501_Lecture_08 - Section 8.1 Inference for a Single...

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Section 8.1 Inference for a Single Proportion
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Recall: Population Proportion Let p be the proportion of “successes” in a population. A random sample of size n is selected, and X is the count of successes in the sample. Suppose n is small relative to the population size, so that X can be regarded as a binomial random variable with and (1 ) μ σ = = - X X np np p
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Recall: Population Proportion We use the sample proportion as an estimator of the population proportion p . is an unbiased estimator of p, with mean and SD: When n is large, is approximately normal. Thus is approximately standard normal. ˆ = X p n ˆ ˆ (1 ) and μ σ - = = p p p p p n p ˆ ˆ (1 ) - = - p p z p p n ˆ p
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CI for a Population Proportion Since is normal. The standard error of is An approximate level C confidence interval for p: where P ( Z ≥ z* ) = (1 – C)/2. ˆ ˆ (1 ) ˆ ( ) p p SE p n - = * * ˆ ˆ (1 ) ˆ ˆ ˆ ( ) p p p z SE p p z n - = ˆ p p ˆ
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CI for a Population Proportion The margin of error is Use this interval when the successes and failures are both at least 15 Use Table A or last row of Table D to find z* . * * ˆ ˆ (1 ) ˆ ( ) p p m z SE p z n - = =
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Example: A news program constructs a call-in poll about a proposed city ban on handguns. 2372 people call in to the show. Of these, 1921 oppose the ban. Construct a 95% confidence interval for the true proportion of people who oppose the ban. What are the possible problems with the study design?
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Solution: Note: Since p is a proportion, if you ever get an upper limit value of > 1 or lower <0 while calculating the CI, replace by 1 and 0 (respectively).
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Choosing a Sample Size If we want to estimate the proportion p within a specified margin of error m , the required sample size is (at least): ( 29 ( 29 ( 29 2 * 2 ˆ ˆ 1 p p z n m - =
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Choosing a Sample Size Since is unknown before the data is collected, we use any prior information we have to get a rough
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This note was uploaded on 02/20/2012 for the course STAT 501 taught by Professor Staff during the Spring '08 term at Purdue University-West Lafayette.

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501_Lecture_08 - Section 8.1 Inference for a Single...

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