5. Estimation - Chap 5 6 Statistical inference methods •...

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Unformatted text preview: Chap. 5, 6: Statistical inference methods • Chapter 5: Estimation (of population parameters) Ex. “Based on GSS data, we’re 95% confident that the population mean of the variable LONELY (no. of days in past week you felt lonely, = 1.5, s = 2.2) falls between 1.4 and 1.6. • Chapter 6: Significance Testing (Making decisions about hypotheses regarding “effects” and associations) • Ex. Article in Science , 2008: “We hypothesized that spending money on other people has a more positive impact on happiness than spending money on oneself.” y 5. Statistical Inference: Estimation Goal : How can we 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 . (Based on sampling distribution of the point estimate) Point Estimators – Most common to use sample values • Sample mean estimates population mean μ ˆ i y y n μ = = ∑ • Sample std. dev. estimates population std. dev. σ 2 ( ) ˆ 1 i y y s n σ- = =- ∑ • Sample proportion estimates population proportion • ˆ π Properties of good estimators • Unbiased : Sampling distribution of the estimator centers around the parameter value ex . Biased estimator: sample range. It cannot be larger than population range. • Efficient : Smallest possible standard error, compared to other estimators Ex . If population is symmetric and approximately normal in shape, sample mean is more efficient than sample median in estimating the population mean and median. (can check this with sampling distribution applet at www.prenhall.com/agresti) 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. Most studies use a 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) • Recall that the sample proportion is a mean when we let y = 1 for observation in category of interest, y = 0 otherwise • Recall the population proportion is mean µ of prob. dist having • The standard deviation of this probability distribution 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 • The sampling distribution of a sample proportion for large random samples is approximately normal (Central Limit Theorem) • So, with probability 0.95, sample proportion falls within 1.96 standard errors of population proportion •within 1....
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This note was uploaded on 07/12/2011 for the course STA 3030 taught by Professor Agresti during the Spring '11 term at University of Florida.

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5. Estimation - Chap 5 6 Statistical inference methods •...

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