Bias and Variability Lecture 27 Section 8.3 Wed, Nov 3, 2004

Unbiased Statistics Unbiased statistic – A statistic whose average value equals the parameter that it is estimating. Unbiased statistic – A statistic whose average value equals the parameter that it is estimating. The variability of a statistic is a measure of how spread out its sampling distribution is. The variability of a statistic is a measure of how spread out its sampling distribution is. All estimators exhibit some variability. All estimators exhibit some variability.

The Parameter The parameter

Unbiased, Low Variability The parameter The sampling distribution

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. An unbiased statistic allows us to make accurate estimates. A low variability statistic allows us to make precise estimates. A low variability statistic allows us to make precise estimates. The best estimator is one that is unbiased and with low variability. The best estimator is one that is unbiased and with low variability. Then we can make estimates that are both accurate and precise. 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. 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, Because n appears in the denominator of the standard deviation, The standard deviation decreases as n increases. The standard deviation decreases as n increases. Therefore, for large samples (large n), p^ has a lower variability than it does for small samples. Therefore, for large samples (large n), p^ has a lower variability than it does for small samples.

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. 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 #1: randBin(50,.3, 200) (30% male) Volunteer #2: randBin(50,.5, 200) (50% male) Volunteer #2: randBin(50,.5, 200) (50% male) Volunteer #3: randBin(50,.7, 200) (70% male) Volunteer #3: randBin(50,.7, 200) (70% male) Volunteer #4: randBin(50,.9, 200) (90% male) Volunteer #4: randBin(50,.9, 200) (90% male) It will take the TI-83 about 6 minutes. It will take the TI-83 about 6 minutes.

Experiment Divide the list by 50 to get proportions. Divide the list by 50 to get proportions. Store the results in list L 1. Store the results in list L 1. STO L 1. STO L 1. Compute the statistics for L 1. Compute the statistics for L 1. 1-Var Stats L 1. 1-Var Stats L 1. What are the means and standard deviations? What are the means and standard deviations? Do they seem to change, depending on the population proportion? Do they seem to change, depending on the population proportion?