Homework 5 Instructions • No late homework accepted under any circumstances. • Must hand in in person to Prof. Sanchez at the beginning of lecture. No email, mailboxes, fax or other way of turning it in will be allowed. Please, do not give it to the TA. If you need to turn in your homework early, please contact Prof. Sanchez and make arrangements with her. • Write your name, ID, Hwk number, date and your section number, day and TA's name on the upper right corner. • Must show work for full credit, or explain depending on the problem. No partial credit is possible if you don't show work • Must answer problems in the order given. • Staple your homework and write in dark pencil or pen. The homework must be clearly written. If it is not possible to read your hand writing, your homework will get 0 points. • Honor code applies. See the course syllabus. Indication that honor code not followed will result in 0 points in the assignment. • There will be no answer key given for this homework before it is due. DO NOT USE CONFIDENCE INTERVALS TO DO HYPOTHESIS TESTING IN THIS HOMEWORK (THIS NOTE IS FOR THOSE THAT HAVE ALREADY TAKEN STATS BEFORE). YOU MUST SHOW THAT YOU KNOW ALL THE JARGON AND CONCEPTS OF HYPOTHESIS TESTING. 1.- Let x1, x2,....., x25 be a sample from a normal distribution with unknown mean µ and s2 =2.3. (a) Find the optimal rejection rule in terms of a pivot and the rejection region at level a=0.01 for the test H0 :µ = 0 Ha :µ = 2 Use the likelihood ratio test approach and simplify the problem to a point where the pivot is familiar. When done with that continue work until you set the decision in terms of a value of the likelihood ration. Show work. 2.- Let be a random sample from an exponential distribution with the density function Derive a likelihood ratio test of and show that the rejection region is of the form 3.- Show that the standard Cauchy distribution and the t distribution with 1 degrees of freedom are the same. 4.-A claim was made that 60% of the adult population thinks that there is too much violence on television. You thought that figure is too high and were asked to set up null and alternative hypotheses to evaluate the claim. Set the hypotheses and state what is the correct test statistic to use to do the test. Include as much as you can that is known in the expression you use for the test. 5.-.- To test the claim that the average home in a certain town is within 5.5 miles of the nearest fire department, and insurance company measured the distances from 25 randomly selected homes to the nearest fire department and found y = 5.8 miles and s = 2.4 miles. Determine what the insurance compan found out with a test of significance. What assumption did you make about the underlying parent distribution? 6.- Consider the problem of using a random sample X1, ....., Xn, with n > 2 from a Poisson distribution with mean µ to test the hypothesis Ho : µ = µo against the alternative hypothesis H1 : µ = µ1 < µo.="" (a)="" show="" that="" the="" best="" critical="" region="" for="" this="" test="" is="" the="" left="" tail="" of="" the="" sampling="" distribution="" of="" xi="" i="1" n="" show="" how="" and="" why="" you="" pass="" from="" each="" step="" of="" your="" computation="" to="" another="" step="" and="" interpret="" the="" result="" (i.e,="" what="" does="" it="" mean="" that="" that="" is="" the="" critical="" region="" in="" the="" context="" of="" the="" test?).="" (b)="" what="" is="" the="" sampling="" distribution="" of="" xi="" i="1" n="" give="" its="" name,="" formula,="" and="" its="" expected="" value="" and="" variance.="" 7.-="" suppose="" they="" observe="" the="" lifetimes="" of="" 40="" electrical="" parts="" and="" it="" is="" conjectured="" that="" these="" observations="" might="" be="" exponentially="" distributed="" with="" mean="" lifetime="" =="" 100.="" suppose,="" in="" a="" particular="" application,="" that="" the="" parts="" will="" be="" unsuitable="" if="" the="" mean="" is="" less="" than="" 100.="" the="" sample="" mean="" is="" x="93.1." carry="" out="" a="" size="" a="0.05" significance="" test="" of="" h0="" :?="100" ha="" :?="">< 100="" using="" as="" pivot="" for="" the="" test="" 2?="" xi="" i="1" n="" .="" show="" work.="" 8.-="" the="" chamber="" of="" commerce="" of="" a="" particular="" city="" claims="" that="" the="" mean="" carbon="" dioxide="" level="" of="" air="" pollution="" is="" no="" greater="" than="" 4.9="" parts="" per="" million="" (ppm).="" a="" random="" sample="" of="" 16="" readings="" resulted="" in="" y="5.6" ppm="" and="" s="2.1" ppm.="" if="" we="" assume="" that="" the="" carbon="" dioxide="" level="" is="" normally="" distributed,="" is="" there="" sufficient="" evidence="" against="" the="" chamber="" of="" commerce’s="" claim?="" state="" null="" and="" alternative="" hypotheses="" to="" evaluate="" the="" conjecture.="" calculate="" the="" t="" statistic="" and="" its="" p-value="" and="" state="" your="" conclusion="" to="" the="" test.="" why="" is="" it="" important="" to="" assume="" that="" the="" parent="" distribution="" is="" normal?="" 9.-="" .-="" according="" to="" a="" recent="" report,="" after="" 5="" years="" on="" the="" job,="" american="" workers="" get="" an="" average="" of="" 24="" days="" of="" paid="" holidays="" and="" vacation="" leave="" each="" year.="" the="" number="" of="" days="" of="" paid="" holidays="" and="" vacation="" leave="" taken="" by="" a="" random="" sample="" of="" 35="" workers="" in="" the="" american="" textile="" industry="" were="" observed.="" the="" average="" number="" of="" days="" of="" paid="" holidays="" and="" vacation="" leave="" in="" the="" sample="" was="" 20.51="" with="" a="" sample="" standard="" deviation="" of="" 9.13="" days="" .="" is="" there="" evidence="" in="" the="" data="" that="" the="" mean="" vacation="" leave="" for="" textile="" workers="" is="" no="" different="" from="" the="" mean="" of="" all="" american="" workers?="" 10.-many="" people="" have="" trouble="" programming="" their="" vcrs,="" so="" a="" company="" has="" developed="" what="" it="" hopes="" will="" be="" easier="" instructions.="" the="" goal="" is="" to="" have="" at="" least="" 90%="" of="" customers="" succeed.="" the="" company="" tests="" the="" new="" system="" on="" 200="" people,="" of="" whom="" 188="" were="" successful.="" is="" this="" strong="" evidence="" that="" the="" new="" system="" fails="" to="" meet="" the="" company’s="" goal?="" a="" student’s="" test="" of="" this="" hypothesis="" is="" shown="" below.="" how="" many="" mistakes="" can="" you="" find?="" cross="" out="" what="" is="" wrong="" and="" write="" the="" correct="" thing="" next="" h0="" :="" p="" =="" 0.9="" ha="" :="" p="" 0.9="" 0.90(200)="">10, 0.1(200) >10 188 200 = 0.94;SD p ? ? ? ? ? ? ? = (0.94)(0.06) 200 = 0.017 z = 0.94 - 0.9 0.017 = 2.35 P - value = P(z > 2.35) = 0.009 A presidential aid said that at least half of the nation agreed with the U.S. invasion of Grenada in 1984. However, a Roper poll of 2,000 Americans found that only 34% supported the invasion. Did the 34% happen simply by chance, or is there statistical evidence that the aide was in error?