand of automobile states that it accelerates from0 to 60 mph in an average of 5.0 seconds. Makers of a competing automobile feelthat the true average number of seconds it
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409H0: = 0Fall 2009Examples for 10/23/2009H 1 : = 1.vs.Likelihood Ratio: ( x 1 , x 2 ,., x n ) =L ( 0 ; x 1 , x 2 ,., x n ).L ( 1 ; x 1 , x 2 ,., x n )Neyman-Pearson Lemma:C = cfw_ ( x 1 , x 2 , , x n ) : ( x 1 , x 2 ,., x n ) k .( Rejec
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 410H0: = 0Examples for 10/26/2009vs.Fall 2009H 1 : = 1.Likelihood Ratio: ( x 1 , x 2 ,., x n ) =L ( 0 ; x 1 , x 2 ,., x n ).L ( 1 ; x 1 , x 2 ,., x n )Neyman-Pearson Theorem:cfw_ ( x 1 , x 2 , , x n ) : ( x 1 , x 2 ,., x n ) k ( Reject H
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 410Examples for 10/26/2009H0: = 0Fall 2009H 1 : = 1.vs.Likelihood Ratio: ( x 1 , x 2 ,., x n ) =L ( 0 ; x 1 , x 2 ,., x n ).L ( 1 ; x 1 , x 2 ,., x n )Neyman-Pearson Theorem:cfw_ ( x 1 , x 2 , , x n ) : ( x 1 , x 2 ,., x n ) k ( Reject H
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 10/28/20091.Let X 1 , X 2 , , X n be a random sample of size n from a N ( 0 , 2 )distribution. We are interested in testing H 0 : = 2 vs. H 1 : = 5.a)Use the likelihood ratio to show that the best rejection region isn
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 10/28/20091.Let X 1 , X 2 , , X n be a random sample of size n from a N ( 0 , 2 )distribution. We are interested in testing H 0 : = 2 vs. H 1 : = 5.a)Use the likelihood ratio to show that the best rejection region isn
University of Illinois, Urbana Champaign - STATISTICS - stat 410
STAT 409Fall 2009Examples for 11/02/2009Example 1:12345p0(x )0.250.150.200.050.35p1(x )0.200.200.200.200.20H 0 : X has p.m.f. p 0 ( x )vs.H 1 : X has p.m.f. p 1 ( x ).1(x)23451.250.7510.251.75Reject H 0 if x = 4 = 0.05,
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