Sampling Distribution of the ProportionTV+More ExampleTheoryLaw of Large Numbers for Sample ProportionsEstimating the Population ProportionQuestions Asked About Population ProportionExamplesSampling Distribution of the Proportion Estimating the Population ProportionChapter 5Sampling Distribution of the ProportionJ.C. WangJC Wang (WMU) Stat2160 S2160, Chapter 5 1 / 25Sampling Distribution of the Proportion Estimating the Population ProportionGoal and Objectivesof Chapter 5To learn about the sampling distribution of the proportionJC Wang (WMU) Stat2160 S2160, Chapter 5 2 / 25Sampling Distribution of the Proportion Estimating the Population ProportioniClicker Question 5.1 Pre-lectureiClicker Question 5.1 Pre-lectureJC Wang (WMU) Stat2160 S2160, Chapter 5 3 / 25Sampling Distribution of the Proportion Estimating the Population ProportionOutline1Sampling Distribution of the ProportionTV+More ExampleTheoryLaw of Large Numbers for Sample Proportions2Estimating the Population ProportionQuestions Asked About Population ProportionExamplesJC Wang (WMU) Stat2160 S2160, Chapter 5 4 / 25Sampling Distribution of the Proportion Estimating the Population ProportionOutline1Sampling Distribution of the ProportionTV+More ExampleTheoryLaw of Large Numbers for Sample Proportions2Estimating the Population ProportionQuestions Asked About Population ProportionExamplesJC Wang (WMU) Stat2160 S2160, Chapter 5 5 / 25Sampling Distribution of the Proportion Estimating the Population ProportionSampling Distribution of the ProportionTV+More ExampleSuppose TV+More sells 60 extended warranties with 300 TV setssold. The warranty sales rate is60300= 0.20.Therefore, let X denote the number of successes out of a sampleof n observations. Then X is a binomial random variable withparameters n and p.The proportion of successes,ˆp =Xnin a sample is also a randomvariable.JC Wang (WMU) Stat2160 S2160, Chapter 5 6 / 25Sampling Distribution of the Proportion Estimating the Population ProportionSampling Distribution of the ProportionTV+More ExampleSuppose TV+More sells 60 extended warranties with 300 TV setssold. The warranty sales rate is60300= 0.20.Therefore, let X denote the number of successes out of a sampleof n observations. Then X is a binomial random variable withparameters n and p.The proportion of successes,ˆp =Xnin a sample is also a randomvariable.JC Wang (WMU) Stat2160 S2160, Chapter 5 6 / 25Sampling Distribution of the Proportion Estimating the Population ProportionSampling Distribution of the ProportionTV+More ExampleSuppose TV+More sells 60 extended warranties with 300 TV setssold. The warranty sales rate is60300= 0.20.Therefore, let X denote the number of successes out of a sampleof n observations. Then X is a binomial random variable withparameters n and p.The proportion of successes,ˆp =Xnin a sample is also a randomvariable.JC Wang (WMU) Stat2160 S2160, Chapter 5 6 / 25Sampling Distribution of the Proportion Estimating the Population ProportionSampling Distributionof the Proportionˆp =Xn= (number of successes) / (sample size)For the binomial, X is expected to be around np give or take√npq, where q = 1 − p.For the proportion,ˆp is expected to be p =npngive or takerpqn=√npqnJC Wang (WMU) Stat2160 S2160, Chapter 5 7 / 25Sampling Distribution of the Proportion Estimating the Population ProportionSampling Distributionof the Proportionˆp =Xn= (number of successes) / (sample size)For the binomial, X is expected to be around np give or take√npq, where q = 1 − p.For the proportion,ˆp is expected to be p =npngive or takerpqn=√npqnJC Wang (WMU) Stat2160 S2160, Chapter 5 7 / 25Sampling Distribution of the Proportion Estimating the Population ProportionSampling Distributionof the Proportionˆp =Xn= (number of successes) / (sample size)For the binomial, X is expected to be around np give or take√npq, where q = 1 − p.For the proportion,ˆp is expected to be p =npngive or takerpqn=√npqnJC Wang (WMU) Stat2160 S2160, Chapter 5 7 / 25Sampling Distribution of the Proportion Estimating the Population ProportionTV+More ExamplerevisitedThe number of warranties sold is expected to be around 60 ± 7The proportion of warranties sold is expected to be around60300±7300or 0.2 ± 0.02.JC Wang (WMU) Stat2160 S2160, Chapter 5 8 / 25Sampling Distribution of the Proportion Estimating the Population ProportionTV+More ExamplerevisitedThe number of warranties sold is expected to be around 60 ± 7The proportion of warranties sold is expected to be around60300±7300or 0.2 ± 0.02.JC Wang (WMU) Stat2160 S2160, Chapter 5 8 / 25Sampling Distribution of the Proportion Estimating the Population ProportionLaw of Large Numbersfor sample proportionsThe sample proportion tends to get closer to the true proportion assample size increases.For TV+More Example:Recall if TV+More sold 300 TV sets thenˆp = .2 and sd = 0.02.If TV+More sold 1200 TV sets andˆp = .2 and nowsd =r0.2 × (1 − 0.2)1200= 0.0115JC Wang (WMU) Stat2160 S2160, Chapter 5 9 / 25Sampling Distribution of the Proportion Estimating the Population ProportionLaw of Large Numbersfor sample proportionsThe sample proportion tends to get closer to the true proportion assample size increases.For TV+More Example:Recall if TV+More sold 300 TV sets thenˆp = .2 and sd = 0.02.If TV+More sold 1200 TV sets andˆp = .2 and nowsd =r0.2 × (1 − 0.2)1200= 0.0115JC Wang (WMU) Stat2160 S2160, Chapter 5 9 / 25Sampling Distribution of the Proportion Estimating the Population ProportionSampling Distribution of Sample Proportionis approximately normalIf TV+More sold 100 TV sets last year, the percentage of sets sold withextended warranties is expected to be around 20% give or take 4%.Estimate the likelihood that more than 25% of TV sets sold last yearhad sold with extended warranties, in other words,P(ˆp> 0.25) =?JC Wang (WMU) Stat2160 S2160, Chapter 5 10 / 25Sampling Distribution of the Proportion Estimating the Population ProportionSample Proportion is approximately normalcontinuedGiven: n = 100 and p = .2P(ˆp> .25) =normalCDF(.25, 9999, .2,q.2×(1−.2)100)= 0.1056proportion ofwarranties sold0.10 0.15 0.20 0.25 0.30JC Wang (WMU) Stat2160 S2160, Chapter 5 11 / 25Sampling Distribution of the Proportion Estimating the Population ProportionOutline1Sampling Distribution of the ProportionTV+More ExampleTheoryLaw of Large Numbers for Sample Proportions2Estimating the Population ProportionQuestions Asked About Population ProportionExamplesJC Wang (WMU) Stat2160 S2160, Chapter 5 12 /