Bias and Variability Lecture 28 Section 8 3 Fri Mar 4 2005 Unbiased Statistics Unbiased statistic A statistic whose average value equals the parameter that it is estimating We have already seen that p is an unbiased estimator of p because p p Would the sample range be an unbiased estimator of the population range Variability of a Statistic The variability of a statistic is a measure of how spread out the sampling distribution of that statistic is All estimators exhibit some variability The less variability the better The Parameter The parameter Unbiased Low Variability The sampling distribution The parameter Unbiased High Variability The parameter Biased High Variability The parameter Biased Low Variability The parameter Accuracy and Precision An unbiased statistic allows us to make accurate estimates A low variability statistic allows us to make precise estimates The best estimator is one that is unbiased and with low variability Then we can make estimates that are both accurate and precise The Sampling Distribution of p Since the mean of p equals p then p is an unbiased estimator of p Because n appears in the denominator of the standard deviation The standard deviation of p decreases as n increases Therefore for large samples large n p has a lower variability than it does for small samples In that respect larger samples are better Experiment I will use randBin 50 1 200 to simulate selecting 50 people from a population that is 10 male 200 times and counting the males Volunteer 1 randBin 50 3 200 30 male Volunteer 2 randBin 50 5 200 50 male Volunteer 3 randBin 50 7 200 70 male Volunteer 4 randBin 50 9 200 90 male It will take the TI 83 about 6 minutes Experiment Divide the list by 50 to get proportions Store the results in list L1 Compute the statistics for L1 STO L1 1 Var Stats L1 What are the means and standard deviations Do they seem to change depending on the population proportion Sampling Distributions and Hypothesis Testing Suppose we choose a sample of n students from an unknown population However we know that the population consists of either 1 3 freshmen or 2 3 freshmen Our purpose is to test the following hypotheses H0 p 1 3 H1 p 2 3 Sampling Distributions and Hypothesis Testing Under H0 the sampling distribution of p should be normal p 1 3 p 1 3 2 3 n 0 4714 n Under H1 the sampling distribution of p should be normal p 2 3 p 2 3 1 3 n 0 4714 n Sampling Distributions and Hypothesis Testing The likelihood of being able to tell the difference based on p will depend on the sample size The larger the sample the more likely it is that we will be able to distinguish between the two hypothetical populations PDFs of p for n 5 H0 H1 PDFs of p for n 10 H0 H1 PDFs of p for n 20 H0 H1 PDFs of p for n 50 H0 H1 PDFs of p for n 100 H0 H1 Let s Do It Let s do it 8 5 p 484 Probabilities about the Proportion of People with Type B Blood Let s do it 8 6 p 485 Estimating the Proportion of Patients with Side Effects Let s do it 8 7 p 487 Testing hypotheses about Smoking Habits See Example 8 5 p 486 Testing Hypotheses about the Proportion of Cracked Bottles
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