CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions


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1 CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Meas ad Proportios Itroductio: We wat to kow the value of a parameter for a populatio. We do t kow the value of this parameter for the etire populatio because we do t have data for the etire populatio. (If we did already kow the value of the parameter, we would t eed to do ay statistical ivestigatio or calculatios.) We will use sample statistics to estimate populatio parameters. Recall from chapter : A parameter is If we do t kow the value of a populatio parameter, we ca estimate it usig a sample statistic. Recall from chapter : A statistic is Usig data from a sample to draw a coclusio about a populatio is called statistics. Two Types of Estimates for populatio parameters: 1) POINT ESTIMATE: A populatio parameter ca be estimated by oe umber: the sample statistic. This is called a poit estimate. (Statistical theory has idetified desirable properties of poit estimates, which are studied i more depth i upper level statistics classes. Oe property usually cosidered desirable is that a poit estimate be ubiased, meaig that the average of the poit estimates from all possible samples would equal the true value of the populatio parameter.) The best poit estimate of a populatio mea µ is The best poit estimate of a populatio proportio p is The best poit estimate of a populatio stadard deviatio is ) CONFIDENCE INTERVAL ESTIMATE: The populatio parameter is estimated by a iterval of umbers that we believe cotais the true (ukow) value of the populatio parameter. We are able to state how cofidet we are that the iterval estimate cotais the true (ukow) value of the parameter. This cofidece iterval estimate is built usig two items: a poit estimate, ad margis of error; the margis of error are also called error bouds. We will use cofidece iterval estimates based o sample data to estimate a populatio average (mea) populatio proportio Cofidece itervals for meas ad proportios are symmetric; the poit estimate is at the ceter of the iterval. The edpoits of the iterval are foud as (poit estimate error boud, poit estimate + error boud ) (For some other parameters, such as stadard deviatio, a cofidece iterval may ot be symmetric about the poit estimate, movig differet distaces above ad below the poit estimate to the eds of the iterval estimate.) We ll lear by example to calculate the poit estimates ad the error bouds ad what they mea. The last 3 pages of these otes has a cocise summary of formulas, procedures, ad iterpretatios. We ll start i class by examiig a jar with beads to determie the proportio of beads i the jar that are blue; after we explore the cocepts, the we ll move o to the mathematical calculatios. Page 1 of 10
2 CHAPTER 8 EXAMPLE 1: CONFIDENCE INTERVAL ESTIMATE for a ukow POPULATION PROPORTION p a. Statistics ad data i this example are based o iformatio from : A tred i urba developmet is to reduce the eed for residets to have a car; city eighborhoods are ofte raked for walkability. I recet studies, the US city with the lowest car owership rate is New York City; a majority (56%) of households are carfree with oly 44% of households owig ay vehicles. Sa Jose has the highest car owership rate of large US cities; oly about 6% of households carfree. Sa Fracisco s percet of carfree households has chaged rapidly i recet years. Suppose a recet study of 100 households i Sa Fracisco showed that 37 households were carfree. Costruct ad iterpret a 95% cofidece iterval for the true proportio of households i Sa Fracisco that are carfree. Use a 95% cofidece level. populatio parameter: p = radom variable p = We are usig sample data to estimate a ukow proportio for the whole populatio HOW TO CALCULATE THE CONFIDENCE INTERVAL Poit Estimate = p Cofidece Level CL is area i the middle Error Boud = (Critical Value)(Stadard Error) Critical Value is Z is the Z value that p'q' EBP = Z creates area of CL i the middle; Z ~N(0,1) Use POSITIVE value of Z Cofidece Iterval = Poit Estimate + Error Boud Cofidece Iterval = p + EBP ivormarea to left, 0,1 Stadard Error p'q' Calculatios ad iterpretatio i cotext of the problem: Page of 10
3 CHAPTER 8 EXAMPLE : Usig Cofidece Itervals Before electios, may polls survey samples of likely voters to try to gauge support for the cadidates for public office i the electio. Sice we ca t poll all voters before the electio, the polls are usig the sample data to try to predict the proportio of the populatio of all actual voters who will vote for each cadidate. A. Suppose a poll idicates 5% of those i survey idicatig they ited to vote for Cadidate A for Presidet. The poll uses a 95% cofidece level ad has a margi of error of +4%. Costruct the cofidece iterval estimate for cadidate A. Based o this cofidece iterval, does the poll give a idicatio (with 95% cofidece) whether Cadidate A will have more tha 50% of the vote. Iterval: Explaatio of Results: B. Suppose a differet poll idicates 55% of those i survey idicatig they ited to vote for Cadidate A for Presidet. The poll uses a 95% cofidece level ad has a margi of error of +3%. Costruct the cofidece iterval estimate for cadidate A. Based o this cofidece iterval, does the poll give a idicatio (with 95% cofidece) whether Cadidate A will have more tha 50% of the vote. Iterval: Explaatio of Results: C. I Example 1 the poit estimate is = ; the cofidece iterval is (i) Ca we coclude with 95% cofidece that more tha 5% of SF households are carfree? (ii) Ca we coclude with 95% cofidece that more tha 30% of SF households are carfree? D. What does it mea whe we say the cofidece level is 95% or that "we are 95% cofidet"? Page 3 of 10
4 CHAPTER 8 EXAMPLE 3: CONFIDENCE INTERVAL ESTIMATE for ukow POPULATION MEAN whe the POPULATION STANDARD DEVIATION is KNOWN A soda bottlig plat fills cas labeled to cotai 1 ouces of soda. The fillig machie varies ad does ot fill each ca with exactly 1 ouces. To determie if the fillig machie eeds adjustmet, each day the quality cotrol maager measures the amout of soda per ca for a radom sample of 50 cas. Experiece shows that its fillig machies have a kow populatio stadard deviatio of 0.35 ouces. I today's sample of 50 cas of soda, the sample average amout of soda per ca is 1.1 ouces. a. Costruct ad iterpret a 90% cofidece iterval estimate for the true populatio average amout of soda cotaied i all cas filled today at this bottlig plat. Use a 90% cofidece level. X = populatio parameter: = radom variable X = We are usig sample data to estimate a ukow mea (average) for the whole populatio HOW TO CALCULATE THE CONFIDENCE INTERVAL for µ Whe IS kow, use the Stadard ormal distributio Z ~ N(0,1) Poit Estimate = x Cofidece Level CL is area i the middle Error Boud = (Critical Value)(Stadard Error) Critical Value is Z is the Z value that EBM = Z creates area of CL i the middle; Z ~N(0,1) Cofidece Iterval = Poit Estimate + Error Boud Use POSITIVE value of Z Cofidece Iterval = x + EBM ivormarea to left, 0,1 Calculatios ad iterpretatio i cotext of the problem: Stadard Error Page 4 of 10
5 CHAPTER 8 EXAMPLE 3: CONFIDENCE INTERVAL ESTIMATE for ukow POPULATION MEAN whe the POPULATION STANDARD DEVIATION is NOT KNOWN a. The traffic commissioer wats to kow the average speed of all vehicles drivig o River Rd. Police use radar to observe the speeds for a sample of 0 vehicles o River Rd. For the vehicles i the sample, the average speed is 31.3 miles per hour with stadard deviatio 7.0 mph. Costruct ad iterpret a 98% cofidece iterval estimate of the true populatio average speed of all vehicles o River Rd. Use a 98% cofidece level. X = populatio parameter: = radom variable X = We are usig sample data to estimate a ukow mea (average) for the whole populatio HOW TO CALCULATE THE CONFIDENCE INTERVAL for µ Whe is NOT kow, use the t distributio with degrees of freedom = sample size 1 : t with df = 1) Poit Estimate = x Error Boud = (Critical Value)(Stadard Error) EBM = t s Cofidece Iterval = Poit Estimate + Error Boud Cofidece Iterval = x + EBM Calculatios ad iterpretatio i cotext of the problem: Cofidece Level CL is area i the middle Critical Value t is the t value that creates a area of CL i the middle; Use t distributio with df = 1 Use POSITIVE value of t TI84: = ivt(area to left, df) t Stadard Error s b. I Example 3, suppose that you were ot give the sample mea ad sample stadard deviatio ad istead you were give a list of data for the speeds (i miles per hour) of the 0 vehicles How would you use the data to do this problem? NOTE: Use of tdistributio requires the uderlyig populatio of idividual values to be approximately ormally distributed. It is OK if this assumptio is violated a little, but if the uderlyig populatio of idividual values has a distributio that differs too much from the ormal distributio, the this cofidece method is ot appropriate, Page 5 of 10 ad statisticias would use other techiques that we do ot study i Math 10.
6 CHAPTER 8 EXAMPLE 4: Workig Backwards: Fidig the Error Boud ad Poit Estimate if we kow oly the cofidece iterval: Jessie wats to estimate the average cost of a ride to the airport, from her apartmet, usig Uber ad usig Lyft ad usig a taxi. For Uber, the 90% cofidece iterval estimate of the average fare is $0 to $7. For Lyft, the 90% cofidece iterval estimate of the average fare is $1 to $5 For a taxi, the 90% cofidece iterval estimate of the average fare is $8 to $31 (NOTE: This iformatio is based o radom samples of fare estimates geerated by fare compariso websites for Uber, Lyft, ad taxis usig oe particular address ad the closest major airport to that address, with requests made at a variety of times. This may ot be represetative of the relatioships betwee average fares for Lyft ad Uber ad Taxi i all locatios.) a. Fid the poit estimate for the true average fare for each. Explai how to use the cofidece iterval to fid the poit estimate. Which has the lowest poit estimate? b. Fid the margi of error (error boud for each). Explai how to use the cofidece iterval to fid the error boud. Which has the lowest margi of error? c. Ca we draw ay coclusios about the true average fare based o the estimates above? Page 6 of 10
7 CHAPTER 8: Cofidece Iterval for a Proportio: Calculatig the Sample Size eeded i a Study Give a desired cofidece level ad a desired margi of error, how large a sample is eeded? EBP = Z p 'q' We kow the error boud EBP that we wat. We kow the cofidece level CL we wat, so we ca fid Z correspodig to the desired CL. We do't kow p' or q' util we do the study, so we will assume for ow that p' = q' = ½ = 0.5 The we ca substitute all these values ito EBP = Solvig EBP = Z p' q' Sample Size Formula to determie sample size eeded whe estimatig a populatio proportio p EXAMPLE 5: Fidig the Sample Size: Z EBP Z for gives. = p' q'. p 'q' ad solve for. The 0.5 appears i the formula because we are assumig that p' = q' = ½ = 0.5 ALWAYS ROUND UP to the ext higher iteger a. Public opiio ad political polls ofte do surveys with a 95% cofidece level ad 3% margi of error. Fid the sample size eeded. Z = (.5) = EBP b. Suppose a margi of error of % was wated with a 95% cofidece level. Fid the sample size eeded. Z = (.5) = EBP c. Suppose a margi of error of 3% was wated with a 90% cofidece level. Fid the sample size eeded. Z = (.5) = EBP d. Suppose a poll uses a sample size of =100, ad a cofidece level of 95%. Estimate the expected error boud usig p' = q' = ½ = 0.5 EBP = Z p'q' Z = (.5) EBP Note the actual error boud will differ after the study is doe because we will kow p' ad q' ad will o loger be estimatig that p'=q'=0.5 e. Is the error boud i part d large or small compared to the examples i parts a, b, c? Explai why this happeed. Page 7 of 10
8 CHAPTER 8 EXTRA PRACTICE EXAMPLES : CALCULATING CONFIDENCE INTERVALS Sources for Practice Examples: #6 based o Chapter 8 Practice i Itroductory Statistics from OpeStax available for dowload for free at /latest/. ; #7based o iformatio from ; #8 based o iformatio from ; #9 based o iformatio from PRACTICE EXAMPLE 6: A supermarket chai is decidig what produce providers to purchase from. A sample of 0 heads of lettuce is selected to estimate the average weight of the lettuce from this provider. The populatio stadard deviatio for the weight is kow to be 0. pouds. The sample of 0 heads of lettuce had a mea weight of. pouds with a sample stadard deviatio of 0.1 pouds. Calculate ad iterpret a cofidece iterval estimate for the true average weight of all heads of lettuce from this provider. Use a 90% cofidece level. PRACTICE EXAMPLE 7: The staff at the All Over Albay (N.Y.) website wated to kow how much ice cream is i a scoop. For a radom sample of 5 ice cream stores, the amout of ice cream cotaied i a scoop was 6.8, 3.9, 5.4, 3.55, 5. ouces. Calculate ad iterpret a 95% cofidece iterval estimate of the average amout of ice cream i a scoop for all ice cream sold at ice cream stores i Albay NY. Use a 95% cofidece level. PRACTICE EXAMPLE 8: Suppose a survey of 300 youg adults age 18 to 4 showed that 81 had used olie datig. Calculate ad iterpret a cofidece iterval estimate for the true proportio of all youg adults age 18 to 4 who ever used olie datig. Use a 95% cofidece level. PRACTICE EXAMPLE 9: Suppose a survey of 500 people age 18 to 34 idicated that 3.% of them live with oe or both of their parets. Calculate ad iterpret a cofidece iterval estimate for the true proportio of all people age 18 to 34 who live with oe or both parets. Use a 94% cofidece level. CHAPTER 8: FLOW CHART VIEW OF FORMULAS FOR CONFIDENCE INTERVAL ESTIMATES 015 R. Bloom For more details, see the presetatio of the formulas i boxes for each case o the ext page. Mea (average) Proportio p Mea (average) if populatio stadard deviatio IS KNOWN Poit Estimate: x Error Boud: EBM = Z Mea (average) if populatio stadard deviatio IS NOT KNOWN Poit Estimate x Error Boud: EBM = t s Poit Estimate p Error Boud: p' q' EBP = Z pq ˆ ˆ Z Distributio: Normal Distributio: Normal Distributio: Studets t t 1 degrees of freedom = 1 Page 8 of 10
9 CHAPTER 8: CONFIDENCE INTERVALS: SUMMARY OF FORMULAS, PROCEDURES, & INTERPRETATIONS Cofidece Iterval for a Proportio p To estimate a populatio proportio p (biomial probability of success). Poit Estimate + Margi of Error (Margi of Error is also called Error Boud) x umber of successes i sample Poit Estimate: Sample Proportio: p ' total umber i sample Error Boud: EBP = (critical value)(stadard error) = Z The critical value Z depeds o the cofidece level. Cofidece Iterval: p + EBP which is p + p'q' Z is the Z value that puts a area equal to the cofidece level (CL) i the middle of stadard ormal distributio N(0,1) Z tells us how may "appropriate stadard deviatios" to eclose about the poit estimate, where the "stadard error" p 'q' is the appropriate stadard deviatio for a proportio Cofidece Iterval for a Mea whe is kow To estimate the populatio average if we already kow the populatio stadard deviatio. Poit Estimate + Margi of Error (Margi of Error is also called Error Boud) Poit Estimate: Sample Average (Sample Mea) x Error Boud: EBM = (critical value)(stadard error) = Z The critical value Z depeds o the cofidece level. Z is the Z value that puts a area equal to the cofidece level (CL) i the middle of stadard ormal distributio N(0,1) Z tells us how may "appropriate stadard deviatios" to eclose about the poit estimate, where the "stadard error" is the appropriate stadard deviatio for the sample mea Cofidece Iterval: x + EBM which is x + Z Cofidece Iterval for a Mea whe is NOT kow To estimate the populatio average ad we do ot kow the populatio stadard deviatio. Use the sample stadard deviatio s to estimate the populatio stadard deviatio Poit Estimate + Margi of Error (Margi of Error is also called Error Boud) Poit Estimate: Sample Average (Sample Mea) x Error Boud: EBM = (critical value)(stadard error) = t s The critical value t depeds o the cofidece level. t is the t value that puts a area equal to the cofidece level (CL) i the middle of the studet t distributio with 1 degrees of freedom t tells us how may "appropriate stadard deviatios" we eed to move away from the poit estimate, where s is a estimate of the stadard error ("appropriate stadard deviatio") for the sample mea Cofidece Iterval: x + EBM which is x + t s YOU ARE NOT PERMITTED TO BRING A PRINTOUT OF THIS PAGE AS YOUR NOTES FOR AN EXAM OR QUIZ. You CAN write whatever iformatio you wat from this page ito your hadwritte otes for exams or quizzes. 016 R. Bloom Page 9 of 10 Z p'q'
10 CHAPTER 8: CONFIDENCE INTERVALS: SUMMARY OF FORMULAS, PROCEDURES, & INTERPRETATIONS Iterpretig the Cofidece Iterval for a PROPORTION ( ways to word it) We estimate with % cofidece that the true proportio of the populatio that describe the populatio parameter i the situatio of this problem is betwee ad We estimate with % cofidece that betwee % ad % of the populatio describe the populatio parameter i the situatio of this problem Iterpretig the Cofidece Iterval for a MEAN (average) We estimate with % cofidece that the true populatio average (or mea) describe the populatio parameter i the situatio of this problem is betwee ad What is the meaig of the Cofidece Level? What does it mea to be CL% cofidet? The cofidece level represets a expected accuracy rate for the cofidece iterval process. It tells us what percet of cofidece itervals calculated i this maer would be good estimates. If we took repeated samples ad calculated may cofidece iterval estimates (oe for each sample), we expect that CL% of the cofidece iterval estimates would be good estimates that successfully eclose (capture) the true value of the populatio parameter that we wat to estimate. If we took repeated samples), we expect that 100% CL% of the cofidece iterval estimates would be bad estimates that would NOT eclose (capture) the true value of the populatio parameter. Note that the cofidece iterval does ot estimate idividual data values. It estimates proportios or averages. It does NOT imply that CL% of the data lies withi the cofidece iterval. Fidig the Poit Estimate ad Error Boud (Margi of Error) if we kow the Cofidece Iterval: The iterval is (lower boud, upper boud) Poit Estimate = (lower boud + upper boud)/ Error Boud = (upper boud lower boud)/ To fid Z that puts the area equal to the cofidece level i the middle CL specifies the area i the middle = 1 CL is area outside, split equally betwee both tails is area i oe tail. To fid Z : ivorm(1 α, 0, 1) OR use ivorm( α, 0, 1) ad take absolute value (drop the " " sig) Without calculator: Use a stadard ormal probability table to fid Z. To fid t that puts the area equal to the cofidece level i the middle CL specifies the area i the middle = 1 CL is area outside, split equally betwee both tails is area i oe tail. df = degrees of freedom = 1 To fid t : TI84+: ivt(1 α, df) OR use ivt ( α, df) ad take absolute value (drop the " " sig) TI83,83+: Use INVT program; ask istructor to dowload it to your calculator: PRGM INVT eter area to the left ad df after prompts: area to left is 1 α ; (if usig α as area to left, drop the " " sig) Without calculator or if calculator does ot have iverse t: Use a studet st distributio probability table. Value of t is foud at the itersectio of the colum for the cofidece level ad row for degrees of freedom YOU ARE NOT PERMITTED TO BRING A PRINTOUT OF THIS PAGE AS YOUR NOTES FOR AN EXAM OR QUIZ. You CAN write whatever iformatio you wat from this page ito your hadwritte otes for exams or quizzes. 016 R. Bloom Page 10 of 10
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