VIEWS: 29 PAGES: 14 POSTED ON: 5/18/2011 Public Domain
Conﬁdence Intervals about a Population Proportion MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2009 J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Motivation Example PRINCETON, NJ – Gallup Poll Daily tracking ﬁnds 41% of Americans describing economic conditions as "poor," down slightly from the 2008 high of 44%, but still more than double the percentage who say the economy is "excellent" or "good" (17%). The vast majority, 85%, perceive the economy to getting worse. – Jeff Jones The results reported here are based on combined data from 1,544 interviews conducted March 19-21, 2008. For results based on this sample, the maximum margin of sampling error is ±3 percentage points. J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Background The point estimate for the population proportion p is the ˆ sample proportion p. If 1544 are surveyed and 633 respond that the economy is getting worse then ˆ p = 633/1544 ≈ 0.410. J. Robert Buchanan Conﬁdence Intervals about a Population Proportion ˆ Sampling Distribution of p Theorem For a simple random sample of size n such that n ≤ 0.05N (that is, the sample is less than or equal to 5% of the population), ˆ the shape of the sampling distribution of p is approximately normal provided np(1 − p) ≥ 10, ˆ the mean of the sampling distribution of p is µp = p, ˆ ˆ the standard deviation of the sampling distribution of p is p(1 − p) σp = ˆ . n J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Conﬁdence Interval for a Sample Proportion Suppose a simple random sample of size n is taken from a population. A (1 − α) · 100% conﬁdence interval for p is given by ˆ ˆ p(1 − p) ˆ Lower and Upper bounds: p ± zα/2 · n ˆ ˆ Note: it must be the case that np(1 − p ) ≥ 10 and n ≤ 0.05N to construct this interval. J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Example Example Of 1500 people surveyed, 850 had eaten pizza within the last month. Construct the 95% conﬁdence interval estimate of the population proportion of people who have eaten pizza within the last month. J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Example Example In the city or area where you live, are you satisﬁed or dissatisﬁed with the quality of water? In the United States 1000 residents aged 15 or older were surveyed and 870 replied they were satisﬁed with the water quality. Construct the 99% conﬁdence interval estimate of all US residents satisﬁed with their water quality. J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Example Example Have recent price increases in gasoline caused any ﬁnancial hardship for you or your household? In the United States 1025 residents aged 18 or older were surveyed and 646 replied “yes”. Construct the 90% conﬁdence interval estimate of all US residents who report that the price increases in gasoline have caused some ﬁnancial hardship for themselves or their household. J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Estimating the Sample Size ˆ ˆ p(1 − p) E = zα/2 · n zα/2 2 ˆ ˆ n = p(1 − p) E ˆ Remark: to use this formula we need a prior estimate of p or we must consider the worst case scenario. J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Worst Case Scenario p1 p 0.25 0.20 0.15 0.10 0.05 p 0.2 0.4 0.6 0.8 1.0 J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Estimating the Sample Size The sample size required to obtain a (1 − α) · 100% conﬁdence interval for p with a margin of error E is given by 2 zα/2 ˆ ˆ n = p(1 − p) E ˆ (rounded up the next integer), where p is a prior estimate of p. If a prior estimate of p is unavailable, the sample size required is 2 zα/2 n = 0.25 E rounded up the next integer. J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Example Example Determine the sample size necessary to estimate the true proportion of college students with blue eyes, if the estimate is to have a margin of error of 0.02 with 90% conﬁdence. J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Example Example An automobile manufacturer purchases bolts from a supplier who claim that the bolts are approximately 5% defective. Determine the sample size necessary to estimate the true proportion of defective bolts if the margin of error is to be 0.01 with 95% conﬁdence. J. Robert Buchanan Conﬁdence Intervals about a Population Proportion Homework Read Section 9.3. Pages 441-443: 5–25 odd J. Robert Buchanan Conﬁdence Intervals about a Population Proportion