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Unformatted text preview: 5. Statistical Inference: Estimation Goal : Use sample data to estimate values of population parameters Point estimate : A single statistic value that is the “best guess” for the parameter value Interval estimate : An interval of numbers around the point estimate, that has a fixed “confidence level” of containing the parameter value. Called a confidence interval . Point Estimators – Most common to use sample values • Sample mean estimates population mean μ ˆ μ = • Sample std. dev. estimates population std. dev. σ ˆ σ = • Sample proportion estimates population proportion : ˆ π Properties of good estimators • Unbiased : Sampling dist of estimator centers around parameter value • Efficient : Smallest possible standard error, compared to other estimators Confidence Intervals • A confidence interval (CI) is an interval of numbers believed to contain the parameter value. • The probability the method produces an interval that contains the parameter is called the confidence level (close to 1, such as 0.95 or 0.99. • Most CIs have the form point estimate ± margin of error with margin of error based on spread of sampling distribution of the point estimator (e.g., margin of error : 2(standard error) for 95% confidence) Confidence Interval for a Proportion (in a particular category) • Sample proportion is a mean when we let y=1 for observation in category of interest, y=0 otherwise • Population proportion is mean µ of prob. dist having • The standard dev. of this prob. dist. is • The standard error of the sample proportion is (1) and (0) 1 P P π π = =  (1 ) (e.g., 0.50 when 0.50) σ π π π = = ˆ / (1 ) / n n π σ σ π π = = ˆ π • Sampling distribution of sample proportion for large random samples is approximately normal (CLT) • So, with probability 0.95, sample proportion falls within 1.96 standard errors of population proportion : • 0.95 probability that • Once sample selected, we’re 95% confident ˆ ˆ ˆ falls between 1.96 and 1.96 π π π π σ π σ + ˆ ˆ ˆ ˆ 1.96 to 1.96 contains π π π σ π σ π + ˆ π Finding a CI in practice • Complication: The true standard error itself depends on the unknown parameter! ˆ / (1 ) / n n π σ σ π π = = In practice, we estimate and then find 95% CI using formula ^ ˆ ˆ 1 (1 ) by se n n π π π π π σ   = = ˆ ˆ 1.96( ) to 1.96( ) se se π π + Example: What percentage of 1822 year old Americans report being “very happy”? Recent GSS data: 35 of n= 164 “very happy” (others report being “pretty happy” or “not too happy”) 95% CI is (i.e., “margin of error” = ) which gives ( , ). We’re 95% confident the population proportion who are “very happy” is between and ....
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 Fall '10
 ALAN
 Normal Distribution

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