STA 291 Spring 2008 Lecture 13 Dustin Lueker

Statistical Inference: Estimation Inferential statistical methods provide predictions about characteristics of a population, based on information in a sample from that population Quantitative variables Usually estimate the population mean Mean household income Qualitative variables Usually estimate population proportions Proportion of people voting for candidate A STA 291 Spring 2008 Lecture 13

Two Types of Estimators Point Estimate A single number that is the best guess for the parameter Sample mean is usually at good guess for the population mean Interval Estimate Point estimator with error bound A range of numbers around the point estimate Gives an idea about the precision of the estimator The proportion of people voting for A is between 67% and 73% STA 291 Spring 2008 Lecture 13

Point Estimator A point estimator of a parameter is a sample statistic that predicts the value of that parameter A good estimator is Unbiased Centered around the true parameter Consistent Gets closer to the true parameter as the sample size gets larger Efficient Has a standard error that is as small as possible (made use of all available information) STA 291 Spring 2008 Lecture 13

Unbiased An estimator is unbiased if its sampling distribution is centered around the true parameter For example, we know that the mean of the sampling distribution of equals μ, which is the true population mean So, is an unbiased estimator of μ Note: For any particular sample, the sample mean may be smaller or greater than the population mean Unbiased means that there is no systematic underestimation or overestimation STA 291 Spring 2008 Lecture 13

Biased A biased estimator systematically underestimates or overestimates the population parameter In the definition of sample variance and sample standard deviation uses n-1 instead of n, because this makes the estimator unbiased With n in the denominator, it would systematically underestimate the variance STA 291 Spring 2008 Lecture 13

Efficient An estimator is efficient if its standard error is small compared to other estimators Such an estimator has high precision A good estimator has small standard error and small bias (or no bias at all) The following pictures represent different estimators with different bias and efficiency Assume that the true population parameter is the point (0,0) in the middle of the picture STA 291 Spring 2008 Lecture 13

Bias and Efficient Note that even an unbiased and efficient estimator does not always hit exactly the population parameter. But in the long run, it is the best estimator. STA 291 Spring 2008 Lecture 13

Confidence Interval Inferential statement about a parameter should always provide the accuracy of the estimate How close is the estimate likely to fall to the true parameter value? Within 1 unit? 2 units? 10 units? This can be determined using the sampling distribution of the estimator/sample statistic In particular, we need the standard error to make a statement about accuracy of the estimator STA 291 Spring 2008 Lecture 13

Confidence Interval Range of numbers that is likely to cover (or capture) the true parameter Probability that the confidence interval captures the true parameter is called the confidence coefficient or more commonly the confidence level Confidence level is a chosen number close to 1, usually 0.90, 0.95 or 0.99 Level of significance = α = 1 – confidence level STA 291 Spring 2008 Lecture 13

Confidence Interval To calculate the confidence interval, we use the Central Limit Theorem Use this formula if σ is known Also, we need a that is determined by the confidence level Formula for 100(1-α)% confidence interval for μ STA 291 Spring 2008 Lecture 13

Common Confidence Intervals Confidence level of 0.90 α=.10 Zα/2=1.645 95% confidence interval Confidence level of 0.95 α=.05 Zα/2=1.96 99% confidence interval Confidence level of 0.99 α=.01 Zα/2=2.576 STA 291 Spring 2008 Lecture 13

Confidence Intervals This interval will contain μ with a 100(1-α)% probability We need to know σ to find a confidence interval If σ is unknown, we may replace it by s (sample standard deviation) but the value Zα/2 needs adjustment to take into account the possible extra variability introduced by s A different distribution is used, the t-distribution, we will get to this later STA 291 Spring 2008 Lecture 13

Example Compute a 95% confidence interval for μ if we know that σ=12 and the sample of size 25 yielded a mean of 7 STA 291 Spring 2008 Lecture 13

Interpreting Confidence Intervals “Probability” means that in the long run 100(1-α)% of the intervals will contain the parameter If repeated samples were taken and confidence intervals calculated then 100(1-α)% of the intervals will contain the parameter For one sample, we do not know whether the confidence interval contains the parameter The 100(1-α)% probability only refers to the method that is being used STA 291 Spring 2008 Lecture 13

Interpreting Confidence Intervals Incorrect statement With 95% probability, the population mean will fall in the interval from 3.5 to 5.2 To avoid the misleading word “probability” we say that we are “confident” We are 95% confident that the true population mean will fall between 3.5 and 5.2 STA 291 Spring 2008 Lecture 13

Confidence Intervals Changing our confidence level will change our confidence interval Increasing our confidence level will increase the length of the confidence interval A confidence level of 100% would require a confidence interval of infinite length Not informative There is a tradeoff between length and accuracy Ideally we would like a short interval with high accuracy (high confidence level) STA 291 Spring 2008 Lecture 13