Confidence Intervals and Sample Size


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1 8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGrawHill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 71 Cofidece Itervals for the Mea Whe Is Kow 7 Cofidece Itervals for the Mea Whe Is Ukow 73 for Proportios 74 Cofidece Itervals for Variaces ad Stadard Deviatios CHAPTER 7 Learig Objectives 1 Fid the cofidece iterval for the mea whe is kow. Determie the miimum sample size for fidig a cofidece iterval for the mea. 3 Fid the cofidece iterval for the mea whe is ukow. 4 Fid the cofidece iterval for a proportio. 5 Determie the miimum sample size for fidig a cofidece iterval for a proportio. 6 Fid a cofidece iterval for a variace ad a stadard deviatio. Copyright 015 The McGrawHill Educatio. Compaies, Permissio Ic. Permissio required required for reproductio for reproductio or display. or display Cofidece Itervals for the Mea Whe Is Kow A poit estimate is a specific umerical value estimate of a parameter. The best poit estimate of the populatio mea µ is the sample mea X. Three Properties of a Good Estimator 1. The estimator should be a ubiased estimator. That is, the expected value or the mea of the estimates obtaied from samples of a give size is equal to the parameter beig estimated. Three Properties of a Good Estimator. The estimator should be cosistet. For a cosistet estimator, as sample size icreases, the value of the estimator approaches the value of the parameter estimated
2 8/7/015 Three Properties of a Good Estimator 3. The estimator should be a relatively efficiet estimator; that is, of all the statistics that ca be used to estimate a parameter, the relatively efficiet estimator has the smallest variace. Cofidece Itervals for the Mea Whe Is Kow A iterval estimate of a parameter is a iterval or a rage of values used to estimate the parameter. This estimate may or may ot cotai the value of the parameter beig estimated. Cofidece Level of a Iterval Estimate The cofidece level of a iterval estimate of a parameter is the probability that the iterval estimate will cotai the parameter, assumig that a large umber of samples are selected ad that the estimatio process o the same parameter is repeated Cofidece Iterval A cofidece iterval is a specific iterval estimate of a parameter determied by usig data obtaied from a sample ad by usig the specific cofidece level of the estimate. Formula for the Cofidece Iterval of the Mea for a Specific a X za/ X za/ For a 90% cofidece iterval: z a / 1.65 For a 95% cofidece iterval: z a / 1.96 For a 99% cofidece iterval: z a /.58 Margi of error The margi of error, also called the maximum error of the estimate, is the maximum likely differece betwee the poit estimate of a parameter ad the actual value of the parameter Bluma, 1
3 8/7/015 95% Cofidece Iterval of the Mea Cofidece Iterval for a Mea Roudig Rule Whe you are computig a cofidece iterval for a populatio mea by usig raw data, roud off to oe more decimal place tha the umber of decimal places i the origial data. Whe you are computig a cofidece iterval for a populatio mea by usig a sample mea ad a stadard deviatio, roud off to the same umber of decimal places as give for the mea. Sectio 71 Example 71 Page # Example 71: Days to Sell a Aveo A researcher wishes to estimate the umber of days it takes a automobile dealer to sell a Chevrolet Aveo. A sample of 50 cars had a mea time o the dealer s lot of 54 days. Assume the populatio stadard deviatio to be 6.0 days. Fid the best poit estimate of the populatio mea ad the 95% cofidece iterval of the populatio mea. The best poit estimate of the mea is 54 days. X 54, 6.0, 50,95% z 1.96 X z X z a a Example 71: Days to Sell a Aveo X 54, 6.0, 50,95% z 1.96 X z X z a a Oe ca say with 95% cofidece that the iterval betwee 5 ad 56 days cotais the populatio mea, based o a sample of 50 automobiles. Sectio 71 Example 7 Page #
4 8/7/015 Example 7: Number of Customers A large departmet store foud that it averages 36 customers per hour. Assume that the stadard deviatio is 9.6 ad a radom sample of 40 hours was used to determie the average. Fid the 99% cofidece iterval of the populatio mea. Example 7: Number of Customers 95% Cofidece Iterval of the Mea Hece, oe ca be 99% cofidet (roudig values) that the mea umber of customers that the store averages is betwee 350 ad 374 customers per hour % Cofidece Iterval of the Mea Fidig z a for 98% CL. Sectio 71 z a.33 Example 73 Page #
5 8/7/015 Example 73: Credit Uio Assets The followig data represet a sample of the assets (i millios of dollars) of 30 credit uios i southwester Pesylvaia. Fid the 90% cofidece iterval of the mea Example 73: Credit Uio Assets Step 1: Fid the mea ad stadard deviatio. Usig techology, we fid X = ad s = Assume Step : Fid α/. 90% CL α/ = Step 3: Fid z α/. 90% CL α/ = 0.05 z.05 = 1.65 Example 73: Credit Uio Assets Step 4: Substitute i the formula. X z X z a a Oe ca be 90% cofidet that the populatio mea of the assets of all credit uios is betwee $6.75 millio ad $ millio, based o a sample of 30 credit uios Commet to computers ad calculator users This chapter ad subsequet chapters iclude examples usig raw data. If you are usig computer or calculator programs to fid the solutios, the aswers you get may vary somewhat from the oes give i the textbook. This is so because computers ad calculators do ot roud the aswers i the itermediate steps ad ca use 1 or more decimal places for computatio. Also, they use more exact values tha those give i the tables i the back of this book. These discrepacies are part ad parcel of statistics. Formula for Miimum Needed for a Iterval Estimate of the Populatio Mea z a where E is the margi of error. If ecessary, roud the aswer up to obtai a whole umber. That is, if there is ay fractio or decimal portio i the aswer, use the ext whole umber for sample size. E Sectio 71 Example 74 Page #
6 8/7/015 Example 74: Depth of a River A scietist wishes to estimate the average depth of a river. He wats to be 99% cofidet that the estimate is accurate withi feet. From a previous study, the stadard deviatio of the depths measured was 4.33 feet. How large a sample is required? a = 0.01(or , za/ =,58 ad E = z a E Therefore, to be 99% cofidet that the estimate is withi feet of the true mea depth, the scietist eeds at least a sample of 3 measuremets Cofidece Itervals for the Mea Whe Is Ukow The value of, whe it is ot kow, must be estimated by usig s, the stadard deviatio of the sample. Whe s is used, especially whe the sample size is small (less tha 30), critical values greater tha the values for z a are used i cofidece itervals i order to keep the iterval at a give level, such as the 95%. These values are take from the Studet t distributio, most ofte called the t distributio. Characteristics of the t Distributio The t distributio is similar to the stadard ormal distributio i these ways: 1. It is bellshaped.. It is symmetric about the mea. 3. The mea, media, ad mode are equal to 0 ad are located at the ceter of the distributio. 4. The curve ever touches the x axis Characteristics of the t Distributio The t distributio differs from the stadard ormal distributio i the followig ways: 1. The variace is greater tha 1.. The t distributio is actually a family of curves based o the cocept of degrees of freedom, which is related to sample size. 3. As the sample size icreases, the t distributio approaches the stadard ormal distributio. Degrees of Freedom The symbol d.f. will be used for degrees of freedom. The degrees of freedom for a cofidece iterval for the mea are foud by subtractig 1 from the sample size. That is, d.f. = 1. Note: For some statistical tests used later i this book, the degrees of freedom are ot equal to 1. Sectio 7 Example 75 Page #
7 8/7/015 Example 75: Usig Table F Fid the t α/ value for a 95% cofidece iterval whe the sample size is. Degrees of freedom are d.f. = 1. Formula for a Specific Cofidece Iterval for the Mea Whe Is Ukow ad < 30 s s X ta X ta The degrees of freedom are 1. Sectio 7 Example 76 Page # Example 76: Ifat Growth A radom sample of 10 childre foud that their average growth for the first year was 9.8 iches. Assume the variable is ormally distributed ad the sample stadard deviatio is 0.96 ich. Fid the 95% cofidece iterval of the populatio mea for growth durig the first year. Therefore, oe ca be 95% cofidet that the populatio mea of the firstyear growth is betwee 9.11 ad iches Sectio 7 Example 77 Page #377 Example 77: Home Fires by Cadles The data represet a sample of the umber of home fires started by cadles for the past several years. Fid the 99% cofidece iterval for the mea umber of home fires started by cadles each year Step 1: Fid the mea ad stadard deviatio. The mea is X = ad stadard deviatio s = Step : Fid t α/ i Table F. The cofidece level is 99%, ad the degrees of freedom d.f. = 6 t.005 =
8 8/7/015 Example 77: Home Fires by Cadles Step 3: Substitute i the formula. s s X ta X ta Oe ca be 99% cofidet that the populatio mea umber of home fires started by cadles each year is betwee ad 997.6, based o a sample of home fires occurrig over a period of 7 years. 7.3 for Proportios p = populatio proportio ˆp (read p hat ) = sample proportio For a sample proportio, X X pˆ ad qˆ or qˆ 1 pˆ where X = umber of sample uits that possess the characteristics of iterest ad = sample size. Sectio 73 Example 78 Page # Example 78: Drivig to Work A radom sample of 00 workers foud that 18 drove to work aloe. Fid ˆp ad ˆq where ˆp is the proportio of workers who drove to work aloe. Sice X = 18 ad = % of the people i the survey drive to work aloe, ad 36% drive with others. Formula for a Specific Cofidece Iterval for a Proportio pq ˆ ˆ pˆ z p pˆ z whe p 5 ad q 5. a a pq ˆ ˆ Roudig Rule: Roud off to three decimal places. Sectio 73 Example 79 Page #
9 8/7/015 Example 79: Coverig College Costs A survey coducted by Sallie Mae ad Gallup of 1404 respodets foud that 33 studets paid for their educatio by studet loas. Fid the 90% cofidece of the true proportio of studets who paid for their educatio by studet loas. Example 79: Coverig College Costs Sice α = = 0.10, z α/ = Example 79: Coverig College Costs You ca be 90% cofidet that the percetage of studets who pay for their college educatio by studet loas is betwee 1.1 ad 4.9%. Determie ˆp ad ˆq Determie the critical value Sectio 73 Example 710 Page #384 Example 710: Law weeds A survey of 1898 people foud that 45% of the adults said that dadelios were the toughest weeds to cotrol i their yards. Fid the 95% cofidece iterval of the true proportio who said that dadelios were the toughest weeds to cotrol i their yards You ca say with 95% cofidece that the percetage of adults who cosider dadelios the toughest weeds to cotrol to be betwee 4.8% ad 47.% Formula for Miimum Needed for Iterval Estimate of a Populatio Proportio z ˆˆ a pq E If ecessary, roud up to the ext whole umber
10 8/7/015 Sectio 73 Example 711 Page #386 Example 711: Home Computers A researcher wishes to estimate, with 95% cofidece, the proportio of people who ow a home computer. A previous study shows that 40% of those iterviewed had a computer at home. The researcher wishes to be accurate withi % of the true proportio. Fid the miimum sample size ecessary. z 1.96 ˆˆ a pq E 0.0 The researcher should iterview a sample of at least 305 people. Sectio 73 Example 71 Page # Example 71: Home Computers I Example 711 assume that o pervious study was doe. Fid the miimum sample size ecessary to be accurate withi % of the true populatio. Here ˆp = 0.5 ad ˆq = people must be iterviewed whe ˆp is ukow. This is 96 more people eeded if ˆp is kow 74 Cofidece Itervals for Variaces ad Stadard Deviatios Whe products that fit together (such as pipes) are maufactured, it is importat to keep the variatios of the diameters of the products as small as possible; otherwise, they will ot fit together properly ad will have to be scrapped. I the maufacture of medicies, the variace ad stadard deviatio of the medicatio i the pills play a importat role i makig sure patiets receive the proper dosage. For these reasos, cofidece itervals for variaces ad stadard deviatios are ecessary. ChiSquare Distributios The chisquare distributio must be used to calculate cofidece itervals for variaces ad stadard deviatios. The chisquare variable is similar to the t variable i that its distributio is a family of curves based o the umber of degrees of freedom. The symbol for chisquare is (Greek letter chi, proouced ki ). A chisquare variable caot be egative, ad the distributios are skewed to the right
11 8/7/015 ChiSquare Distributios At about 100 degrees of freedom, the chisquare distributio becomes somewhat symmetric. The area uder each chisquare distributio is equal to 1.00, or 100%. Example 713: Usig Table G Use the 0.95 ad 0.05 colums ad the row correspodig to 4 d.f. i Table G. Sectio 74 Example 713 Page #393 The value is ; the value is right left Example 713: Usig Table G Fid the values for right ad left for a 90% cofidece iterval whe = 5. To fid right, subtract = Divide by to get To fid left, subtract to get Formula for the Cofidece Iterval for a Variace 1 s 1 s, d.f. = 1 right Formula for the Cofidece Iterval for a Stadard Deviatio 1 s 1 s, d.f. = 1 right left left Cofidece Iterval for a Variace or Stadard Deviatio Roudig Rule Whe you are computig a cofidece iterval for a populatio variace or stadard deviatio by usig raw data, roud off to oe more decimal places tha the umber of decimal places i the origial data. Whe you are computig a cofidece iterval for a populatio variace or stadard deviatio by usig a sample variace or stadard deviatio, roud off to the same umber of decimal places as give for the sample variace or stadard deviatio
12 8/7/015 Sectio 74 Example 714 Page #40 Example 714: Nicotie Cotet Fid the 95% cofidece iterval for the variace ad stadard deviatio of the icotie cotet of cigarettes maufactured if a sample of 0 cigarettes has a stadard deviatio of 1.6 milligrams. To fid right, subtract = Divide by to get To fid left, subtract to get I Table G, the 0.05 ad colums with the d.f. 19 row yield values of 3.85 ad 8.907, respectively. Example 714: Nicotie Cotet 1 s 1 s right left You ca be 95% cofidet that the true variace for the icotie cotet is betwee 1.5 ad 5.5 milligrams You ca be 95% cofidet that the true stadard deviatio is betwee 1. ad.3 milligrams Sectio 74 Example 715 Page #395 Example 715: Named Storms Fid the 90% cofidece iterval for the variace ad stadard deviatio for the umber of amed storms per year i the Atlatic basi. A radom sample of 10 years has bee used. Assume the distributio is approximately ormal Usig techology, we fid the variace of the data is s = I Table G, the 0.05 ad 0.95 colums with the d.f. 9 row yield a value of ad 3.35, respectively. Example 715: Named Storms You ca be 90% cofidet that the true variace for the cost of ski lift tickets is betwee 15.0 ad You ca be 95% cofidet that the stadard deviatio is betwee 3.0 ad 6.8 for amed storms i a sample 10 years
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