Probability and Statistics Estimation Chapter 7 Section 3 Estimating p in the Binomial Distribution

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1 Probability ad Statistics Estimatio Chapter 7 Sectio 3 Estimatig p i the Biomial Distributio Essetial Questio: How are cofidece itervals used to determie the rage for the value of p? Studet Objectives: The studet will determie the cofidece iterval for proportios usig the formula. The studet will determie the cofidece iterval for proportios usig the calculator. The studet will determie the miimum size required for a give cofidece level where the proportios are kow. The studet will determie the miimum size required for a give cofidece level where the proportios are ukow. Terms: Cofidece iterval E p ˆp q ˆq r Roudig up ceilig fuctio Sample size! z c Graphig Calculator Skills: Calculatig the cofidece itervals usig the stat package from the graphig calculator Eter a list of data Eter the stats of the sample data Performig a 1-ProbZIt (optio A) o the calculator

2 Key Cocepts: For large samples of size < 30. ˆp = r The estimate are GOOD estimates if ˆ p > 5 ad ˆ q > 5. P (!z c < z < z c ) = c ad P( 0 < z < z c ) = z 2 " P$ ˆp! z c < p < ˆp + z c % ' & = c E = z c ˆp! z c ˆp! E < p < ˆp + E < p < ˆp + z c ˆp = sample proportio of success E = z c, where ˆq = 1! ˆp c = cofidece level 0 < c < 1 z c = critical value = sample size

3 Determiig the sample size based o the proportio of the sample. = pq z 2! c $ " E % &, if we have a prelimiary estimate for p OR = 1 2! z c $ 4 " E % &, if we do NOT have a pelimiary estimate for p E = Error, how far ca you differ from the actual mea c = cofidece level 0 < c < 1 z c = critical value p = proportio of the sample q = 1! p = sample size Sample Questios: 1. A survey of LVC studets idicated that out of the graduates 128 studets had chaged their majors at least oce while they were erolled. Form a 95% cofidece iterval for the proportio of studets at LVC that will chaged their major at least oce before graduatio.

4 2. Determie a 90% cofidece iterval usig the iformatio i questio 1 by usig your calculator. 3. We wish to do a study of studets at LVC to determie a 90% cofidece iterval for the proportio of studets who chage their major at least oce while erolled at LVC. We eed to be withi 0.01 of the true proportio. If o prelimiary study was available, how large should we make our sample?

5 4. Suppose we do our prelimiary study for the estimatio of p, the proportio of studets at LVC who have chaged their majors. We fid that out of 1,527 studets i the study that 1,038 have chaged their majors at least oce. We ow decide to chage our cofidece level to 85%. How may studets would we eed i our sample i order to be withi 0.01 of the true proportio? How may more studets would we eed to iclude i out sample? Homework: Pages Exercises 1-25, odd Exercises 2-26, eve (Exercises 16, 17, 18, 19, 20, 21 are by calculator)

6 Aswers to the Sample Questios: 1. A survey of LVC studets idicated that out of the graduates 128 studets had chaged their majors at least oce while they were erolled. Form a 95% cofidece iterval for the proportio of studets at LVC that will chaged their major at least oce before graduatio. ˆp = 128 ˆq = 247 ˆp = ˆq = ! ( ) = z 0.95 = ˆp! z c ( ) ! ( ) ! < p < ˆp + z c < p < ! ( ) ( ) < p < ! ( ) ( ) < p < ! ( ) < p < ! ! ! < p < ! < p < We ca say with 95% cofidece level that the populatio proportio of LVC studets that chage their major at least oce is betwee ad Determie a 90% cofidece iterval usig the iformatio i questio 1 by usig your calculator. STAT TESTS 1-PropZIt 1! PropZIt x: 128 : C-Level: 90 Calculate 1! PropZIt ( , ) ˆp= = We ca say with 90% cofidece level that the populatio proportio of LVC studets that chage their major at least oce is betwee ad

7 3. We wish to do a study of studets at LVC to determie a 90% cofidece iterval for the proportio of studets who chage their major at least oce while erolled at LVC. We eed to be withi 0.01 of the true proportio. If o prelimiary study was available, how large should we make our sample? E = 0.01 z 0.90 = = 1! $ 4 " 0.01 % & = 1 ( )2 2 We would eed a sample size of 6,765 LVC studets i order to costruct a 90% cofidece level iterval with a error of 0.01 ad o prelimiary study for the populatio proportio of studets who chage their major at least oce while erolled at LVC. 4. Suppose we do our prelimiary study for the estimatio of p, the proportio of studets at LVC who have chaged their majors. We fid that out of 1,527 studets i the study that 1,038 have chaged their majors at least oce. We ow decide to chage our cofidece level to 85%. How may studets would we eed i our sample i order to be withi 0.01 of the true proportio? How may more studets would we eed to iclude i out sample? E = 0.01 z 0.90 = ˆp = ˆp = ˆq = ˆq = We would eed a sample size of 4,511 LVC studets i order to costruct a 85% cofidece level iterval with a error of 0.01 give a prelimiary study of 1,038 out of 1,527 studets idicatig that they have chaged their major at least oce for the populatio proportio of the studets that have chaged their major at least oce while attedig LVC. A additioal 2,984 LVC studets would eed to be icluded i the study. = 1 ( ) = = 6765 = ( ) = ! " 0.01 $ % & ( ) ( ) 2 ( ) ( ) = = = 4,511 Additioal = 4,511!1,527 Additioal = 2,984 2