22 - BUAD 310 Applied Business Statistics 1 Outline for...

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September 22, 2010 BUAD 310: Applied Business Statistics 1
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Outline for Today Statistical inference Confidence intervals Confidence intervals for a proportion, p Confidence intervals for a mean, μ (σ known or unknown) Margin of error and sample size determination 2
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3 Statistical inference
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A Quick Review Our major goal - make an inference about a population based on information from a sample We need to be able to quantify the uncertainty inherent in our inferences We do this by studying sampling distributions Estimation of population parameters ( e.g., population mean µ and population proportion p ) Point estimate: one number estimate such as the sample mean and the sample proportion The sampling distributions of and : approximately normal Interval estimate: confidence interval (CI) 4
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Confidence Interval We want to make a statement (inference) about a population parameter (e.g. μ or p ; unknown value) using information from observed sample data (statistic; an estimate such as ) CI: an interval of likely estimates that contains the true value of the population parameter which is unknown to us with a given confidence level. Idea: construct a (1-α)100% CI for µ of the form point estimate ± margin of error (e.g., ) (1-α)100% is called the confidence level ˆ or x p 5
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Motivating Example: Before deciding to offer an affinity credit card to alumni of a university, the credit company wants to know how many customers will accept the offer and how large a balance they will carry? Use confidence intervals to answer such questions They convey information about the precision of the estimates Confidence Interval 6
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Two Parameters of Interest p, the proportion who will return the application for the credit card μ, the average monthly balance that those who accept the credit card will carry Confidence Interval 7
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Summary Statistics ( n = 1000) Confidence Interval 8
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Confidence Interval for the Proportion A confidence interval is a range of plausible values for a parameter based on a sample. Constructing confidence intervals relies on the sampling distribution of the statistic. Confidence Interval 9
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Confidence Interval for the Proportion The Central Limit Theorem implies a normal model for the sampling distribution of . E( ) = p and SE( ) = p ˆ p ˆ p ˆ n p p / ) ˆ 1 ( ˆ - Confidence Interval 10
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CI for a Population Proportion A (1 - α)100% confidence interval for population proportion p using sample proportion is 11
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Checklist for Confidence Interval for p SRS condition. The sample is a simple random sample from the relevant population. Sample size condition (for proportion). Both n and n are larger than 10. p ˆ ) ˆ 1 ( p - CI for a Population Proportion 12
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Credit Card Example The standard error is SE( ) = = 0.01097 The 95% confidence interval is 0.14 ± 1.96(0.01097) = [0.1185 to 0.1615] p ˆ 1000 ) 14 . 0 1 ( 14 . 0 - CI for a Population Proportion 13
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Credit Card Example With 95% confidence, the population proportion that will accept the offer is between about 12% and 16%.
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