Download Confidence Intervals About a Population Proportion - Lecture Slides | MATH 130 and more Assignments Statistics in PDF only on Docsity! Confidence Intervals about a Population Proportion MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Spring 2008 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 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 Lower and Upper bounds: p̂ − zα/2 · √ p̂(1 − p̂) n Note: it must be the case that np̂(1− p̂) ≥ 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 Worst Case Scenario 0.2 0.4 0.6 0.8 1.0 p ` 0.05 0.10 0.15 0.20 0.25 p ` H1-p ` L J. Robert Buchanan Confidence Intervals about a Population Proportion Estimating the Sample Size (cont.) The sample size required to obtain a (1 − α) · 100% confidence interval for p with a margin of error E is given by n = p̂(1 − p̂) ( zα/2 E )2 (rounded up the next integer), where p̂ is a prior estimate of p. If a prior estimate of p is unavailable, the sample size required is n = 0.25 ( zα/2 E )2 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
