Confidence Interval for a Population Proportion

Transcription

1 Coexios module: m Cofidece Iterval for a Populatio Proportio OpeStax College This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese 3.0 Durig a electio year, we see articles i the ewspaper that state codece itervals i terms of proportios or percetages. For example, a poll for a particular cadidate ruig for presidet might show that the cadidate has 40% of the vote withi three percetage poits (if the sample is large eough). Ofte, electio polls are calculated with 95% codece, so, the pollsters would be 95% codet that the true proportio of voters who favored the cadidate would be betwee 0.37 ad 0.43: ( , ). Ivestors i the stock market are iterested i the true proportio of stocks that go up ad dow each week. Busiesses that sell persoal computers are iterested i the proportio of households i the Uited States that ow persoal computers. Codece itervals ca be calculated for the true proportio of stocks that go up or dow each week ad for the true proportio of households i the Uited States that ow persoal computers. The procedure to d the codece iterval, the sample size, the error boud, ad the codece level for a proportio is similar to that for the populatio mea, but the formulas are dieret. How do you kow you are dealig with a proportio problem? First, the uderlyig distributio is abiomial distributio. (There is o metio of a mea or average.) If X is a biomial radom variable, the X B(, p) where is the umber of trials ad p is the probability of a success. To form a proportio, take X, the radom variable for the umber of successes ad divide it by, the umber of trials (or the sample size). The radom variable P (read "P prime") is that proportio, P = X (Sometimes the radom variable is deoted as ^P, read "P hat".) Whe is large ad p is ot close to zero or oe, we ca use the ormal distributio to approximate the biomial. X N ( p, pq ) If we divide the radom variable, the mea, ad the stadard deviatio by, we get a ormal distributio of proportios with P, called the estimated proportio, as the radom variable. (Recall that a proportio as the umber of successes divided by.) X = P ~ N ( p, pq ) pq Usig algebra to simplify : = pq P X follows a ormal distributio for proportios: ( p, ) pq = P ~ N The codece iterval has the form (p EBP, p + EBP). EBP is error boud for the proportio. p = x p = the estimated proportio of successes (p is a poit estimate for p, the true proportio.) Versio 1.5: Nov 5, 013 5:37 pm

2 Coexios module: m46999 x = the umber of successes = the size of the sample The error boud for a proportio is EBP = ( ) ( ) p z q α where q = 1 p This formula is similar to the error boud formula for a mea, except that the "appropriate stadard deviatio" is dieret. For a mea, whe the populatio stadard deviatio is kow, the appropriate stadard deviatio that we use is σ. For a proportio, the appropriate stadard deviatio is pq. p q However, i the error boud formula, we use as the stadard deviatio, istead of pq. I the error boud formula, the sample proportios p ad q are estimates of the ukow populatio proportios p ad q. The estimated proportios p ad q are used because p ad q are ot kow. The sample proportios p ad q are calculated from the data: p is the estimated proportio of successes, ad q is the estimated proportio of failures. The codece iterval ca be used oly if the umber of successes p ad the umber of failures q are both greater tha ve. : For the ormal distributio of proportios, the z-score formula is as follows. If P ~N ( p, ) pq the the z-score formula is z = p p pq Example 1 Suppose that a market research rm is hired to estimate the percet of adults livig i a large city who have cell phoes. Five hudred radomly selected adult residets i this city are surveyed to determie whether they have cell phoes. Of the 500 people surveyed, 41 respoded yes - they ow cell phoes. Usig a 95% codece level, compute a codece iterval estimate for the true proportio of adult residets of this city who have cell phoes. Solutio A Solutio A The rst solutio is step-by-step (Solutio A). The secod solutio uses a fuctio of the TI-83, 83+ or 84 calculators (Solutio B). Let X = the umber of people i the sample who have cell phoes. X is biomial. X B ( 500, 500) 41. To calculate the codece iterval, you must d p, q, ad EBP. = 500 x = the umber of successes = 41 p = x = = 0.84 p = 0.84 is the sample proportio; this is the poit estimate of the populatio proportio. q = 1 p = = Sice CL = 0.95, the α = 1 CL = = 0.05 ( ) α = The z α = z 0.05 = 1.96 Use the TI-83, 83+, or 84+ calculator commad ivnorm(0.975,0,1) to d z Remember that the area to the right of z 0.05 is 0.05 ad the area to the left of z 0.05 is This ca also be foud usig appropriate commads o other calculators, usig a computer, or usig a Stadard Normal probability table. EBP = ( ) p z q α = (1.96) (0.84)(0.158) 500 = 0.03 p' EBP = = 0.81 p + EBP = = The codece iterval for the true biomial populatio proportio is (p EBP, p + EBP) = (0.810, 0.874).

3 Coexios module: m Iterpretatio We estimate with 95% codece that betwee 81% ad 87.4% of all adult residets of this city have cell phoes. Explaatio of 95% Codece Level Niety-ve percet of the codece itervals costructed i this way would cotai the true value for the populatio proportio of all adult residets of this city who have cell phoes. Solutio B Solutio B : Press STAT ad arrow over to TESTS. Arrow dow to A:1-PropZit. Press ENTER. Arrow dow to x ad eter 41. Arrow dow to ad eter 500. Arrow dow to C-Level ad eter.95. Arrow dow to Calculate ad press ENTER. The codece iterval is ( , ). : Exercise (Solutio o p..) Suppose 50 radomly selected people are surveyed to determie if they ow a tablet. Of the 50 surveyed, 98 reported owig a tablet. Usig a 95% codece level, compute a codece iterval estimate for the true proportio of people who ow tablets. Example For a class project, a political sciece studet at a large uiversity wats to estimate the percet of studets who are registered voters. He surveys 500 studets ad ds that 300 are registered voters. Compute a 90% codece iterval for the true percet of studets who are registered voters, ad iterpret the codece iterval. Solutio A The rst solutio is step-by-step (Solutio A). The secod solutio uses a fuctio of the TI-83, 83+, or 84 calculators (Solutio B). Solutio A x = 300 ad = 500 p = x = = q = 1 p = = Sice CL = 0.90, the α = 1 CL = = 0.10 ( ) α = 0.05 z α = z 0.05 = Use the TI-83, 83+, or 84+ calculator commad ivnorm(0.95,0,1) to d z Remember that the area to the right of z 0.05 is 0.05 ad the area to the left of z 0.05 is This ca also be foud usig appropriate commads o other calculators, usig a computer, or usig a stadard ormal probability table. EBP = ( z α ) p q = (1.645) (0.60)(0.40) 500 = 0.036

4 Coexios module: m p EBP = = p + EBP = = The codece iterval for the true biomial populatio proportio is (p EBP, p + EBP) = (0.564,0.636). Iterpretatio We estimate with 90% codece that the true percet of all studets that are registered voters is betwee 56.4% ad 63.6%. Alterate Wordig: We estimate with 90% codece that betwee 56.4% ad 63.6% of ALL studets are registered voters. Explaatio of 90% Codece Level Niety percet of all codece itervals costructed i this way cotai the true value for the populatio percet of studets that are registered voters. Solutio B Solutio B : Press STAT ad arrow over to TESTS. Arrow dow to A:1-PropZit. Press ENTER. Arrow dow to x ad eter 300. Arrow dow to ad eter 500. Arrow dow to C-Level ad eter Arrow dow to Calculate ad press ENTER. The codece iterval is (0.564, 0.636). : Exercise 4 (Solutio o p..) A studet polls his school to see if studets i the school district are for or agaist the ew legislatio regardig school uiforms. She surveys 600 studets ad ds that 480 are agaist the ew legislatio. a. Compute a 90% codece iterval for the true percet of studets who are agaist the ew legislatio, ad iterpret the codece iterval. Exercise 5 (Solutios o p..) b. I a sample of 300 studets, 68% said they ow a ipod ad a smart phoe. Compute a 97% codece iterval for the true percet of studets who ow a ipod ad a smartphoe.

5 Coexios module: m Plus Four Codece Iterval for p There is a certai amout of error itroduced ito the process of calculatig a codece iterval for a proportio. Because we do ot kow the true proportio for the populatio, we are forced to use poit estimates to calculate the appropriate stadard deviatio of the samplig distributio. Studies have show that the resultig estimatio of the stadard deviatio ca be awed. Fortuately, there is a simple adjustmet that allows us to produce more accurate codece itervals. We simply preted that we have four additioal observatios. Two of these observatios are successes ad two are failures. The ew sample size, the, is + 4, ad the ew cout of successes is x +. Computer studies have demostrated the eectiveess of this method. It should be used whe the codece level desired is at least 90% ad the sample size is at least te. Example 3 A radom sample of 5 statistics studets was asked: Have you smoked a cigarette i the past week? Six studets reported smokig withi the past week. Use the plus-four method to d a 95% codece iterval for the true proportio of statistics studets who smoke. A: Solutio A Solutio A Six studets out of 5 reported smokig withi the past week, so x = 6 ad = 5. Because we are usig the plus-four method, we will use x = 6 + = 8 ad = = 9. p = x = q = 1 p = = 0.74 Sice CL = 0.95, we kow α = = 0.05 ad α = z 0.05 = 1.96 EP B = ( z α ) p q = (1.96) 0.76(0.74) p EPB = = p + EPB = = We are 95% codet that the true proportio of all statistics studets who smoke cigarettes is betwee ad Solutio B Solutio B : Press STAT ad arrow over to TESTS. Arrow dow to A:1-PropZit. Press ENTER. : Remember that the plus-four method assume a additioal four trials: two successes ad two failures. You do ot eed to chage the process for calculatig the codece iterval; simply update the values of x ad to reect these additioal trials. Arrow dow to x ad eter eight. Arrow dow to ad eter 9. Arrow dow to C-Level ad eter Arrow dow to Calculate ad press ENTER. The codece iterval is (0.113, 0.439).

6 Coexios module: m : Exercise 7 (Solutios o p..) Out of a radom sample of 65 freshme at State Uiversity, 31 studets have declared a major. Use the plus-four method to d a 96% codece iterval for the true proportio of freshme at State Uiversity who have declared a major. Example 4 The Berkma Ceter for Iteret & Society at Harvard recetly coducted a study aalyzig the privacy maagemet habits of tee iteret users. I a group of 50 tees, 13 reported havig more tha 500 frieds o Facebook. Use the plus four method to d a 90% codece iterval for the true proportio of tees who would report havig more tha 500 Facebook frieds. Solutio A Solutio A Usig plus-four, we have x = 13 + = 15 ad = = 54. p ' = q ' = 1 p ' = = 0.7 Sice CL = 0.90, we kow α = = 0.10 ad α = z 0.05 = EP B = ( ) ( ) p z q α ( = (1.645) ) (0.78)(0.7) p EPB = = p + EPB = = We are 90% codet that betwee 17.8% ad 37.8% of all tees would report havig more tha 500 frieds o Facebook. Solutio B Solutio B : Press STAT ad arrow over to TESTS. Arrow dow to A:1-PropZit. Press ENTER. Arrow dow to x ad eter 15. Arrow dow to ad eter 54. Arrow dow to C-Level ad eter Arrow dow to Calculate ad press ENTER. The codece iterval is (0.178, 0.378). : Exercise 9 (Solutios o p. 3.) The Berkma Ceter Study refereced i Example 4 talked to tees i smaller focus groups, but also iterviewed additioal tees over the phoe. Whe the study was complete, 588 tees had aswered the questio about their Facebook frieds with 159 sayig

7 Coexios module: m that they have more tha 500 frieds. Use the plus-four method to d a 90% codece iterval for the true proportio of tees that would report havig more tha 500 Facebook frieds based o this larger sample. Compare the results to those i Example 4. Calculatig the Sample Size If researchers desire a specic margi of error, the they ca use the error boud formula to calculate the required sample size. The error boud formula for a populatio proportio is EBP = ( ) ( ) p z q α Solvig for gives you a equatio for the sample size. = z α (p q ) EBP Example 5 Suppose a mobile phoe compay wats to determie the curret percetage of customers aged 50+ who use text messagig o their cell phoes. How may customers aged 50+ should the compay survey i order to be 90% codet that the estimated (sample) proportio is withi three percetage poits of the true populatio proportio of customers aged 50+ who use text messagig o their cell phoes. Solutio From the problem, we kow that EBP = 0.03 (3%=0.03) ad z α z 0.05 = because the codece level is 90%. However, i order to d, we eed to kow the estimated (sample) proportio p. Remember that q = 1 p. But, we do ot kow p yet. Sice we multiply p ad q together, we make them both equal to 0.5 because p q = (0.5)(0.5) = 0.5 results i the largest possible product. (Try other products: (0.6)(0.4) = 0.4; (0.3)(0.7) = 0.1; (0.)(0.8) = 0.16 ad so o). The largest possible product gives us the largest. This gives us a large eough sample so that we ca be 90% codet that we are withi three percetage poits of the true populatio proportio. To calculate the sample size, use the formula ad make the substitutios. = z p q EBP gives = (0.5)(0.5) 0.03 = Roud the aswer to the ext higher value. The sample size should be 75 cell phoe customers aged 50+ i order to be 90% codet that the estimated (sample) proportio is withi three percetage poits of the true populatio proportio of all customers aged 50+ who use text messagig o their cell phoes. : Exercise 11 (Solutio o p. 3.) Suppose a iteret marketig compay wats to determie the curret percetage of customers who click o ads o their smartphoes. How may customers should the compay survey i order to be 90% codet that the estimated proportio is withi ve percetage poits of the true populatio proportio of customers who click o ads o their smartphoes?

8 Coexios module: m : Class Time: Names: Studet Learig Outcomes The studet will calculate the 90% codece iterval for the mea cost of a home i the area i which this school is located. The studet will iterpret codece itervals. The studet will determie the eects of chagig coditios o the codece iterval. Collect the Data Check the Real Estate sectio i your local ewspaper. Record the sale prices for 35 radomly selected homes recetly listed i the couty. : May ewspapers list them oly oe day per week. Also, we will assume that homes come up for sale radomly. 1.Complete the table: Describe the Data Table 1 1.Compute the followig: a.x = b.s x = c. =.I words, dee the radom variable X. 3.State the estimated distributio to use. Use both words ad symbols. Fid the Codece Iterval 1.Calculate the codece iterval ad the error boud. a.codece Iterval: b.error Boud:.How much area is i both tails (combied)? α = 3.How much area is i each tail? α =

9 Coexios module: m Fill i the blaks o the graph with the area i each sectio. The, ll i the umber lie with the upper ad lower limits of the codece iterval ad the sample mea. Figure 1 5.Some studets thik that a 90% codece iterval cotais 90% of the data. Use the list of data o the rst page ad cout how may of the data values lie withi the codece iterval. What percet is this? Is this percet close to 90%? Explai why this percet should or should ot be close to 90%. Describe the Codece Iterval 1.I two to three complete seteces, explai what a codece iterval meas (i geeral), as if you were talkig to someoe who has ot take statistics..i oe to two complete seteces, explai what this codece iterval meas for this particular study. Use the Data to Costruct Codece Itervals 1.Usig the give iformatio, costruct a codece iterval for each codece level give. Codece level EBM/Error Boud Codece Iterval 50% 80% 95% 99% Table.What happes to the EBM as the codece level icreases? Does the width of the codece iterval icrease or decrease? Explai why this happes. : Class Time: Names: Studet Learig Outcomes The studet will calculate the 90% codece iterval the proportio of studets i this school who were bor i this state. The studet will iterpret codece itervals.

10 Coexios module: m The studet will determie the eects of chagig coditios o the codece iterval. Collect the Data 1.Survey the studets i your class, askig them if they were bor i this state. Let X = the umber that were bor i this state. a. = b.x =.I words, dee the radom variable P. 3.State the estimated distributio to use. Fid the Codece Iterval ad Error Boud 1.Calculate the codece iterval ad the error boud. a.codece Iterval: b.error Boud:.How much area is i both tails (combied)? α = 3.How much area is i each tail? α = 4.Fill i the blaks o the graph with the area i each sectio. The, ll i the umber lie with the upper ad lower limits of the codece iterval ad the sample proportio. Figure Describe the Codece Iterval 1.I two to three complete seteces, explai what a codece iterval meas (i geeral), as though you were talkig to someoe who has ot take statistics..i oe to two complete seteces, explai what this codece iterval meas for this particular study. 3.Costruct a codece iterval for each codece level give. Codece level EBP/Error Boud Codece Iterval 50% 80% 95% 99% Table 3 4.What happes to the EBP as the codece level icreases? Does the width of the codece iterval icrease or decrease? Explai why this happes.

11 Coexios module: m : Class Time: Names: Studet Learig Outcomes The studet will calculate a 90% codece iterval usig the give data. The studet will determie the relatioship betwee the codece level ad the percetage of costructed itervals that cotai the populatio mea. Give: Heights of 100 Wome (i Iches) Table 4 1.Table 4: Heights of 100 Wome (i Iches) lists the heights of 100 wome. Use a radom umber geerator to select te data values radomly..calculate the sample mea ad the sample stadard deviatio. Assume that the populatio stadard deviatio is kow to be 3.3 iches. With these values, costruct a 90% codece iterval for your sample of te values. Write the codece iterval you obtaied i the rst space of Table 5: 90% Codece Itervals. 3.Now write your codece iterval o the board. As others i the class write their codece itervals o the board, copy them ito Table 5: 90% Codece Itervals.

12 Coexios module: m % Codece Itervals Discussio Questios Table 5 1.The actual populatio mea for the 100 heights give Table 4: Heights of 100 Wome (i Iches) is µ = Usig the class listig of codece itervals, cout how may of them cotai the populatio mea µ; i.e., for how may itervals does the value of µ lie betwee the edpoits of the codece iterval?.divide this umber by the total umber of codece itervals geerated by the class to determie the percet of codece itervals that cotais the mea µ. Write this percet here:. 3.Is the percet of codece itervals that cotai the populatio mea µ close to 90%? 4.Suppose we had geerated 100 codece itervals. What do you thik would happe to the percet of codece itervals that cotaied the populatio mea? 5.Whe we costruct a 90% codece iterval, we say that we are 90% codet that the true populatio mea lies withi the codece iterval. Usig complete seteces, explai what we mea by this phrase. 6.Some studets thik that a 90% codece iterval cotais 90% of the data. Use the list of data give (the heights of wome) ad cout how may of the data values lie withi the codece iterval that you geerated based o that data. How may of the 100 data values lie withi your codece iterval? What percet is this? Is this percet close to 90%? 7.Explai why it does ot make sese to cout data values that lie i a codece iterval. Thik about the radom variable that is beig used i the problem. 8.Suppose you obtaied the heights of te wome ad calculated a codece iterval from this iformatio. Without kowig the populatio mea µ, would you have ay way of kowig for certai if your iterval actually cotaied the value of µ? Explai. 3 Refereces Jese, Tom. Democrats, Republicas Divided o Opiio of Music Icos. Public Policy Pollig. Available olie at (accessed July, 013). Madde, Mary, Amada Lehart, Sadra Coresi, Urs Gasser, Maeve Dugga, Aaro Smith, ad Meredith Beato. Tees, Social Media, ad Privacy. PewIteret, 013. Available olie at Social-Media-Ad-Privacy.aspx (accessed July, 013). Price Survey Research Associates Iteratioal. 013 Tee ad Privacy Maagemet Survey. Pew Research Ceter: Iteret ad America Life Project. Available olie at /media//files/questio (accessed July, 013).

13 Coexios module: m Saad, Lydia. Three i Four U.S. Workers Pla to Work Pas Retiremet Age: Slightly more say they will do this by choice rather tha ecessity. Gallup Ecoomy, 013. Available olie at four-workers-pla-work-past-retiremet-age.aspx (accessed July, 013). The Field Poll. Available olie at (accessed July, 013). Zogby. New SUNYIT/Zogby Aalytics Poll: Few Americas Worry about Emergecy Situatios Occurrig i Their Commuity; Oly oe i three have a Emergecy Pla; 70% Support Ifrastructure `Ivestmet' for Natioal Security. Zogby Aalytics, 013. Available olie at americas-either-worried-or-prepared-i-case-of-a-disaster-suyit-zogby-aalytics-poll (accessed July, 013). 5% Say Big-Time College Athletics Corrupt Educatio Process. Rasmusse Reports, 013. Available olie at (accessed July, 013). 4 Chapter Review Some statistical measures, like may survey questios, measure qualitative rather tha quatitative data. I this case, the populatio parameter beig estimated is a proportio. It is possible to create a codece iterval for the true populatio proportio followig procedures similar to those used i creatig codece itervals for populatio meas. The formulas are slightly dieret, but they follow the same reasoig. Let p represet the sample proportio, x/, where x represets the umber of successes ad represets the sample size. Let q = 1 p. The the codece iterval for a populatio proportio is give by the followig formula: (lower boud, upper boud) = (p EBP, p + EBP ) = ( p z p q, p + z ) p q The plus four method for calculatig codece itervals is a attempt to balace the error itroduced by usig estimates of the populatio proportio whe calculatig the stadard deviatio of the samplig distributio. Simply imagie four additioal trials i the study; two are successes ad two are failures. Calculate p = x+ +4, ad proceed to d the codece iterval. Whe sample sizes are small, this method has bee demostrated to provide more accurate codece itervals tha the stadard formula used for larger samples. 5 Formula Review p = x / where x represets the umber of successes ad represets the sample size. The variable p is the sample proportio ad serves as the poit estimate for the true populatio proportio. q = 1 p p N ( p, ) pq The variable p has a biomial distributio that ca be approximated with the ormal distributio show here. EBP = the error boud for a proportio = z α p q Codece iterval for a proportio: ( lower boud, upper boud) = (p EBP, p + EBP ) = ( p z p q, p + z ) p q = z α p q EBP provides the umber of participats eeded to estimate the populatio proportio with codece 1 - α ad margi of error EBP. Use the ormal distributio for a sigle populatio proportio p = x EBP = ( ) p q z α p + q = 1 The codece iterval has the format (p EBP, p + EBP). x is a poit estimate for µ p is a poit estimate for ρ s is a poit estimate for σ

14 Coexios module: m Use the followig iformatio to aswer the ext two exercises: Marketig compaies are iterested i kowig the populatio percet of wome who make the majority of household purchasig decisios. Exercise 1 Whe desigig a study to determie this populatio proportio, what is the miimum umber you would eed to survey to be 90% codet that the populatio proportio is estimated to withi 0.05? Exercise 13 (Solutio o p. 3.) If it were later determied that it was importat to be more tha 90% codet ad a ew survey were commissioed, how would it aect the miimum umber you eed to survey? Why? Use the followig iformatio to aswer the ext ve exercises: Suppose the marketig compay did do a survey. They radomly surveyed 00 households ad foud that i 10 of them, the woma made the majority of the purchasig decisios. We are iterested i the populatio proportio of households where wome make the majority of the purchasig decisios. Exercise 14 Idetify the followig: a. x = b. = c. p = Exercise 15 (Solutio o p. 3.) Dee the radom variables X ad P i words. Exercise 16 Which distributio should you use for this problem? Exercise 17 (Solutio o p. 3.) Costruct a 95% codece iterval for the populatio proportio of households where the wome make the majority of the purchasig decisios. State the codece iterval, sketch the graph, ad calculate the error boud. Exercise 18 List two diculties the compay might have i obtaiig radom results, if this survey were doe by . Use the followig iformatio to aswer the ext ve exercises: Of 1,050 radomly selected adults, 360 idetied themselves as maual laborers, 80 idetied themselves as o-maual wage earers, 50 idetied themselves as mid-level maagers, ad 160 idetied themselves as executives. I the survey, 8% of maual laborers preferred trucks, 6% of o-maual wage earers preferred trucks, 54% of mid-level maagers preferred trucks, ad 6% of executives preferred trucks. Exercise 19 (Solutio o p. 4.) We are iterested i dig the 95% codece iterval for the percet of executives who prefer trucks. Dee radom variables X ad P i words. Exercise 0 Which distributio should you use for this problem? Exercise 1 (Solutio o p. 4.) Costruct a 95% codece iterval. State the codece iterval, sketch the graph, ad calculate the error boud.

15 Coexios module: m Exercise Suppose we wat to lower the samplig error. What is oe way to accomplish that? Exercise 3 (Solutio o p. 4.) The samplig error give i the survey is ±%. Explai what the ±% meas. Use the followig iformatio to aswer the ext ve exercises: A poll of 1,00 voters asked what the most sigicat issue was i the upcomig electio. Sixty-ve percet aswered the ecoomy. We are iterested i the populatio proportio of voters who feel the ecoomy is the most importat. Exercise 4 Dee the radom variable X i words. Exercise 5 (Solutio o p. 4.) Dee the radom variable P i words. Exercise 6 Which distributio should you use for this problem? Exercise 7 (Solutio o p. 4.) Costruct a 90% codece iterval, ad state the codece iterval ad the error boud. Exercise 8 What would happe to the codece iterval if the level of codece were 95%? Use the followig iformatio to aswer the ext 16 exercises: The Ice Chalet oers dozes of dieret begiig ice-skatig classes. All of the class ames are put ito a bucket. The 5 P.M., Moday ight, ages 8 to 1, begiig ice-skatig class was picked. I that class were 64 girls ad 16 boys. Suppose that we are iterested i the true proportio of girls, ages 8 to 1, i all begiig ice-skatig classes at the Ice Chalet. Assume that the childre i the selected class are a radom sample of the populatio. Exercise 9 (Solutio o p. 5.) What is beig couted? Exercise 30 I words, dee the radom variable X. Exercise 31 (Solutio o p. 5.) Calculate the followig: a. x = b. = c. p = Exercise 3 State the estimated distributio of X. X Exercise 33 (Solutio o p. 5.) Dee a ew radom variable P. What is p estimatig? Exercise 34 I words, dee the radom variable P. Exercise 35 (Solutio o p. 5.) State the estimated distributio of P. Costruct a 9% Codece Iterval for the true proportio of girls i the ages 8 to 1 begiig ice-skatig classes at the Ice Chalet. Exercise 36 How much area is i both tails (combied)?

16 Coexios module: m Exercise 37 (Solutio o p. 5.) How much area is i each tail? Exercise 38 Calculate the followig: a. lower limit b. upper limit c. error boud Exercise 39 (Solutio o p. 5.) The 9% codece iterval is. Exercise 40 Fill i the blaks o the graph with the areas, upper ad lower limits of the codece iterval, ad the sample proportio. Figure 3 Exercise 41 (Solutio o p. 5.) I oe complete setece, explai what the iterval meas. Exercise 4 Usig the same p ad level of codece, suppose that were icreased to 100. Would the error boud become larger or smaller? How do you kow? Exercise 43 (Solutio o p. 5.) Usig the same p ad = 80, how would the error boud chage if the codece level were icreased to 98%? Why? Exercise 44 If you decreased the allowable error boud, why would the miimum sample size icrease (keepig the same level of codece)?

17 Coexios module: m Homework Exercise 45 (Solutio o p. 5.) Isurace compaies are iterested i kowig the populatio percet of drivers who always buckle up before ridig i a car. a. Whe desigig a study to determie this populatio proportio, what is the miimum umber you would eed to survey to be 95% codet that the populatio proportio is estimated to withi 0.03? b. If it were later determied that it was importat to be more tha 95% codet ad a ew survey was commissioed, how would that aect the miimum umber you would eed to survey? Why? Exercise 46 Suppose that the isurace compaies did do a survey. They radomly surveyed 400 drivers ad foud that 30 claimed they always buckle up. We are iterested i the populatio proportio of drivers who claim they always buckle up. a. i. x = ii. = iii. p = b. Dee the radom variables X ad P, i words. c. Which distributio should you use for this problem? Explai your choice. d. Costruct a 95% codece iterval for the populatio proportio who claim they always buckle up. i. State the codece iterval. ii. Sketch the graph. iii. Calculate the error boud. e. If this survey were doe by telephoe, list three diculties the compaies might have i obtaiig radom results. Exercise 47 (Solutio o p. 5.) Accordig to a recet survey of 1,00 people, 61% feel that the presidet is doig a acceptable job. We are iterested i the populatio proportio of people who feel the presidet is doig a acceptable job. a. Dee the radom variables X ad P i words. b. Which distributio should you use for this problem? Explai your choice. c. Costruct a 90% codece iterval for the populatio proportio of people who feel the presidet is doig a acceptable job. i. State the codece iterval. ii. Sketch the graph. iii. Calculate the error boud. Exercise 48 A article regardig iterracial datig ad marriage recetly appeared i the Washigto Post. Of the 1,709 radomly selected adults, 315 idetied themselves as Latios, 33 idetied themselves as blacks, 54 idetied themselves as Asias, ad 779 idetied themselves as whites. I this survey, 86% of blacks said that they would welcome a white perso ito their families. Amog Asias, 77% would welcome a white perso ito their families, 71% would welcome a Latio, ad 66% would welcome a black perso.

18 Coexios module: m a. We are iterested i dig the 95% codece iterval for the percet of all black adults who would welcome a white perso ito their families. Dee the radom variables X ad P, i words. b. Which distributio should you use for this problem? Explai your choice. c. Costruct a 95% codece iterval. i. State the codece iterval. ii. Sketch the graph. iii. Calculate the error boud. Exercise 49 (Solutio o p. 5.) Refer to the iformatio i Exercise. a. Costruct three 95% codece itervals. i. percet of all Asias who would welcome a white perso ito their families. ii. percet of all Asias who would welcome a Latio ito their families. iii. percet of all Asias who would welcome a black perso ito their families. b. Eve though the three poit estimates are dieret, do ay of the codece itervals overlap? Which? c. For ay itervals that do overlap, i words, what does this imply about the sigicace of the diereces i the true proportios? d. For ay itervals that do ot overlap, i words, what does this imply about the sigicace of the diereces i the true proportios? Exercise 50 Staford Uiversity coducted a study of whether ruig is healthy for me ad wome over age 50. Durig the rst eight years of the study, 1.5% of the 451 members of the 50-Plus Fitess Associatio died. We are iterested i the proportio of people over 50 who ra ad died i the same eight-year period. a. Dee the radom variables X ad P i words. b. Which distributio should you use for this problem? Explai your choice. c. Costruct a 97% codece iterval for the populatio proportio of people over 50 who ra ad died i the same eightyear period. i. State the codece iterval. ii. Sketch the graph. iii. Calculate the error boud. d. Explai what a 97% codece iterval meas for this study. Exercise 51 (Solutio o p. 6.) A telephoe poll of 1,000 adult Americas was reported i a issue of Time Magazie. Oe of the questios asked was What is the mai problem facig the coutry? Twety percet aswered crime. We are iterested i the populatio proportio of adult Americas who feel that crime is the mai problem. a. Dee the radom variables X ad P i words. b. Which distributio should you use for this problem? Explai your choice. c. Costruct a 95% codece iterval for the populatio proportio of adult Americas who feel that crime is the mai problem. i. State the codece iterval.

19 Coexios module: m ii. Sketch the graph. iii. Calculate the error boud. d. Suppose we wat to lower the samplig error. What is oe way to accomplish that? e. The samplig error give by Yakelovich Parters, Ic. (which coducted the poll) is ±3%. I oe to three complete seteces, explai what the ±3% represets. Exercise 5 Refer to Exercise. Aother questio i the poll was [How much are] you worried about the quality of educatio i our schools? Sixty-three percet respoded a lot. We are iterested i the populatio proportio of adult Americas who are worried a lot about the quality of educatio i our schools. a. Dee the radom variables X ad P i words. b. Which distributio should you use for this problem? Explai your choice. c. Costruct a 95% codece iterval for the populatio proportio of adult Americas who are worried a lot about the quality of educatio i our schools. i. State the codece iterval. ii. Sketch the graph. iii. Calculate the error boud. d. The samplig error give by Yakelovich Parters, Ic. (which coducted the poll) is ±3%. I oe to three complete seteces, explai what the ±3% represets. Use the followig iformatio to aswer the ext three exercises: Accordig to a Field Poll, 79% of Califoria adults (actual results are 400 out of 506 surveyed) feel that educatio ad our schools is oe of the top issues facig Califoria. We wish to costruct a 90% codece iterval for the true proportio of Califoria adults who feel that educatio ad the schools is oe of the top issues facig Califoria. Exercise 53 (Solutio o p. 6.) A poit estimate for the true populatio proportio is: a b. 1.7 c d. 400 Exercise 54 A 90% codece iterval for the populatio proportio is. a. (0.761, 0.80) b. (0.15, 0.188) c. (0.755, 0.86) d. (0.130, 0.183) Exercise 55 (Solutio o p. 6.) The error boud is approximately. a b c d

20 Coexios module: m Use the followig iformatio to aswer the ext two exercises: Five hudred ad eleve (511) homes i a certai souther Califoria commuity are radomly surveyed to determie if they meet miimal earthquake preparedess recommedatios. Oe hudred sevety-three (173) of the homes surveyed met the miimum recommedatios for earthquake preparedess, ad 338 did ot. Exercise 56 Fid the codece iterval at the 90% Codece Level for the true populatio proportio of souther Califoria commuity homes meetig at least the miimum recommedatios for earthquake preparedess. a. (0.975, ) b. (0.670, ) c. (0.3041, ) d. (0.604, 0.705) Exercise 57 (Solutio o p. 6.) The poit estimate for the populatio proportio of homes that do ot meet the miimum recommedatios for earthquake preparedess is. a b c. 173 d. 338 Exercise 58 O May 3, 013, Gallup reported that of the 1,005 people surveyed, 76% of U.S. workers believe that they will cotiue workig past retiremet age. The codece level for this study was reported at 95% with a ±3% margi of error. a. Determie the estimated proportio from the sample. b. Determie the sample size. c. Idetify CL ad α. d. Calculate the error boud based o the iformatio provided. e. Compare the error boud i part d to the margi of error reported by Gallup. Explai ay diereces betwee the values. f. Create a codece iterval for the results of this study. g. A reporter is coverig the release of this study for a local ews statio. How should she explai the codece iterval to her audiece? Exercise 59 (Solutio o p. 6.) A atioal survey of 1,000 adults was coducted o May 13, 013 by Rasmusse Reports. It cocluded with 95% codece that 49% to 55% of Americas believe that big-time college sports programs corrupt the process of higher educatio. a. Fid the poit estimate ad the error boud for this codece iterval. b. Ca we (with 95% codece) coclude that more tha half of all America adults believe this? c. Use the poit estimate from part a ad = 1,000 to calculate a 75% codece iterval for the proportio of America adults that believe that major college sports programs corrupt higher educatio. d. Ca we (with 75% codece) coclude that at least half of all America adults believe this?

21 Coexios module: m Exercise 60 Public Policy Pollig recetly coducted a survey askig adults across the U.S. about music prefereces. Whe asked, 80 of the 571 participats admitted that they have illegally dowloaded music. a. Create a 99% codece iterval for the true proportio of America adults who have illegally dowloaded music. b. This survey was coducted through automated telephoe iterviews o May 6 ad 7, 013. The error boud of the survey compesates for samplig error, or atural variability amog samples. List some factors that could aect the survey's outcome that are ot covered by the margi of error. c. Without performig ay calculatios, describe how the codece iterval would chage if the codece level chaged from 99% to 90%. Exercise 61 (Solutio o p. 6.) You pla to coduct a survey o your college campus to lear about the political awareess of studets. You wat to estimate the true proportio of college studets o your campus who voted i the 01 presidetial electio with 95% codece ad a margi of error o greater tha ve percet. How may studets must you iterview? Exercise 6 I a recet Zogby Iteratioal Poll, ie of 48 respodets rated the likelihood of a terrorist attack i their commuity as likely or very likely. Use the plus four method to create a 97% codece iterval for the proportio of America adults who believe that a terrorist attack i their commuity is likely or very likely. Explai what this codece iterval meas i the cotext of the problem.

22 Coexios module: m46999 Solutios to Exercises i this Module to Exercise (p. 3) (0.3315, 0.455) to Exercise (p. 4) (0.7731, 0.869); We estimate with 90% codece that the true percet of all studets i the district who are agaist the ew legislatio is betwee 77.31% ad 8.69%. A to Exercise (p. 4): Solutio A Sixty-eight percet (68%) of studets ow a ipod ad a smart phoe. p = 0.68 q = 1 p = = 0.3 Sice CL = 0.97, we kow α = = 0.03 ad α = The area to the left of z is 0.015, ad the area to the right of z is = Usig the TI 83, 83+, or 84+ calculator fuctio IvNorm(.985,0,1), z =.17 EP B = ( ) p z q 0.68 (0.3) α = p EPB = = p + EPB = = We are 97% codet that the true proportio of all studets who ow a ipod ad a smart phoe is betwee ad B to Exercise (p. 4): Solutio B : Press STAT ad arrow over to TESTS. Arrow dow to A:1-PropZit. Press ENTER. Arrow dow to x ad eter 300*0.68. Arrow dow to ad eter 300. Arrow dow to C-Level ad eter Arrow dow to Calculate ad press ENTER. The codece iterval is (0.6531, ). A to Exercise (p. 6): Solutio A Usig plus four, we have x = 31 + = 33 ad = = 69. p ' = q = 1 p = = 0.5 Sice CL = 0.96, we kow α = = 0.04 ad α = 0.0. z 0.0 =.054 EP B = ( z α ) p q = (.054) ( ) (0.478)(0.5) p EPB = = p + EPB = = 0.60 We are 96% codet that betwee 35.4% ad 60.% of all freshme at State U have declared a major. B to Exercise (p. 6): Solutio B : Press STAT ad arrow over to TESTS. Arrow dow to A:1-PropZit. Press ENTER. Arrow dow to x ad eter 33. Arrow dow to ad eter 69. Arrow dow to C-Level ad eter 0.96.

23 Coexios module: m Arrow dow to Calculate ad press ENTER. The codece iterval is (0.355, 0.60). A to Exercise (p. 6): Solutio A Usig plus-four, we have x = = 161 ad = = 59. p = q = 1 p = = 0.78 Sice CL = 0.90, we kow ) α = = 0.10 ad ( ) α = 0.05 EP B = ( z α ) ( p ' q ' = (1.645) (0.7)(0.78) p EPB = = 0.4 p + EPB = = 0.30 We are 90% codet that betwee 4.% ad 30.% of all tees would report havig more tha 500 frieds o Facebook. B to Exercise (p. 6): Solutio B : Press STAT ad arrow over to TESTS. Arrow dow to A:1-PropZit. Press ENTER. Arrow dow to x ad eter 161. Arrow dow to ad eter 59. Arrow dow to C-Level ad eter Arrow dow to Calculate ad press ENTER. The codece iterval is (0.4, 0.30). Coclusio: The codece iterval for the larger sample is arrower tha the iterval from Example 4. Larger samples will always yield more precise codece itervals tha smaller samples. The plus four method has a greater impact o the smaller sample. It shifts the poit estimate from 0.6 (13/50) to 0.78 (15/54). It has a smaller impact o the EPB, chagig it from 0.10 to I the larger sample, the poit estimate udergoes a smaller shift: from 0.70 (159/588) to 0.7 (161/59). It is easy to see that the plus-four method has the greatest impact o smaller samples. Solutio to Exercise (p. 7) 71 customers should be surveyed.check the Real Estate sectio i your local Solutio to Exercise (p. 14) It would decrease, because the z-score would decrease, which reducig the umerator ad lowerig the umber. Solutio to Exercise (p. 14) X is the umber of successes where the woma makes the majority of the purchasig decisios for the household. P is the percetage of households sampled where the woma makes the majority of the purchasig decisios for the household. Solutio to Exercise (p. 14) CI: (0.531, )

24 Coexios module: m Figure 4 EBM: Solutio to Exercise (p. 14) X is the umber of successes where a executive prefers a truck. P is the percetage of executives sampled who prefer a truck. Solutio to Exercise (p. 14) CI: (0.1943, ) Figure 5 EBM: Solutio to Exercise (p. 15) The samplig error meas that the true mea ca be % above or below the sample mea. Solutio to Exercise (p. 15) P is the proportio of voters sampled who said the ecoomy is the most importat issue i the upcomig electio.

25 Coexios module: m Solutio to Exercise (p. 15) CI: (0.6735, ) EBM: Solutio to Exercise (p. 15) The umber of girls, ages 8 to 1, i the 5 P.M. Moday ight begiig ice-skatig class. Solutio to Exercise (p. 15) a. x = 64 b. = 80 c. p = 0.8 Solutio to Exercise (p. 15) p Solutio to Exercise (p. 15) P N ( 0.8, (0.8)(0.) 80 ). (0.7171, ). Solutio to Exercise (p. 16) 0.04 Solutio to Exercise (p. 16) (0.7; 0.88) Solutio to Exercise (p. 16) With 9% codece, we estimate the proportio of girls, ages 8 to 1, i a begiig ice-skatig class at the Ice Chalet to be betwee 7% ad 88%. Solutio to Exercise (p. 16) The error boud would icrease. Assumig all other variables are kept costat, as the codece level icreases, the area uder the curve correspodig to the codece level becomes larger, which creates a wider iterval ad thus a larger error. Solutio to Exercise (p. 17) a. 1,068 b. The sample size would eed to be icreased sice the critical value icreases as the codece level icreases. Solutio to Exercise (p. 17) a. X = the umber of people who feel that the presidet is doig a acceptable job; P ( = the proportio of people i a sample who feel that the presidet is doig a acceptable job. ) b. N 0.61, (0.61)(0.39) 100 c. i. CI: (0.59, 0.63) ii. Check studet's solutio iii. EBM: 0.0 Solutio to Exercise (p. 18) a. i. (0.7, 0.8) ii. (0.65, 0.76) iii. (0.60, 0.7) b. Yes, the itervals (0.7, 0.8) ad (0.65, 0.76) overlap, ad the itervals (0.65, 0.76) ad (0.60, 0.7) overlap. c. We ca say that there does ot appear to be a sigicat dierece betwee the proportio of Asia adults who say that their families would welcome a white perso ito their families ad the proportio of Asia adults who say that their families would welcome a Latio perso ito their families.

26 Coexios module: m d. We ca say that there is a sigicat dierece betwee the proportio of Asia adults who say that their families would welcome a white perso ito their families ad the proportio of Asia adults who say that their families would welcome a black perso ito their families. Solutio to Exercise (p. 18) a. X = the umber of adult Americas who feel that crime is the mai problem; P = the proportio of adult Americas who feel that crime is the mai problem b. Sice we are estimatig a proportio, give P = 0. ad = 1000, the distributio we should use is N ( 0., (0.)(0.8) 1000 ). c. i. CI: (0.18, 0.) ii. Check studet's solutio. iii. EBM: 0.0 d. Oe way to lower the samplig error is to icrease the sample size. e. The stated ± 3% represets the maximum error boud. This meas that those doig the study are reportig a maximum error of 3%. Thus, they estimate the percetage of adult Americas who feel that crime is the mai problem to be betwee 18% ad %. Solutio to Exercise (p. 19) c Solutio to Exercise (p. 19) d Solutio to Exercise (p. 0) a Solutio to Exercise (p. 0) a. p ( ) = = 0.5; EBP = = 0.03 b. No, the codece iterval icludes values less tha or equal to It is possible that less tha half of the populatio believe this. c. CL = 0.75, so α = = 0.5 ad α = 0.15 z α = (The area to the right of this z is 0.15, so the area to the left is = ) EBP = (1.150) 0.5(0.48) 1, (p - EBP, p + EBP) = ( , ) = (0.50, 0.538) Alterate Solutio : STAT TESTS A: 1-PropZiterval with x = (0.5)(1,000), = 1,000, CL = Aswer is (0.50, 0.538) d. Yes this iterval does ot fall less tha 0.50 so we ca coclude that at least half of all America adults believe that major sports programs corrupt educatio but we do so with oly 75% codece. Solutio to Exercise (p. 1) CL = 0.95 α = = 0.05 α = 0.05 z α = Use p = q = 0.5. = z α p q EBP = 1.96 (0.5)(0.5) 0.05 = You eed to iterview at least 385 studets to estimate the proportio to withi 5% at 95% codece. Glossary Deitio 1: Biomial Distributio a discrete radom variable (RV) which arises from Beroulli trials; there are a xed umber,, of idepedet trials. Idepedet meas that the result of ay trial (for example, trial 1) does ot aect the results of the followig trials, ad all trials are coducted uder the same coditios.

27 Coexios module: m Uder these circumstaces the biomial RV X is deed as the umber of successes i trials. The otatio is: X B(,p). The mea is µ = p ad the stadard deviatio is σ = pq. The probability of exactly x successes i trials is P (X = x) = ( x ) p x q x. Deitio : Error Boud for a Populatio Proportio (EBP) the margi of error; depeds o the codece level, the sample size, ad the estimated (from the sample) proportio of successes.