Confidence intervals for proportions

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Cofidece itervals for roortios Studet Activity 7 8 9 0 2 TI-Nsire Ivestigatio Studet 60 mi Itroductio From revious activity This activity assumes kowledge of the material covered i the activity

1 Cofidece itervals for roortios Studet Activity TI-Nsire Ivestigatio Studet 60 mi Itroductio From revious activity This activity assumes kowledge of the material covered i the activity Distributio of samle roortios. That activity focused o key features of the samlig distributio of samle roortios. Through simulated radom samlig for various samle sizes,, ad oulatio roortios,, it was foud that: The samlig distributio of the samle roortio, ˆP, is cetred at. For a give samle size, the sread ad symmetry of the samlig distributio of ˆP deeds o the oulatio roortio,. The sread is greatest for 0.5. The samlig distributio also becomes more symmetric closer to 0.5. For a give value of, the sread ad symmetry of the samlig distributio of ˆP deeds o the samle size,. The sread decreases as the samle size icreases, while symmetry icreases with samle size. For large samles, the samlig distributio of ˆP ca be modelled by a ormal distributio, 2 ˆ P Pˆ, where ˆ EPˆ P ad ˆ SDPˆ P N,. Overview of this activity I this activity you will ivestigate the recisio of the estimator, ˆ, ad cofidece itervals for the oulatio roortio,. Through simulatio, you will exlore variatios i cofidece itervals betwee samles, ad come to uderstad the sigificace of cofidece itervals for. Iterval estimates of the oulatio roortio If we take a radom samle from a large oulatio, such as a oiio oll, we will obtai a sigle value, ˆ, that rovides us with a estimate of the true oulatio roortio,. But it is ulikely that ˆ will be exactly equal to, so it is valuable to combie the estimate with iformatio about the recisio of the estimate. Simulatios for the recisio of the estimator ˆ Oe the TI-Nsire documet CI_roortios. Navigate to Page.2 ad follow the istructios to seed the radom umber geerator. Navigate to Page.3. Suose that i a large oulatio the roortio of the oulatio with a articular attribute is 0.3. Samles of size 40 will be draw from this oulatio. Later, you will chage the values of ad o Page.3, ad reeat the simulatio usig the ew values. Simulatio of 00 samles with = 40 ad = 0.3 Texas Istrumets 205. You may coy, commuicate ad modify this material for o-commercial educatioal uroses rovided all ackowledgemets associated with this material are maitaied. Author: Frak Moya

2 Distributio of samle roortios - Studet Worksheet 2 Navigate to Page.4. The left vertical boudary idicates the value of The vertical lie at 0.30 idicates the value of The right vertical boudary idicates the value of The variable vertical lie idicates the mea of ˆP 2 2 Questio What is the sigificace of the values of 2 ad 2? Use the slider o the to left-had corer of Page.4 to simulate drawig radom samles of size from a large oulatio. Each time you click the right or left slider arrow, a ew samle is draw, ad the samle roortio for that samle is added to the sreadsheet ad to the grah. The umber of samles draw is show o the to right-had side. Look for samle roortios, ˆ, with values outside the boudaries 2. Sto whe the umber of samles draw is 00. Questio 2 a. From the 00 samles draw, how may of the observed values of ˆ were outside the boudaries defied by 2? b. I the log ru, what is the aroximate ercetage of observed values of ˆ that you would exect to lie outside of these boudaries? c. Exlai your aswer to art b. above. Simulatio of 00 samles with differet values of ad Navigate back to Page.3. Choose differet values for ad by editig the value i the Mathbox the ressig [Eter]. Navigate to Page.4. Reset the simulatio by selectig the formula cell A= of the sreadsheet, the ress [Eter][Eter]. The boudaries for 2 will have bee automatically recalculated ad show o the grah. Chage the widow settigs for the grah, as aroriate. (To chage widow settigs, click o a emty art of the grah, the [Ctrl]+[Meu] > Zoom > Widow settigs.) Use the slider o the to right-had corer to simulate drawig 00 radom samles, as described earlier. Questio 3 a. From the 00 samles draw, how may of the observed values of ˆ were outside the boudaries i this case? Is the result cosistet with your exectatios? Texas Istrumets 205. You may coy, commuicate ad modify this material for o-commercial educatioal uroses rovided all ackowledgemets associated with this material are maitaied. Author: Frak Moya

. Usig this grah, or otherwise, show that, where is a oulatio roortio. 4 c. Hece show that for ay value of, SD ˆ. 2 d. Discuss the sigificace of the result i art c. above.

3 Distributio of samle roortios - Studet Worksheet 3 b. A istructio o Page.3 suggests that whe choosig values of ad, the followig should be met: 0 ad 0. Why do you thik that these criteria eed to be met? Plaig the recisio of the estimator Questio 4 a. If 0.3, fid the miimum samle size required to esure that SD ˆ 0.0. b. Navigate to Page 2.. Usig this grah, or otherwise, show that, where is a oulatio roortio. 4 c. Hece show that for ay value of, SD ˆ. 2 d. Discuss the sigificace of the result i art c. above. Stadard error of ˆ I the simulatios from Page.3, we saw that if a radom samle of size is draw from a large oulatio, where the oulatio roortio is, the it is very likely (aroximately 95%) that the samle roortio, ˆ, will be withi two stadard deviatios of. That is Pr 2 ˆ However, there is a ractical roblem with usig the stadard deviatio of ˆ, SD ˆ measure of the recisio of the estimator. Namely, SD ˆ deeds o kowig the value of. But whe we are usig ˆ as a estimate of, it is because is ukow to us (otherwise, why would we eed to estimate it?). SD ca be. 2 SD ˆ by usig ˆ as a aroximatio to. From Questio 4 c., we kow that SD ˆ, so this the maximum that ˆ We ca, however, obtai a better estimate of This gives the stadard error of the estimator, SE ˆ, where ˆ ˆ SE ˆ. The stadard error ca be used i lace of the stadard deviatio, which it estimates. I articular, ˆ ˆ ˆ Pr 2SE 2SE 0.95., as a Simulatio to comare SD ˆ ad SE ˆ I this simulatio, you will observe a comariso of the itervals 2SD ˆ, 2SD ˆ 2SE ˆ, 2SE ˆ. I Page ad 3., the default values have bee set to 00, 0.6. Navigate to Page 3.2. The grah shows the boudaries for the iterval 2SD ˆ, 2SD ˆ as aroximately0.50,0.70. Use the slider arrows to draw a ew samle. The iterval for 2SE ˆ, 2SE ˆ is recalculated for each samle, ad show o the grah ad i the Mathbox below the grah. Texas Istrumets 205. You may coy, commuicate ad modify this material for o-commercial educatioal uroses rovided all ackowledgemets associated with this material are maitaied. Author: Frak Moya

4 Distributio of samle roortios - Studet Worksheet 4 The simulatio ca be reeated with ew values of ad selected o Page 2.. Questio 5 From the results of the simulatio, discuss the reasoableess of usig stadard error, rather tha stadard deviatio, as a idicator of the recisio of the estimator, ˆ. Cofidece itervals for the oulatio roortio, I the revious sectio we saw that the stadard error is a idicator of the recisio of the samle statistic: the samle roortio, ˆ. However, what we are much more iterested i is the recisio to which we ca estimate the oulatio arameter: the oulatio roortio,. We are lookig for is a rage of values - that is, a iterval that we are reasoably sure cotais the true value of. This is called a cofidece iterval. I the simulatios that will be used to ivestigate cofidece itervals, we will assume that values of ad are aroriate for the samlig distributio of ˆP to be modelled by a ormal distributio, 2 ˆ P Pˆ, where ˆ EPˆ P ad ˆ SDPˆ P N,. Levels of cofidece ad quatiles for the stadard ormal distributio Z N 0,, the stadard Normal Navigate to Page 4.2. The grah of radom variable, is show. Use the slider to adjust the ercetage of the area uder the curve to be shaded, betwee 50% ad 99%, symmetrically about the origi. Questio 6 a. What is the relatioshi betwee the slider value ad the shaded area? b. What is the sigificace of the value labelled z? c. What is the sigificace of the gree lie segmet? From Page 4.2 we ca see that Z Pr , where z.96 is the quatile of the stadard Normal distributio that symmetrically shades aroximately 95% of the area uder the curve. Questio 7 Use the slider o Page 4.2 to hel you comlete the followig table, statig corresodig value of z to two decimal laces. Cofidece level Stadard Normal (ercetage of total area that is shaded) quatile ( z ) 50% 75% 90% 95%.96 99% Stadard Normal aroximatio of the distributio of ˆP Assume the aroximatio ˆ 2 N ˆ, ˆ Pˆ ˆ Pˆ P Stadardisig gives Z ˆ P, where ad ˆP P P P. Pˆ, ad let N0, Z. Texas Istrumets 205. You may coy, commuicate ad modify this material for o-commercial educatioal uroses rovided all ackowledgemets associated with this material are maitaied. Author: Frak Moya

Distributio of samle roortios - Studet Worksheet 5 95% cofidece iterval for the true oulatio roortio, From Page 4.2 we kow that: Pr.96 Z.96 0.95, therefore ˆ Pr P.96.96 0.95. Rearragig the iequalities gives ˆ ˆ Pr.

5 Distributio of samle roortios - Studet Worksheet 5 95% cofidece iterval for the true oulatio roortio, From Page 4.2 we kow that: Pr.96 Z , therefore ˆ Pr P Rearragig the iequalities gives ˆ ˆ Pr.96 P P However, whe usig samlig to estimate a oulatio roortio (for examle, i a oiio oll), we have a sigle observed value ˆ of the radom variable ˆP. Furthermore, as discussed earlier, the stadard deviatio eeds to be aroximated by the stadard ˆ ˆ error: ˆ SE P, as is ukow. The aroximate 95% cofidece iterval for is calculated as ˆ ˆ ˆ ˆ ˆ.96, ˆ.96. Margi of error For a cofidece iterval, the rage of values above ad below the samle roortio, ˆ, is called the margi of error. For a 95% cofidece iterval, the margi of error ca be writte course, if we kew the value of SDP ˆ, we could use M.96SD Pˆ ˆ ˆ.96 M, rather tha the stadard error aroximatio. Questio 8 a. Refer back to Questio 4 c. For a 95% cofidece iterval with samle size, write a exressio, i terms of, for the greatest ossible value of the margi of error. b. Oiio olls usually reort a margi of error, for a 95% cofidece iterval, of aroud 2 or 3 ercet. Suose that a ollster surveyed a radom samle of 600 eole ad calculated a 95% cofidece iterval. What is the maximum margi of error that could be obtaied? Write the aswer as a ercetage, correct to two decimal laces. Uderstadig the sigificace of a 95% cofidece itervals for Suose that i a articular city 30% of households are coected to a ew iteret etwork, BNB. Navigate to Page 5.2. Use the slider arrow to simulate drawig a radom samle of size 200 from this oulatio, where 0.3. The value of the samle roortio, ˆ, is used to calculate ad grah a 95% cofidece itervals for. Note that i the scree-shot show, the iterval cotais the value 0.3. For the samle, the umerical statistics are show o the left-had ael of the scree, where cl = cofidece level, x = umber of successes i the samle, se = stadard error, me = margi of error ad c_low ad c_u are the edoits of the cofidece iterval.. Of Texas Istrumets 205. You may coy, commuicate ad modify this material for o-commercial educatioal uroses rovided all ackowledgemets associated with this material are maitaied. Author: Frak Moya

Distributio of samle roortios - Studet Worksheet 6 Navigate to Page 5.3. As you were drawig samles o Page 5.2, the sreadsheet was beig oulated with the cofidece iterval edoits.

How may of the 00 cofidece itervals geerated cotai the value of? Reset the slider value to (click o slider, the [Ctrl]+[Meu] > Settigs). Draw a further 00 samles.

6 Distributio of samle roortios - Studet Worksheet 6 Navigate to Page 5.3. As you were drawig samles o Page 5.2, the sreadsheet was beig oulated with the cofidece iterval edoits. Colum C records yes if the iterval cotais, ad o otherwise. Cells E2 ad E4 record ad umber ad ercetage of yes. Questio 9 a. Usig the slider, draw 00 samles. How may of the 00 cofidece itervals geerated cotai the value of? Reset the slider value to (click o slider, the [Ctrl]+[Meu] > Settigs). Draw a further 00 samles. The sreadsheet will cotiue to be oulated. b. From your 200 samles, what ercetage of cofidece itervals cotaied the value of? c. For a cofidece level of 0.95, i the log ru, what ercetage of cofidece itervals do you thik will cotai the true value of the oulatio roortio? I the revious simulatio we saw that a cofidece iterval is obtaied from the radom variable ˆP, so the iterval itself ca be cosidered a radom iterval; the iterval varies from oe samle to the ext, just as the value of the radom variable does. Questio 0 Suose that i a simulatio similar to that o Page 5.2 we geerate 00 ideedet 95% cofidece itervals for, where the value of is ukow to us. Let Y be the umber of itervals that cotai the value of. a. Exlai why Y ca be regarded as a biomially distributed radom variable. Bi, E Y? b. If Y a b, what are the values of a, b ad Navigate to Page 6.. Carry out the followig calculatios o the bottom Calculator aels of Pages 6. ad 6.2. c. Suose that 00 ew 95% cofidece itervals are to be geerated. i. Fid the robability, correct to two decimal laces, that exactly 5 cofidece itervals do ot cotai the value of true value of. ii. Fid the robability, correct to two decimal laces, that at least 5 cofidece itervals do ot cotai the value of true value of. Chagig the level of cofidece A aroximate C % cofidece iterval for is calculated as ˆ ˆ ˆ ˆ ˆ z, ˆ z, where z is the quatile of the stadard Normal distributio. Questio Refer to Page 4.2 ad Questio 7 to comlete the followig. Cofidece level Stadard Normal quatile ( z ) C % cofidece iterval 50% ˆ ˆ ˆ ˆ ˆ, ˆ 75% ˆ ˆ ˆ ˆ ˆ, ˆ 99% ˆ ˆ ˆ ˆ ˆ, ˆ Texas Istrumets 205. You may coy, commuicate ad modify this material for o-commercial educatioal uroses rovided all ackowledgemets associated with this material are maitaied. Author: Frak Moya

Distributio of samle roortios - Studet Worksheet 7 Comarig C % cofidece itervals for Navigate to Page 7.2. Use the slider arrow to simulate drawig a radom samle of size 200 from this oulatio, where 0.

Note that i the scree-shot show, the iterval cotais the value 0.3. Navigate to Page 7.3. As you were drawig samles o Page 7.

7 Distributio of samle roortios - Studet Worksheet 7 Comarig C % cofidece itervals for Navigate to Page 7.2. Use the slider arrow to simulate drawig a radom samle of size 200 from this oulatio, where 0.3. The value of the samle roortio, ˆ, is used to calculate ad grah 4 differet cofidece itervals for, with cofidece levels of 50%, 75%, 95% ad 99%. Note that i the scree-shot show, the iterval cotais the value 0.3. Navigate to Page 7.3. As you were drawig samles o Page 7.2, the sreadsheet was beig oulated with the cofidece iterval edoits for the four cofidece levels. The ercetage of cofidece itervals cotaiig the value of is recorded i Colum C: Cell C 50% cofidece to Cell C4 99% cofidece. Navigate to Page 8.3. Problem 8 is the same as Problem 7, excet that the cofidece iterval data for 00 samles has already bee catured o Page 8.3. Use the slider arrows o Page 8.2 to select a additioal 50 samles. The data o Page 8.3 is automatically udated. Use the grahs ad data to aswer the followig questios. Questio 2 a. For a cofidece level of C %, i the log ru, what ercetage of cofidece itervals k will cotai the true value of the oulatio roortio? b. Cosider the followig statemets, made by two statisticias with regards to cofidece itervals: there is a trade-off betwee the level of cofidece ad the recisio of the iterval; there is a trade-off betwee margi of error ad level of cofidece. Exlai what these statemet meas. END OF ACTIVITY Texas Istrumets 205. You may coy, commuicate ad modify this material for o-commercial educatioal uroses rovided all ackowledgemets associated with this material are maitaied. Author: Frak Moya

Distribution of Sample Proportions

Distribution of Sample Proportions Distributio of Samle Proortios Probability ad statistics Aswers & Teacher Notes TI-Nsire Ivestigatio Studet 90 mi 7 8 9 10 11 12 Itroductio From revious activity: This activity assumes kowledge of the