Please also complete the optional MatLab exercises.
APMA 1650 HW 8 Due Friday, July 30 at 11pm Eastern. You may work with your classmates, but hand in your own assignment via Gradescope. Please note your collaborators and show your work where appropriate. 1. (a) Suppose one hundred Burger King double whoppers are randomly selected and their weights are measured. The sample mean and standard deviation are x̄ = 10.5, s = 2 (in ounces). Construct a 95% confidence interval for the true mean weight of a Burger King double whopper. (b) How many double whoppers do you need to sample if you want the total length of the 95% confidence interval to be at most 0.2 ounces? You can assume that the sample standard deviation remains roughly 2 ounces. 2. Fifteen students who intended to major in engineering were compared with fifteen students who intended to major in literature. Given below are the means and stan- dard deviations of the scores on the verbal and mathematics portion of the SAT for the two groups of students. Verbal Math Engineering ȳ = 446 s = 42 ȳ = 548 s = 57 Literature ȳ = 534 s = 45 ȳ = 517 s = 52 (a) Construct a 95% confidence interval for the difference in average verbal scores of students majoring in engineering and of those majoring in literature. (b) Construct a 95% confidence interval for the difference in average math scores of students majoring in engineering and of those majoring in literature. (c) What would you conclude from the results in (a) and (b)? What assumptions are necessary for your confidence intervals and conclusions to be valid? 3. An experimenter has prepared a drug dosage level that she claims will induce sleep for 80% of people suffering from insomnia. After examining the dosage, a review panel feels that her claims regarding the effectiveness are inflated. In an attempt to disprove 1 her claim, they administer her prescribed dosage to 20 insomniacs and observe Y , the number for who the drug induces sleep. We wish to test the hypothesis H0 : p = .8 versus the alternative Ha : p < .8.="" assume="" that="" the="" rejection="" region="" {y="" ≤="" 13}="" is="" used.="" (a)="" in="" terms="" of="" this="" problem,="" what="" is="" type="" i="" error?="" (b)="" find="" α.="" (c)="" in="" terms="" of="" this="" problem,="" what="" is="" type="" ii="" error?="" (d)="" find="" β="" when="" p=".6." (e)="" find="" β="" when="" p=".4." (f)="" now="" find="" the="" appropriate="" rejection="" region="" such="" that="" α="" ≈="" 0.01.="" (g)="" for="" the="" rejection="" region="" found="" in="" (f),="" find="" β="" when="" p=".6." (h)="" for="" the="" rejection="" region="" in="" part="" (f),="" find="" β="" when="" p=".4." 4.="" (a)="" a="" study="" compared="" traditional="" and="" activity-oriented="" methods="" for="" teaching="" biol-="" ogy.="" pretests="" were="" given="" to="" students="" who="" were="" subsequently="" taught="" using="" one="" of="" the="" two="" methods.="" for="" the="" 368="" students="" who="" were="" subsequently="" taught="" using="" the="" traditional="" method,="" the="" mean="" and="" standard="" deviation="" of="" the="" pretest="" scores="" were="" 14.06="" and="" 5.45="" respectively.="" for="" the="" 372="" students="" subsequently="" taught="" with="" the="" activity-oriented="" method,="" the="" mean="" and="" standard="" deviations="" of="" the="" pretest="" scores="" were="" 13.38="" and="" 5.59.="" do="" the="" data="" provide="" support="" for="" the="" conjecture="" that="" the="" mean="" pretest="" scores="" do="" not="" differ="" for="" students="" subsequently="" taught="" using="" the="" two="" methods?="" test="" using="" α="0.01." (b)="" a="" gallup="" poll="" posed="" the="" question="" “how="" would="" you="" rate="" the="" overall="" quality="" of="" the="" environment="" in="" this="" country="" today="" –="" excellent,="" good,="" fair="" or="" poor?”="" of="" 1060="" adults="" nationwide,="" 46%="" gave="" a="" rating="" of="" excellent="" or="" good.="" is="" this="" convincing="" evi-="" dence="" that="" a="" majority="" of="" the="" nation’s="" adults="" think="" the="" quality="" of="" the="" environment="" is="" fair="" or="" poor?="" test="" using="" α="0.05." 5.="" the="" output="" voltage="" for="" an="" electric="" circuit="" is="" specified="" to="" be="" 130.="" a="" sample="" of="" 40="" independent="" readings="" on="" the="" voltage="" for="" this="" circuit="" gave="" a="" sample="" mean="" 128.6="" and="" standard="" deviation="" 2.1.="" (a)="" test="" the="" hypothesis="" that="" the="" average="" output="" voltage="" is="" 130="" against="" the="" alternative="" that="" it="" is="" less="" than="" 130.="" use="" a="" test="" with="" level="" 0.05.="" (b)="" using="" the="" rejection="" region="" found="" above,="" find="" β="" for="" the="" specific="" alternative="" hy-="" pothesis="" ha="" :="" µ="129." (c)="" approximately="" what="" sample="" size="" would="" be="" necessary="" to="" achieve="" both="" α="0.01" and="" β="0.01" with="" the="" alternative="" hypothesis="" ha="" :="" µ="129?" (d)="" what="" is="" the="" 95%="" confidence="" interval="" such="" that="" the="" test="" in="" part="" (a)="" is="" equivalent="" to="" rejecting="" the="" null="" hypothesis="" if="" this="" interval="" does="" not="" include="" 130?="" 2="" 6.="" (a)="" a="" large="" sample="" α="" level="" test="" of="" h0="" :="" θ="θ0" versus="" ha="" :="" θ=""> θ0 rejects the null hypothesis if θ̂ − θ0 σθ̂ > zα Show that this is equivalent to rejecting H0 if θ0 is less than the large sample 100(1− α)% lower confidence bound for θ. (b) A large sample α level test of H0 : θ = θ0 versus Ha : θ < θ0="" rejects="" the="" null="" hypothesis="" if="" θ̂="" −="" θ0="" σθ̂="">< −zα="" show="" that="" this="" is="" equivalent="" to="" rejecting="" h0="" if="" θ0="" is="" greater="" than="" the="" large="" sample="" 100(1−="" α)%="" upper="" confidence="" bound="" for="" θ.="" (c)="" (fun="" challenge,="" 0="" points)="" suppose="" you="" have="" a="" large="" sample="" test="" h0="" :="" θ="θ0" versus="" ha="" :="" θ="">< θ0.="" if="" you="" are="" given="" the="" p-value="" associated="" with="" a="" particular="" θ̂,="" what="" upper="" or="" lower="" confidence="" bound="" would="" you="" be="" able="" to="" give="" without="" further="" calculation?="" give="" the="" exact="" bound="" and="" associated="" confidence="" coefficient.="" 7.="" (optional="" matlab="" exercises,="" 1="" point="" extra="" credit="" for="" each.)="" you="" may="" wish="" to="" use="" the="" functions="" binocdf,="" binoinv,="" norminv="" and="" normcdf="" in="" the="" following="" problems.="" (a)="" use="" matlab="" to="" do="" the="" computations="" in="" 3(b),(d)="" and="" (e)="" and="" compare="" your="" answers="" with="" the="" appropriate="" book="" table.="" (b)="" use="" matlab="" to="" find="" the="" rejection="" region="" from="" 3(f).="" do="" your="" answers="" from="" matlab="" and="" the="" book="" table="" agree?="" (c)="" write="" a="" matlab="" script="" to="" perform="" the="" calculation="" in="" problem="" 5(a).="" your="" script="" should="" have="" variables="" for="" the="" null="" hypothesis="" voltage,="" observed="" sample="" mean,="" standard="" deviation,="" sample="" size,="" and="" desired="" alpha.="" try="" to="" have="" your="" script="" output="" zα,="" the="" calculated="" value="" of="" the="" test="" statistic,="" and="" “reject”="" or="" “accept”="" as="" appropriate.="" submit="" your="" code="" and="" command="" window="" output.="" (d)="" save="" a="" new="" version="" of="" the="" above="" as="" a="" function="" taking="" the="" variables="" listed="" above="" as="" arguments.="" submit="" your="" code="" and="" command="" window="" output="" with="" the="" appro-="" priate="" variable="" values.="" (e)="" add="" a="" variable="" for="" a="" specific="" alternative="" hypothesis="" voltage="" (say="" va)="" and="" modify="" your="" function="" to="" calculate="" β="" with="" this="" alternative="" hypothesis.="" does="" your="" output="" agree="" with="" 5(b)?="" (f)="" provide="" output="" of="" your="" function="" keeping="" the="" data="" fixed="" but="" varying="" α="" and/or="" va.="" explain="" why="" you="" chose="" these="" values="" and="" what="" your="" results="" illustrate.="" 3="" 9780495614777.pdf="" moment-="" generating="" distribution="" probability="" function="" mean="" variance="" function="" uniform="" f="" (y)="1" θ2="" −="" θ1="" ;="" θ1="" ≤="" y="" ≤="" θ2="" θ1="" +="" θ2="" 2="" (θ2="" −="" θ1)2="" 12="" etθ2="" −="" etθ1="" t="" (θ2="" −="" θ1)="" normal="" f="" (y)="1" σ="" √="" 2π="" exp="" [="" −="" (="" 1="" 2σ="" 2="" )="" (y="" −="" µ)2="" ]="" µ="" σ="" 2="" +="" t="" 2σ="" 2="" 2="" )="" −∞="">< y="">< +∞="" exponential="" f="" (y)="1" β="" e−y/β="" ;="" β=""> 0 β β2 (1 − βt)−1 0 < y="">< ∞="" gamma="" f="" (y)="[" 1="" �(α)βα="" ]="" yα−1e−y/β="" ;="" αβ="" αβ2="" (1="" −="" βt)−α="" 0="">< y="">< ∞="" chi-square="" f="" (y)="(y)" (v/2)−1e−y/2="" 2v/2�(v/2)="" ;="" v="" 2v="" (1−2t)−v/2="" y2=""> 0 Beta f (y) = [ �(α + β) �(α)�(β) ] yα−1(1 − y)β−1; α α + β αβ (α + β)2(α + β + 1) does not exist in closed form 0 < y="">< 1="" continuous="" distributions="" exp="" µt="" moment-="" generating="" distribution="" probability="" function="" mean="" variance="" function="" binomial="" p(y)="(" n="" y="" )="" py(1="" −="" p)n−y="" ;="" np="" np(1="" −="" p)="" [pet="" +="" (1="" −="" p)]n="" y="0," 1,="" .="" .="" .="" ,="" n="" geometric="" p(y)="p(1" −="" p)y−1;="" 1="" p="" 1="" −="" p="" p2="" pet="" 1="" −="" (1="" −="" p)et="" y="1," 2,="" .="" .="" .="" hypergeometric="" p(y)="(" r="" y="" )="" (="" n−r="" n−y="" )="" (="" n="" n="" )="" ;="" nr="" n="" n="" (="" r="" n="" )(="" n="" −="" r="" n="" )(="" n="" −="" n="" n="" −="" 1="" )="" y="0," 1,="" .="" .="" .="" ,="" n="" if="" n="" ≤="" r="" ,="" y="0," 1,="" .="" .="" .="" ,="" r="" if="" n=""> r Poisson p(y) = λ ye−λ y! ; λ λ exp[λ(et − 1)] y = 0, 1, 2, . . . Negative binomial p(y) = ( y−1 r−1 ) pr (1 − p)y−r ; r p r(1 − p) p2 [ pet 1 − (1 − p)et ]r y = r, r + 1, . . . Discrete Distributions M A T H E M A T I C A L S T A T I S T I C S W I T H A P P L I C A T I O N S This page intentionally left blank S E V E N T H E D I T I O N Mathematical Statistics with Applications Dennis D. Wackerly University of Florida William Mendenhall III University of Florida, Emeritus Richard L. Scheaffer University of Florida, Emeritus Australia • Brazil • Canada • Mexico • Singapore • Spain United Kingdom • United States Mathematical Statistics with Applications, Seventh Edition Dennis D. Wackerly, William Mendenhall III, Richard L. 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