Computing the Sampling Distribution of The Central Limit Theorem for ProportionsApplicationsAssignmentSampling Distribution of a Sample ProportionLecture 26Section 8.4Robb T. KoetherHampden-Sydney CollegeMon, Mar 5, 2012Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 1 / 35Outline1Computing the Sampling Distribution ofˆp2The Central Limit Theorem for Proportions3Applications4AssignmentRobb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 2 / 35Outline1Computing the Sampling Distribution ofˆp2The Central Limit Theorem for Proportions3Applications4AssignmentRobb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 3 / 35Sampling DistributionsDefinition (Sampling Distribution of a Statistic)The sampling distribution of a statistic is the distribution of values ofthat statistic over all possible samples of a given size n from thepopulation.We may sample with or without replacement.For our purposes, it will be simpler to sample with replacement.Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 4 / 35The Sample ProportionWe will work out the sampling distribution forˆp for sample sizes of1, 2, and 3.Then I will show you the sampling distribution forˆp for samplesizes of 4, 5, and 10.Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 5 / 35ExampleSuppose that 45% of all people approve of President Obama’sperformance.Suppose that we select one person at random.We may diagram the 2 possibilities.Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 6 / 35ExampleYN0.550.45Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 7 / 35ExampleNow we take a sample of 2 people, sampling with replacement.Find the sampling distribution ofˆp.Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 8 / 35ExampleYN0.550.45NYNN10(0.55)(0.45) = 0.2475(0.55)2 = 0.3025YN0.550.45YN0.550.45YN 1 (0.45)(0.55) = 0.2475YY 2 (0.45)2 = 0.2025Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 9 / 35ExampleLet x be the number of people (out of 2) who strong disapprove ofPresident Obama’s performance.The probability distribution ofˆp isˆp P(ˆp)0 0.20251/2 0.49501 0.3025Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 10 / 35ExampleNow we take a sample of 3 people, sampling with replacement.Find the sampling distribution ofˆp.Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 11 / 35ExampleYN0.550.45NYYNNY21(0.55)(0.45)2 = 0.1114(0.55)2(0.45) = 0.1114YN0.550.45YNY 2 (0.45)(0.55)(0.45) = 0.1114YYY 3 (0.45)3 = 0.0911YN0.550.45YN0.550.45YN0.550.45YN0.550.45YN0.550.45NYNNNN10(0.55)(0.45)(0.55) = 0.1361(0.55)3 = 0.1664YNN 1 (0.45)(0.55)2 = 0.1361YYN 2 (0.45)2(0.55) = 0.1114Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 12 / 35ExampleLetˆp be the sample proportion of people who strong disapproveof President Obama’s performance.The sampling distribution ofˆp isˆp P(ˆp)0 0.16641/3 0.40842/3 0.33411 0.0911Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 13 / 35Samples of Size n = 4If we sample 4 people, then the sampling distribution of thesample proportion isˆp P(ˆp)0 0.09151/4 0.29952/4 0.36753/4 0.20051 0.0410Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 14 / 35Samples of Size n = 5If we sample 5 people, then the sampling distribution of thesample proportion isˆp P(ˆp)0 0.05031/5 0.20592/5 0.33693/5 0.27574/5 0.11281 0.0185Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 15 / 35Samples of Size n = 6If we sample 6 people, then the sampling distribution of thesample proportion isˆp P(ˆp)0 0.02771/6 0.13592/6 0.27803/6 0.30324/6 0.18615/6 0.06091 0.0083Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 16 / 35Samples of Size n = 10If we sample 10 people, then the sampling distribution of thesample proportion isˆp P(ˆp)0.00 0.00250.10 0.02070.20 0.07630.30 0.16650.40 0.23840.50 0.23400.60 0.15960.70 0.07460.80 0.02290.90 0.00421.00 0.0003Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 17 / 35Outline1Computing the Sampling Distribution ofˆp2The Central Limit Theorem for Proportions3Applications4AssignmentRobb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 18 / 35The Central Limit Theorem for ProportionsTheorem (The Central Limit Theorem for Proportions)For any population and any sample size, the sampling distributionofˆp has the following mean and standard deviation:µˆp= pσˆp=rp(1 − p)n.Furthermore, the sampling distribution ofˆp is approximatelynormal, provided n is large enough.Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 19 / 35The Central Limit Theorem for ProportionsThe Sample Sizen is large enough ifnp ≥ 5 and n(1 − p) ≥ 5.If n is small, then we have to work out the distribution by hand.Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 20 / 35Outline1Computing the Sampling Distribution ofˆp2The Central Limit Theorem for Proportions3Applications4AssignmentRobb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 21 / 35ApplicationsSuppose that 60% of all high-school students own a cell phone.If we survey 3 high-school students, how likely is it that we will findthat at least 2 of them own a cell phone?Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 22 / 35ApplicationsSuppose that 60% of all high-school students own a cell phone.If we survey 150 high-school students, how likely is it that we willfind that at least 65% of them own a cell phone?Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 5, 2012 23 /
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