STAT101_Chap5 - 5. Statistical Inference: Estimation Goal :...

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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 = Sample std. dev. estimates population std. dev. = Sample proportion estimates population proportion : 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 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 = = - (1 ) (e.g., 0.50 when 0.50) =- = / (1 ) / n n = =- 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, were 95% confident falls between 1.96 and 1.96 - + 1.96 to 1.96 contains - + Finding a CI in practice Complication: The true standard error itself depends on the unknown parameter! / (1 ) / n n = =- In practice, we estimate and then find 95% CI using formula ^ 1 (1 ) by se n n - - = = 1.96( ) to 1.96( ) se se - + Example: What percentage of 18-22 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 ( , ). Were 95% confident the population proportion who are very happy is between and ....
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STAT101_Chap5 - 5. Statistical Inference: Estimation Goal :...

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