CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions

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1 CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Meas ad Proportios Itroductio: I this chapter we wat to fid out the value of a parameter for a populatio. We do t kow the value of this parameter for the etire populatio. If we did already kow it, we would t have to do ay statistical ivestigatio or calculatios. But usually we ca t fid out all iformatio about the etire populatio, so the true value of the parameter is usually ot kow. 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.) ) CONFIDENCE INTERVAL ESTIMATE: The populatio parameter is estimated by a iterval of umbers, a rage of umbers that we believe cotais the true (ukow) value of the populatio parameter. Also, we are able to state how cofidet we are that the iterval estimate cotais the true 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, a cofidece iterval may ot be symmetric about the poit estimate, movig differet distaces above ad below the poit estimate to fid 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 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 car-free 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 car-free. Sa Fracisco s percet of car-free households has chaged rapidly i recet years. Suppose a recet study of 100 households i Sa Fracisco showed that 37 households were car-free. Costruct ad iterpret a 95% cofidece iterval for the true proportio of households i Sa Fracisco that are car-free. 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 area to left, 0, 1 ivorm ( ) Stadard Error p'q' Calculatios ad iterpretatio i cotext of the problem: Page of 10

3 CHAPTER 8 EXAMPLE 1 Cotiued: b. Our sample proportio was p =0.31. A city official had thought the percet would be 33%. Based o the cofidece iterval ca we coclude that the true proportio of car-free households i Sa Fracisco is differet tha 33%. Explai. c. Ca we coclude that more tha 5% of Sa Fracisco households are car-free. Explai. d. What does it mea whe we say the cofidece level is 95% or whe we say that we are "95% cofidet"? Page 3 of 10

4 CHAPTER 8 EXAMPLE : CONFIDENCE INTERVAL ESTIMATE for ukow POPULATION MEAN µ whe the POPULATION STANDARD DEVIATION σ 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: µ = 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 σ 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 σ creates area of CL i the middle; Z ~N(0,1) Use POSITIVE value of Z Cofidece Iterval = Poit Estimate + Error Boud Cofidece Iterval = x + EBM ivorm ( area to left, 0, 1) Calculatios ad iterpretatio i cotext of the problem: Stadard Error σ Page 4 of 10

5 CHAPTER 8 EXAMPLE 3: CONFIDENCE INTERVAL ESTIMATE for ukow POPULATION MEAN µ whe the POPULATION STANDARD DEVIATION σ 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: µ = 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 σ 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 TI-84: = 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 t-distributio 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, ad statisticias would use other techiques that we do ot study i Math 10. Page 5 of 10

6 CHAPTER 8 EXAMPLE 4: Workig Backwards: Fidig the Error Boud ad Poit Estimate if we kow the cofidece iterval: The average ightly cost of hotel rooms for two popular vacatio areas are compared. Large radom samples of hotel room costs are collected for each city. The resultig cofidece iterval estimates are reported i a hotel idustry joural. The 90% cofidece iterval estimate for the true populatio average ightly cost of all hotel rooms i Surf City is $134 to $159 per ight. The 90% cofidece iterval estimate for the true populatio average ightly cost of all hotel rooms i Ski Village is $13 to $141 per ight. a. Fid the poit estimate for the true average ightly cost of a hotel room i each city. Which city has a higher poit estimate? b. Fid the error boud for each city. Which city has a smaller margi of error? c. Based o the cofidece itervals oly, would it be reasoable to coclude that the true average ightly cost of a hotel rooms are differet i Surf City ad i Ski Village? d. Would it be true that 90% of hotel rooms cost betwee $134 ad $159 per ight i Surf City ad that 90% of hotel rooms cost betwee $13 ad $141 per ight i Ski Village? 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 PROBLEMS : CALCULATING CONFIDENCE INTERVALS Practice examples 6, 7, 8 are based o examples from Chapter 8 Practice ad Chapter 9 homework i Itroductory Statistics from OpeStax available for dowload for free at /latest/. Practice example 9 is based o iformatio from Bureau of Labor Statistics 01 cited o 8/31/015 at 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: Salaried employees (paid weekly, ot hourly) geerally do ot eed to report the umber of hours they work per week. A start-up compay wats to estimate the average umber of hours its egieerig employees work per week. For a sample of 10 egieerig employees, the hours they report they worked i a typical week were 70, 45, 55, 60, 65, 55, 55, 60, 50, 55 Calculate ad iterpret a 95% cofidece iterval estimate of the average umber of hours worked per week by all egieerig employees at this compay. Use a 95% cofidece level. PRACTICE EXAMPLE 8: Suppose a compay did a market research survey of 00 radomly selected households ad foud that i 10 of them the woma made the majority of purchasig decisios for their products. Calculate ad iterpret a cofidece iterval estimate for the true proportio of all households i which wome make the majority of purchasig decisios for their products. Use a 95% cofidece level. PRACTICE EXAMPLE 9: Suppose a survey of 500 households of married couples where both parters have paid work foud that oly 9% of the wome ear more tha their husbads. Calculate ad iterpret a cofidece iterval estimate for the true proportio of all such households i which the wome ear more tha their husbads. Use a 9% 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 µ (average) Proportio p Mea µ (average) if populatio stadard deviatio σ IS KNOWN Poit Estimate: x Error Boud: EBM = Z σ Mea µ (average) if populatio stadard deviatio σ 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 We wat 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 depeds o the cofidece level. Cofidece Iterval: p + EBP which is p + p'q' Z is the Z value that will put to 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 µ whe σ is kow We wat to estimate the populatio average µ ad we already kow the populatio stadard deviatio σ. Poit Estimate + Margi of Error 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. 007 R. Bloom Page 9 of 10 Z p'q' (Margi of Error is also called Error Boud) Poit Estimate: Sample Average (Sample Mea) x Error Boud: EBM = (critical value)(stadard error) = Z σ The critical value depeds o the cofidece level. Z is the Z value that will put to 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" σ is the appropriate stadard deviatio for the sample mea Cofidece Iterval: x + EBM which is x + Z σ Cofidece Iterval for a Mea µ whe σ is NOT kow We wat to estimate the populatio average µ ad we do ot kow the populatio stadard deviatio σ. We use the sample stadard deviatio s to estimate the populatio stadard deviatio σ Poit Estimate + Margi of Error Poit Estimate: Sample Average (Sample Mea) x Error Boud: EBM = (critical value)(stadard error) = The critical value depeds o the cofidece level. (Margi of Error is also called Error Boud) t s t is the t value that will put to 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

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? 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 eclose (capture) the true value of the populatio parameter we are estimatig. 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 we are estimatig. Note that the cofidece iterval is about proportios or averages. It is ot about idividual data values. 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 tells use the area i the middle = 1 CL is outside, split equally betwee both tails is i oe tail. To fid Z : ivorm(1, 0, 1) OR use ivorm(, 0, 1) ad take absolute value (drop the " " sig) Z 0 Z 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 tells use the area i the middle = 1 CL is outside, split equally betwee both tails is i oe tail. df = degrees of freedom = 1 To fid t : TI-84+: ivt(1, df) t 0 OR use ivt (, df) ad take absolute value (drop the " " sig) t TI-83,83+: 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 ; (if usig as area to left, drop the " " sig) Without calculator or if calculator does ot have iverse t: Use a studet s-t 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. 007 R. Bloom Page 10 of 10

CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions

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