Subject: Advanced Statistics and quantitative Methods. O
ne question, 240 min to do it,Chapter 11 Two-Sample Tests of Hypothesis Chapter 11 McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. * 11-* GOALS Conduct a test of a hypothesis about the difference between two independent population means. Conduct a test of a hypothesis about the difference between two population proportions. Conduct a test of a hypothesis about the mean difference between paired or dependent observations. Understand the difference between dependent and independent samples. * 11-* Comparing two populations – Some Examples Is there a difference in the mean value of residential real estate sold by male agents and female agents in south Florida? Is there a difference in the mean number of defects produced on the day and the afternoon shifts at Kimble Products? Is there a difference in the mean number of days absent between young workers (under 21 years of age) and older workers (more than 60 years of age) in the fast-food industry? Is there is a difference in the proportion of Ohio State University graduates and University of Cincinnati graduates who pass the state Certified Public Accountant Examination on their first attempt? Is there an increase in the production rate if music is piped into the production area? * 11-* Comparing Two Population Means No assumptions about the shape of the populations are required. The samples are from independent populations. The formula for computing the value of z is: * 11-* Comparing Two Population Means - Example The U-Scan facility was recently installed at the Byrne Road Food-Town location. The store manager would like to know if the mean checkout time using the standard checkout method is longer than using the U-Scan. She gathered the following sample information. The time is measured from when the customer enters the line until their bags are in the cart. Hence the time includes both waiting in line and checking out. * 11-* EXAMPLE 1 continued Step 1: State the null and alternate hypotheses. (keyword: “longer than”) H0: µS ≤ µU H1: µS > µU Step 2: Select the level of significance. The .01 significance level is stated in the problem. Step 3: Determine the appropriate test statistic. Because both population standard deviations are known, we can use z-distribution as the test statistic. * 11-* Example 1 continued Step 4: Formulate a decision rule. Reject H0 ifZ > Z Z > 2.33 * 11-* Example 1 continued Step 5: Compute the value of z and make a decision The computed value of 3.13 is larger than the critical value of 2.33. Our decision is to reject the null hypothesis. The difference of .20 minutes between the mean checkout time using the standard method is too large to have occurred by chance. We conclude the U-Scan method is faster. * 11-* Two-Sample Tests about Proportions EXAMPLES The vice president of human resources wishes to know whether there is a difference in the proportion of hourly employees who miss more than 5 days of work per year at the Atlanta and the Houston plants. General Motors is considering a new design for the Pontiac Grand Am. The design is shown to a group of potential buyers under 30 years of age and another group over 60 years of age. Pontiac wishes to know whether there is a difference in the proportion of the two groups who like the new design. A consultant to the airline industry is investigating the fear of flying among adults. Specifically, the company wishes to know whether there is a difference in the proportion of men versus women who are fearful of flying. * 11-* Two Sample Tests of Proportions We investigate whether two samples came from populations with an equal proportion of successes. The two samples are pooled using the following formula. * 11-* Two Sample Tests of Proportions continued The value of the test statistic is computed from the following formula. * 11-* Manelli Perfume Company recently developed a new fragrance that it plans to market under the name Heavenly. A number of market studies indicate that Heavenly has very good market potential. The Sales Department at Manelli is particularly interested in whether there is a difference in the proportions of younger and older women who would purchase Heavenly if it were marketed. Samples are collected from each of these independent groups. Each sampled woman was asked to smell Heavenly and indicate whether she likes the fragrance well enough to purchase a bottle. Two Sample Tests of Proportions - Example * 11-* Step 1: State the null and alternate hypotheses. (keyword: “there is a difference”) H0: 1 = 2 H1: 1 ≠ 2 Step 2: Select the level of significance. The .05 significance level is stated in the problem. Step 3: Determine the appropriate test statistic. We will use the z-distribution Two Sample Tests of Proportions - Example * 11-* Step 4: Formulate the decision rule. Reject H0 ifZ > Z/2 or Z < -="" z/2="" z=""> Z.05/2 or Z < -="" z.05/2="" z=""> 1.96 or Z < -1.96="" two="" sample="" tests="" of="" proportions="" -="" example="" *="" 11-*="" step="" 5:="" select="" a="" sample="" and="" make="" a="" decision="" the="" computed="" value="" of="" -2.21="" is="" in="" the="" area="" of="" rejection.="" therefore,="" the="" null="" hypothesis="" is="" rejected="" at="" the="" .05="" significance="" level.="" to="" put="" it="" another="" way,="" we="" reject="" the="" null="" hypothesis="" that="" the="" proportion="" of="" young="" women="" who="" would="" purchase="" heavenly="" is="" equal="" to="" the="" proportion="" of="" older="" women="" who="" would="" purchase="" heavenly.="" two="" sample="" tests="" of="" proportions="" -="" example="" let="" p1="young" women="" p2="older" women="" *="" 11-*="" two="" sample="" tests="" of="" proportions="" –="" example="" (minitab="" solution)="" *="" 11-*="" comparing="" population="" means="" with="" unknown="" population="" standard="" deviations="" (the="" pooled="" t-test)="" the="" t="" distribution="" is="" used="" as="" the="" test="" statistic="" if="" one="" or="" more="" of="" the="" samples="" have="" less="" than="" 30="" observations.="" the="" required="" assumptions="" are:="" 1.="" both="" populations="" must="" follow="" the="" normal="" distribution.="" 2.="" the="" populations="" must="" have="" equal="" standard="" deviations.="" 3.="" the="" samples="" are="" from="" independent="" populations.="" *="" 11-*="" small="" sample="" test="" of="" means="" continued="" finding="" the="" value="" of="" the="" test="" statistic="" requires="" two="" steps.="" pool="" the="" sample="" standard="" deviations.="" use="" the="" pooled="" standard="" deviation="" in="" the="" formula.="" *="" 11-*="" owens="" lawn="" care,="" inc.,="" manufactures="" and="" assembles="" lawnmowers="" that="" are="" shipped="" to="" dealers="" throughout="" the="" united="" states="" and="" canada.="" two="" different="" procedures="" have="" been="" proposed="" for="" mounting="" the="" engine="" on="" the="" frame="" of="" the="" lawnmower.="" the="" question="" is:="" is="" there="" a="" difference="" in="" the="" mean="" time="" to="" mount="" the="" engines="" on="" the="" frames="" of="" the="" lawnmowers?="" the="" first="" procedure="" was="" developed="" by="" longtime="" owens="" employee="" herb="" welles="" (designated="" as="" procedure="" 1),="" and="" the="" other="" procedure="" was="" developed="" by="" owens="" vice="" president="" of="" engineering="" william="" atkins="" (designated="" as="" procedure="" 2).="" to="" evaluate="" the="" two="" methods,="" it="" was="" decided="" to="" conduct="" a="" time="" and="" motion="" study.="" a="" sample="" of="" five="" employees="" was="" timed="" using="" the="" welles="" method="" and="" six="" using="" the="" atkins="" method.="" the="" results,="" in="" minutes,="" are="" shown="" on="" the="" right.="" is="" there="" a="" difference="" in="" the="" mean="" mounting="" times?="" use="" the="" .10="" significance="" level.="" comparing="" population="" means="" with="" unknown="" population="" standard="" deviations="" (the="" pooled="" t-test)="" *="" 11-*="" step="" 1:="" state="" the="" null="" and="" alternate="" hypotheses.="" (keyword:="" “is="" there="" a="" difference”)="" h0:="" µ1="µ2" h1:="" µ1="" ≠="" µ2="" step="" 2:="" state="" the="" level="" of="" significance.="" the="" 0.10="" significance="" level="" is="" stated="" in="" the="" problem.="" step="" 3:="" find="" the="" appropriate="" test="" statistic.="" because="" the="" population="" standard="" deviations="" are="" not="" known="" but="" are="" assumed="" to="" be="" equal,="" we="" use="" the="" pooled="" t-test.="" comparing="" population="" means="" with="" unknown="" population="" standard="" deviations="" (the="" pooled="" t-test)="" -="" example="" *="" 11-*="" step="" 4:="" state="" the="" decision="" rule.="" reject="" h0="" if="" t=""> t/2,n1+n2-2 or t < -="" t/2,="" n1+n2-2="" t=""> t.05,9 or t < -="" t.05,9="" t=""> 1.833 or t < -="" 1.833="" comparing="" population="" means="" with="" unknown="" population="" standard="" deviations="" (the="" pooled="" t-test)="" -="" example="" *="" 11-*="" step="" 5:="" compute="" the="" value="" of="" t="" and="" make="" a="" decision="" (a)="" calculate="" the="" sample="" standard="" deviations="" comparing="" population="" means="" with="" unknown="" population="" standard="" deviations="" (the="" pooled="" t-test)="" -="" example="" (b)="" calculate="" the="" pooled="" sample="" standard="" deviation="" *="" 11-*="" step="" 5:="" compute="" the="" value="" of="" t="" and="" make="" a="" decision="" the="" decision="" is="" not="" to="" reject="" the="" null="" hypothesis,="" because="" -0.662="" falls="" in="" the="" region="" between="" -1.833="" and="" 1.833.="" we="" conclude="" that="" there="" is="" no="" difference="" in="" the="" mean="" times="" to="" mount="" the="" engine="" on="" the="" frame="" using="" the="" two="" methods.="" comparing="" population="" means="" with="" unknown="" population="" standard="" deviations="" (the="" pooled="" t-test)="" -="" example="" (c)="" determine="" the="" value="" of="" t="" -0.662="" *="" 11-*="" comparing="" population="" means="" with="" unknown="" population="" standard="" deviations="" (the="" pooled="" t-test)="" -="" example="" *="" 11-*="" comparing="" population="" means="" with="" unequal="" population="" standard="" deviations="" compute="" the="" t-statistic="" shown="" on="" the="" right="" if="" it="" is="" not="" reasonable="" to="" assume="" the="" population="" standard="" deviations="" are="" equal.="" the="" sample="" standard="" deviations="" s1="" and="" s2="" are="" used="" in="" place="" of="" the="" respective="" population="" standard="" deviations.="" in="" addition,="" the="" degrees="" of="" freedom="" are="" adjusted="" downward="" by="" a="" rather="" complex="" approximation="" formula.="" the="" effect="" is="" to="" reduce="" the="" number="" of="" degrees="" of="" freedom="" in="" the="" test,="" which="" will="" require="" a="" larger="" value="" of="" the="" test="" statistic="" to="" reject="" the="" null="" hypothesis.="" *="" 11-*="" comparing="" population="" means="" with="" unequal="" population="" standard="" deviations="" -="" example="" personnel="" in="" a="" consumer="" testing="" laboratory="" are="" evaluating="" the="" absorbency="" of="" paper="" towels.="" they="" wish="" to="" compare="" a="" set="" of="" store="" brand="" towels="" to="" a="" similar="" group="" of="" name="" brand="" ones.="" for="" each="" brand="" they="" dip="" a="" ply="" of="" the="" paper="" into="" a="" tub="" of="" fluid,="" allow="" the="" paper="" to="" drain="" back="" into="" the="" vat="" for="" two="" minutes,="" and="" then="" evaluate="" the="" amount="" of="" liquid="" the="" paper="" has="" taken="" up="" from="" the="" vat.="" a="" random="" sample="" of="" 9="" store="" brand="" paper="" towels="" absorbed="" the="" following="" amounts="" of="" liquid="" in="" milliliters.="" 8="" 8="" 3="" 1="" 9="" 7="" 5="" 5="" 12="" an="" independent="" random="" sample="" of="" 12="" name="" brand="" towels="" absorbed="" the="" following="" amounts="" of="" liquid="" in="" milliliters:="" 12="" 11="" 10="" 6="" 8="" 9="" 9="" 10="" 11="" 9="" 8="" 10="" use="" the="" .10="" significance="" level="" and="" test="" if="" there="" is="" a="" difference="" in="" the="" mean="" amount="" of="" liquid="" absorbed="" by="" the="" two="" types="" of="" paper="" towels.="" *="" 11-*="" comparing="" population="" means="" with="" unequal="" population="" standard="" deviations="" -="" example="" the="" following="" dot="" plot="" provided="" by="" minitab="" shows="" the="" variances="" to="" be="" unequal.="" the="" following="" output="" provided="" by="" minitab="" shows="" the="" descriptive="" statistics="" *="" 11-*="" step="" 1:="" state="" the="" null="" and="" alternate="" hypotheses.="" h0:="" 1="2" h1:="" 1="" ≠="" 2="" step="" 2:="" state="" the="" level="" of="" significance.="" the="" .10="" significance="" level="" is="" stated="" in="" the="" problem.="" step="" 3:="" find="" the="" appropriate="" test="" statistic.="" we="" will="" use="" unequal="" variances="" t-test="" comparing="" population="" means="" with="" unequal="" population="" standard="" deviations="" -="" example="" *="" 11-*="" step="" 4:="" state="" the="" decision="" rule.="" reject="" h0="" if="" t=""> t/2d.f. or t < -="" t/2,d.f.="" t=""> t.05,10 or t < -="" t.05,="" 10="" t=""> 1.812 or t < -1.812="" step="" 5:="" compute="" the="" value="" of="" t="" and="" make="" a="" decision="" the="" computed="" value="" of="" t="" is="" less="" than="" the="" lower="" critical="" value,="" so="" our="" decision="" is="" to="" reject="" the="" null="" hypothesis.="" we="" conclude="" that="" the="" mean="" absorption="" rate="" for="" the="" two="" towels="" is="" not="" the="" same.="" comparing="" population="" means="" with="" unequal="" population="" standard="" deviations="" -="" example="" *="" 11-*="" minitab="" *="" 11-*="" two-sample="" tests="" of="" hypothesis:="" dependent="" samples="" dependent="" samples="" are="" samples="" that="" are="" paired="" or="" related="" in="" some="" fashion.="" for="" example:="" if="" you="" wished="" to="" buy="" a="" car="" you="" would="" look="" at="" the="" same="" car="" at="" two="" (or="" more)="" different="" dealerships="" and="" compare="" the="" prices.="" if="" you="" wished="" to="" measure="" the="" effectiveness="" of="" a="" new="" diet="" you="" would="" weigh="" the="" dieters="" at="" the="" start="" and="" at="" the="" finish="" of="" the="" program.="" *="" 11-*="" hypothesis="" testing="" involving="" paired="" observations="" use="" the="" following="" test="" when="" the="" samples="" are="" dependent:="" where="" is="" the="" mean="" of="" the="" differences="" sd="" is="" the="" standard="" deviation="" of="" the="" differences="" n="" is="" the="" number="" of="" pairs="" (differences)="" *="" 6.unknown="" 11-*="" nickel="" savings="" and="" loan="" wishes="" to="" compare="" the="" two="" companies="" it="" uses="" to="" appraise="" the="" value="" of="" residential="" homes.="" nickel="" savings="" selected="" a="" sample="" of="" 10="" residential="" properties="" and="" scheduled="" both="" firms="" for="" an="" appraisal.="" the="" results,="" reported="" in="" $000,="" are="" shown="" on="" the="" table="" (right).="" at="" the="" .05="" significance="" level,="" can="" we="" conclude="" there="" is="" a="" difference="" in="" the="" mean="" appraised="" values="" of="" the="" homes?="" hypothesis="" testing="" involving="" paired="" observations="" -="" example="" *="" 11-*="" step="" 1:="" state="" the="" null="" and="" alternate="" hypotheses.="" h0:="" d="0" h1:="" d="" ≠="" 0="" step="" 2:="" state="" the="" level="" of="" significance.="" the="" .05="" significance="" level="" is="" stated="" in="" the="" problem.="" step="" 3:="" find="" the="" appropriate="" test="" statistic.="" we="" will="" use="" the="" t-test="" hypothesis="" testing="" involving="" paired="" observations="" -="" example="" *="" 11-*="" step="" 4:="" state="" the="" decision="" rule.="" reject="" h0="" if="" t=""> t/2, n-1 or t