24.1. Confidence Intervals and Margins of Error. Engage Confidence Intervals and Margins of Error. Learning Objective

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1 24.1 Cofidece Itervals ad Margis of Error Essetial Questio: How do you calculate a cofidece iterval ad a margi of error for a populatio proportio or populatio mea? Resource Locker LESSON 24.1 Cofidece Itervals ad Margis of Error Alamy Explore Idetifyig Likely Populatio Proportios I a previous lesso, you took samples from a populatio whose parameter of iterest is kow i order to see how well a sample statistic estimated that parameter. I this lesso, you will estimate a populatio parameter usig a statistic obtaied from a radom sample, ad you will quatify the accuracy of that estimate. Suppose you survey a radom sample of 50 studets at your high school ad fid that 40% of those surveyed atteded the football game last Saturday. Although you caot survey the etire populatio of studets, you would still like to kow what populatio proportios are reasoably likely i this situatio. Suppose the proportio p of the populatio that atteded last Saturday s game is 30%. Fid the reasoably likely values of the sample proportio p. 0.3 I this case, p =? ad =?. 50 μ p = p =? ad p = 0.3; 0.3 p (1- p)? (1 -? ) =?? The reasoably likely values of p fall withi 2 stadard deviatios of μ p μ p - 2 p =? - 2 (? ) =? μ p + 2 p =? + 2 (? ) =? Module Lesso 1 Professioal Developmet Learig Progressios I the previous lesso, studets leared that the idividual sample meas or sample proportios withi a samplig distributio vary, formig a approximately ormal distributio havig a stadard deviatio that they leared how to calculate. I the previous lesso, the populatio parameters (populatio mea ad stadard deviatio for umerical data ad populatio proportio for categorical data) were kow. This lesso exteds their kowledge to populatios whose mea or proportio is ot kow. Studets will lear how to determie, with a certai degree of cofidece, that the populatio mea or proportio is withi a particular rage of values. Learig Objective Studets will fid cofidece itervals for populatio proportios ad populatio meas, ad determie the sample size eeded to obtai a desired margi of error. Math Processes ad Practices MPP4 Mathematical Modelig Laguage Objective Work with a parter to label the formula for the cofidece iterval for a populatio proportio. Olie Resources A extra example for each Explai sectio is available olie. Egage Essetial Questio: How do you calculate a cofidece iterval ad a margi of error for a populatio proportio or populatio mea? To calculate a cofidece iterval, use the formula p^ - z c p^ (1 - p^) p p^ + z c p^ (1 - p^), where p is the proportio of successes i a populatio, p^ is the sample proportio, is the sample size, ad z c depeds o the desired degree of cofidece. To calculate a cofidece iterval for a populatio mea, use the formula x - z c µ x + z c, where μ is the mea of a ormally distributed populatio, x is the sample mea, is the populatio stadard deviatio, ad z c depeds o the desired degree of cofidece. For both the populatio proportio ad the populatio mea, the margi of error is half the legth of the cofidece iterval. Lesso

2 Preview: Lesso Performace Task Draw the graph o grid paper, the draw a horizotal lie segmet at the level of 0.3 o the vertical axis to represet the iterval of likely values of p you foud i Step B. View the Egage sectio olie. Discuss the photo ad how differet proportios from samples might represet populatios where the proportios are equal. The preview the Lesso Performace Task. Explore Idetifyig Likely Populatio Proportios Itegrate Techology Studets have the optio of completig the cofidece iterval activity either i the book or olie. Proportio of Successes i Populatio, p Proportio of Successes i Sample, pˆ Now repeat the process for p = 0.35, 0.4, 0.45, ad so o to complete the graph i Step C. You may wish to divide up the work with other studets ad pool your fidigs. See graph above. Draw a vertical lie at 0.4 o the horizotal axis. This represets p = 0.4. The lie segmets that this vertical lie itersects are the populatio proportios for which a sample proportio of 0.4 is reasoably likely. See graph above. 1 Questioig Strategies How does the sample size affect the reasoably likely values of the sample proportio p^? As the sample size icreases, the rage of the reasoably likely values of the sample proportio p^ decreases. If you drew a coclusio that would be reasoably likely for the sample size of 50 studets, could you draw this same coclusio if all umbers i the give situatio stayed the same except that the sample size was 100 istead of 50? Explai your reasoig. No; a populatio proportio would o loger be reasoably likely. If you drew the graph for a sample size of 100 studets, the vertical lie segmet might ot itersect the horizotal lie segmet o the graph. It is both possible ad likely. Sice p = 0.3, the iterval of likely sample proportios Reflect for a sample of 50 studets is 0.17 to 0.43, which icludes p = Discussio Is it possible that 30% of all studets at your school atteded last Saturday s football game? Is it likely? Explai. 2. Discussio Is it possible that 60% of all studets at your school atteded last Saturday s football game? Is it likely? Explai. See margi. 3. Discussio Based o your graph, which populatio proportios do you thik are reasoably likely? Why? See margi. Explai 1 Fidig a Cofidece Iterval for a Populatio Proportio A cofidece iterval is a approximate rage of values that is likely to iclude a ukow populatio parameter. The level of a cofidece iterval, such as 95%, gives the probability that the iterval icludes the true value of the parameter. Recall that whe data are ormally distributed, 95% of the values fall withi 2 stadard deviatios of the mea. Usig this idea i the Explore, you foud a 95% cofidece iterval for the proportio of all studets who atteded the football game last Saturday. Module Lesso 1 Aswers Houghto Miffli Harcourt Publishig Compay Auditory Cues The probability otatio p^, a letter p with a caret or circumflex above it, is sometimes referred to as p-hat. Help studets to uderstad that this otatio coects the probability to cofidece itervals. 2. It is possible but ulikely. Sice p = 0.6, the iterval of likely sample proportios for a sample of 50 studets is 0.46 to 0.74, which does ot iclude p = The proportios from 30% to 50% are reasoably likely because these populatio proportios have itervals of likely sample proportios that iclude p = Cofidece Itervals ad Margis of Error

3 The graph that you completed i the Explore is show. You ca see from the graph that whe the horizotal lie segmet at p = 0.4 is rotated 90 about the poit (0.4, 0.4), it becomes a vertical lie segmet that captures all of the likely populatio proportios. Sice you already kow the iterval o the horizotal axis that defies the horizotal segmet, you ca fid the iterval o the vertical axis that defies the vertical segmet by usig the fact that μ p = p ad iterchagig the variables p o the horizotal axis) ad p (o the vertical axis). So, the horizotal axis iterval μ p - 2 p p μ p + 2 p becomes the vertical axis iterval as follows: Replace μ p with p. Iterchage p ad p. Proportio of Successes i Populatio, p p - 2 p p p + 2 p p - 2 p p p + 2 p I this case, the vertical axis iterval is the 95% cofidece iterval for p. This result ca be geeralized to a c% cofidece iterval for p Proportio of Successes i Sample, pˆ Explai 1 Fidig a Cofidece Iterval for a Populatio Proportio Questioig Strategies Based o the give radom sample, two differet researchers made differet, reasoable statemets about the data. What could be the reaso for the discrepacy? Possible aswer: The researchers made their statemets with differet cofidece itervals. Avoid Commo Errors Some studets may be cofused about why they substitute p for µ p^ whe fidig a cofidece iterval. Poit out that this is doe so that the horizotal lie segmet is used to fid the iterval o the vertical lie segmet that captures all the likely populatio proportios. Houghto Miffli Harcourt Publishig Compay A Cofidece Iterval for a Populatio Proportio A c% cofidece iterval for the proportio p of successes i a populatio is give p (1 - p ) p (1 - p ), where p is the sample proportio, is the sample size, ad z c depeds upo the desired degree of cofidece. by p - z c p p + z c I order for this iterval to describe the value of p reasoably accurately, three coditios must be met: 1. There are oly two possible outcomes associated with the parameter of iterest. The populatio proportio for oe outcome is p, ad the proportio for the other outcome is 1 - p. 2. p ad (1 - p ) must both be at least The size of the populatio must be at least 10 times the size of the sample, ad the sample must be radom. Module Lesso 1 Collaborative Learig Small Group Activity Have studets discuss the formula used to fid a cofidece iterval for a populatio mea, ad how it applies oly whe the scores are ormally distributed. The have them share ad discuss examples of scores that are distributed so that the give formula for a cofidece iterval should ot be used ad explai why. Possible aswer: Examples iclude distributios of the scores that are uiform, skewed left, or skewed right. Whe a distributio is skewed, coclusios based o the use of the give formula for a cofidece iterval are ivalid. Lesso

4 Aswers 4. Yes, because a child either ca or caot write his or her ame, p = 100 (0.76) = 76 ad (1 - p) = 100(0.24) = 24, which are both greater tha 10, ad the populatio of four-year-olds i the Uited States is far greater tha 10 = 10 (100) = Icreasig the value of c icreases the rage of values because the amout that is beig added or subtracted from p icreases. This makes sese because i order to have a higher level of cofidece that you ve caught the populatio proportio, you eed to cast a wider et. Use the values i the table for z c. Note that you should use 1.96 rather tha 2 for z 95% for greater accuracy. Desired degree of cofidece 90% 95% 99% Value of z c Example 1 I a radom sample of 100 four-year-old childre i the Uited States, 76 were able to write their ame. Fid the specified cofidece iterval for the proportio p of four-year-olds i the Uited States who ca write their ame. Fid a 95% cofidece iterval. Idetify the sample size, the proportio p of four-year-olds i the sample who ca write their ame, ad the value of z c for a 95% cofidece iterval. = 100 p = 0.76 z c = 1.96 Substitute the values of, p, ad z c ito the formulas for the edpoits of the cofidece iterval. Simplify ad roud to two decimal places. p - z c p(1 - p) = (1-0.76) p + z c p(1 - p) = (1-0.76) So, you ca state with 95% cofidece that the proportio of all four-year-olds i the Uited States who ca write their ame lies betwee 68% ad 84%. Fid a 99% cofidece iterval. Idetify the sample size, the proportio p of four-year-olds i the sample who ca write their ame, ad the value of z c for a 99% cofidece iterval. = 100 p = 0.76 z c = Substitute the values of, p, ad z c ito the formulas for the edpoits of the cofidece iterval. Simplify ad roud to two decimal places. p - z c p(1 - p) p + z c p(1 - p) = = ( ) ( ) So, you ca state with 99% cofidece that the proportio of all four-year-olds i the Uited States who ca write their ame lies betwee 65% ad 87%. Reflect Houghto Miffli Harcourt Publishig Compay 4. Discussio Do the data from the sample of four-year-old childre satisfy the three coditios for usig the cofidece iterval formula? Explai. See margi. 5. Discussio Does icreasig the value of c icrease or decrease the rage of values for a cofidece iterval of a populatio proportio? Explai why it happes mathematically ad why it makes sese. See margi. Module Lesso 1 Differetiate Istructio Commuicatig Math Studets may beefit from seeig some real-world situatios i which statistics are reported with a specified level of cofidece. Academic jourals i fields ragig from sciece ad medicie to psychology ad busiess ofte display detailed iformatio resultig from surveys i a table format, ofte with a cofidece level idicated at the bottom. Although the statistics used may go beyod what is covered i this lesso, studets ca see that level of cofidece is a compoet of may statistical aalyses. 856 Cofidece Itervals ad Margis of Error

5 Houghto Miffli Harcourt Publishig Compay Your Tur 6. Isabelle surveys a radom sample of 80 voters i her large tow ad fids that 46 support raisig property taxes i order to build a ew library. Fid a 95% cofidece iterval for the proportio p of all voters i Isabelle s tow who support raisig property taxes i order to build a ew library. betwee 47% ad 68% Explai 2 Fidig a Cofidece Iterval for a Populatio Mea You ca use reasoig similar to the argumet i the Explore to develop a formula for a cofidece iterval for a populatio mea. A Cofidece Iterval for a Populatio Mea A c% cofidece iterval for the mea μ i a ormally distributed populatio is give by x - z c μ x + z c, where x is the sample mea, is the sample size, is the populatio stadard deviatio, ad z c depeds upo the desired degree of cofidece. Note that it is assumed that the populatio is ormally distributed ad that you kow the populatio stadard deviatio. I a more advaced statistics course, you ca develop a cofidece iterval that does ot deped upo a ormally distributed populatio or kowig the populatio stadard deviatio. Example 2 For the give situatio, fid the specified cofidece iterval for the populatio mea. I a radom sample of 20 studets at a large high school, the mea score o a stadardized test is 610. Give that the stadard deviatio of all scores at the school is 120, fid a 99% cofidece iterval for the mea score amog all studets at the school. Idetify the sample size, the sample mea x, the populatio stadard deviatio, ad the value of z c for a 99% cofidece iterval. = 20 x = 610 = 120 z c = Substitute the values of, x,, ad z c ito the formulas for the edpoits of the cofidece iterval. Simplify ad roud to the earest whole umber. x - z c = x + z c 20 = So, you ca state with 99% cofidece that the mea score amog all studets lies betwee 541 ad 679. I a radom sample of 30 studets at a large high school, the mea score o a stadardized test is Give that the stadard deviatio of all scores at the school is 141, fid a 95% cofidece iterval for the mea score amog all studets at the school. Idetify the sample size, the sample mea x, the populatio stadard deviatio, ad the value of z c for a 95% cofidece iterval. = 30 x = 1514 = 141 z c = 1.96 Explai 2 Fidig a Cofidece Iterval for a Populatio Mea Questioig Strategies Suppose the cofidece level i a give situatio is reduced. Would the cofidece iterval for the mea test score amog all studets be wider or arrower tha the cofidece iterval for the mea test score amog all studets whe the cofidece level is higher? Explai your reasoig. The cofidece iterval for a lower cofidece level will be arrower tha the cofidece iterval for a higher cofidece level. As the cofidece iterval becomes wider, it is more likely that the mea test score amog all studets will fall withi that cofidece iterval; therefore, the cofidece level icreases. Avoid Commo Errors Remid studets that the formula used will produce valid results oly whe the populatio is ormally distributed. Module Lesso 1 Laguage Support Commuicate Math Have studets work i pairs to idetify the parts of this formula for a populatio proportio, askig them to idetify ad label what the p values, as well as ad z, represet: p^- z c p^ (1 - p^) p p^ + z c p^ (1 - p^) Lesso

6 Explai 3 Choosig a Sample Size Questioig Strategies A researcher wats to decrease the margi of error i a study surveyig a radom sample of studets that represets a much larger populatio of studets. Due to cost cosideratios, the sample size caot be icreased. How could the researcher accomplish the goal of decreasig the margi of error? The researcher could decrease the cofidece level for the results of the survey, which will decrease the margi of error. Itegrate Math Processes ad Practices Focus o Abstract ad Quatitative Reasoig MPP2 Explai to studets that whe desigig a survey, researchers ca use a formula that allows them to calculate the sample size they eed to obtai a desired margi of error ad cofidece level. Have studets discuss why they would wat to use a smaller rather tha a larger sample size, such as the cost of the survey, coveiece i coductig the survey, ad time cosideratios. Have studets cosider how the advatages of choosig a larger sample size may outweigh the disadvatages, despite the fact that choosig a smaller sample size saves moey, effort, ad time. Substitute the values of, x,, ad z c ito the formulas for the edpoits of the cofidece c iterval. Simplify ad roud to the earest whole umber. 141 x - z c = x + z c = So, you ca state with 95% cofidece that the mea score amog all studets at the school lies betwee 1464 ad Reflect 7. What must you assume about the test scores of all studets to use the formula for the cofidece iterval? You must assume that the scores are ormally distributed. Your Tur 8. I a radom sample of 42 employees i a large compay, the mea weekly umber of miutes spet exercisig is 86. Give that the stadard deviatio of all employees is 22.4, fid a 99% cofidece iterval for the mea weekly umber of miutes spet exercisig amog all employees i the compay. betwee 77 miutes ad 95 miutes Explai 3 Choosig a Sample Size I Part B of Example 2, you foud the 95% cofidece iterval 1464 x 1564, which is a rage of values cetered at x = You ca write the cofidece iterval as 1514 ± 50, where 50 is called the margi of error. The margi of error is half the legth of a cofidece iterval. Margi of Error for a Populatio Proportio 30 The margi of error E for the proportio of successes i a populatio with sample p(1 - p) proportio p ad sample size is give by E = z c, where z c depeds o the degree of the cofidece iterval. Margi of Error for a Populatio Mea The margi of error E for the mea i a ormally distributed populatio with stadard deviatio, sample mea x, ad sample size is give by E = z c, where z c depeds o the degree of the cofidece iterval. From the formulas you ca see that the margi of error decreases as the sample size icreases. This suggests usig a sample that is as large as possible. However, it is ofte more practical to determie a margi of error that is acceptable ad the calculate the required sample size. Houghto Miffli Harcourt Publishig Compay Module Lesso Cofidece Itervals ad Margis of Error

7 Example 3 Fid the appropriate sample size for the give situatio. A researcher wats to kow the percet of teeagers i the Uited States who have social etworkig profiles. She is aimig for a 90% cofidece iterval ad a margi of error of 4%. What sample size should she use? Step 1 Rewrite the margi-of-error formula for a populatio proportio by solvig for. E = z c p (1 - p ) E 2 = z 2 c p (1 - p ) Write the formula. Square both sides. E 2 = z 2 c p (1 - p ) Multiply both sides by. = z 2 c p (1 - p ) E 2 Step 2 Estimate the value of p. Divide both sides by E 2. The researcher has ot coducted the survey ad is tryig to fid p. So, she must estimate p as 0.5, which is the value of p that makes the expressio p (1 - p ) as large as possible. Step 3 Idetify the values of E ad z c. E is the margi of error writte as a decimal ad z c is the z-score that correspods to a 90% cofidece iterval. So, E = 0.04 ad z c = Step 4 Substitute the values of p, E, ad z c i the rewritte margi-of-error formula from Step 1. = z 2 c p (1 - p ) 2 = (1.645) (1-0.5) E (0.04) So, the researcher should survey a radom sample of 423 teeagers. Caleb is a restaurat maager ad wats to kow the mea umber of secods it takes to complete a customer s order. He is aimig for a 95% cofidece iterval ad a margi of error of 6 secods. Based o past experiece, Caleb estimates the populatio stadard deviatio to be 21 secods. What sample size should he use? Step 1 Rewrite the margi-of-error formula for a populatio mea by solvig for. E = z c Write the formula. E 2 = z 2 c 2 Square both sides. E 2 = z 2 c 2 Multiply both sides by. = z c 2 2 E 2 Divide both sides by E 2. Module Lesso 1 Lesso

8 Elaborate Itegrate Math Processes ad Practices Focus o Usig ad Evaluatig Logical Reasoig MPP3 Make sure that studets are comfortable with aalyzig the problem statemet for the data set ad determiig which of the formulas i this uit may apply to the problem. Have them highlight the key words i the problem statemet that lead them to choose a formula, ad the ask them to give the restrictios or parameters that allow them to use the formula. Coect Vocabulary Have studets use ote cards to create a list of words that are associated with fidig cofidece itervals ad margi of error for a populatio proportio or populatio mea. This list ca be used to choose or elimiate formulas that do ot apply for a give sample populatio or for a problem statemet about the populatio. Summarize The Lesso Have studets make a graphic orgaizer outliig the processes for calculatig a cofidece iterval for a populatio proportio ad a populatio mea. Iclude iformatio about calculatig the margi of error, as well as steps for choosig a sample size. Step 2 Idetify the values of E,, ad z c. E is the margi of error, is the populatio stadard deviatio, ad z c is the z score that correspods to a 95% cofidece iterval. So, E =6, = 21 ad z c = Step 3 Substitute the values of E,, ad z c i the margi of error for a populatio mea formula that was solved for. ( ) 2 ( ) 2 2 ( 6 ) = Reflect So, Caleb should survey a radom sample of 47 orders. smaller value of E 2 results i a larger value of. 9. Discussio I Part A, do you expect the sample size to icrease or decrease if the researcher decides she wats a smaller margi of error? Explai usig the margi-of-error formula for a populatio proportio. 10. Discussio I Part B, do you expect the sample size to icrease or decrease if Caleb decides he wats a 99% cofidece iterval istead of a 95% cofidece iterval? Explai usig the margi-of-error formula for a populatio mea. The sample size must icrease. A higher level of cofidece meas that the value of z c is greater, which meas that the value of z 2 c is also greater. Your Tur Multiplyig 2 by a greater value of z 2 E 2 c results i a greater value of. 11. Zoe is a editor of a ewspaper i a state capital ad wats to kow the percet of residets i her state who are i favor of baig the use of hadheld cell phoes while drivig, a bill that is beig cosidered i the state legislature. After researchig similar polls coducted i other states, she estimates that p = She is aimig for a 95% cofidece iterval ad a margi of error of 5%. What sample size should Zoe use? radom sample of 350 state residets Elaborate 47 The sample size must icrease. A smaller margi of error meas that the value of E is smaller, which meas that the value of E 2 is also smaller. Dividig z c 2 p(1 - p) by a 12. How ca a iterval that captures 95% of the sample proportios i a samplig distributio be used to fid a 95% cofidece iterval for a populatio proportio? See below. 13. Describe how icreasig the sample size affects the cofidece iterval of a populatio mea. See below. 14. Essetial Questio Check-i What is the relatioship betwee a cofidece iterval ad a margi of error for a populatio proportio or populatio mea? The margi of error is half the legth of the cofidece iterval. 12. Sice the iterval µ p - 2 p p µ p + 2 p captures 95% of the sample proportios i a samplig distributio, the 95% cofidece iterval for a populatio proportio is foud by substitutig p for µ p ad iterchagig p ad p. This yields p - 2 p p p + 2 p, which ca also be writte as p - 2 p(1 - p) p p + 2 p(1 - p). 13. Icreasig the sample size decreases the value of z c, which meas that the edpoits of the cofidece iterval become closer to each other. Houghto Miffli Harcourt Publishig Compay Module Lesso Cofidece Itervals ad Margis of Error