Bias and Variability Lecture 28 Section 8.3 Tue, Oct 25, 2005

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. We have already seen that p ^ is an unbiased estimator of p, because  p^ = p. 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? 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. The variability of a statistic is a measure of how spread out the sampling distribution of that statistic is. All estimators exhibit some variability. All estimators exhibit some variability. The less variability, the better. The less variability, the better.

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 of p ^ decreases as n increases. 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. Therefore, for large samples (large n), p ^ has a lower variability than it does for small samples. In that respect, larger samples are better. 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. 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?

Sampling Distributions and Hypothesis Testing Suppose we choose a sample of n students from an unknown population. 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. However, we know that the population consists of either 1/3 freshmen or 2/3 freshmen. Our purpose is to test the following hypotheses: Our purpose is to test the following hypotheses: H 0 : p = 1/3. H 0 : p = 1/3. H 1 : p = 2/3. H 1 : p = 2/3.

Sampling Distributions and Hypothesis Testing Under H 0, the sampling distribution of p ^ should be Under H 0, the sampling distribution of p ^ should be normal, normal,  p^ = 1/3,  p^ = 1/3,  p^ =  ((1/3)(2/3)/n) = /  n.  p^ =  ((1/3)(2/3)/n) = /  n. Under H 1, the sampling distribution of p ^ should be Under H 1, the sampling distribution of p ^ should be normal, normal,  p^ = 2/3,  p^ = 2/3,  p^ =  ((2/3)(1/3)/n) = /  n.  p^ =  ((2/3)(1/3)/n) = /  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 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. 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 H0H0 H1H1

PDFs of p ^ for n = 10 H0H0 H1H1

PDFs of p ^ for n = 20 H0H0 H1H1

PDFs of p ^ for n = 50 H0H0 H1H1

PDFs of p ^ for n = 100 H0H0 H1H1

Let’s Do It! Let’s do it! 8.5, p. 521 – Probabilities about the Proportion of People with Type B Blood. Let’s do it! 8.5, p. 521 – Probabilities about the Proportion of People with Type B Blood. Let’s do it! 8.6, p. 523 – Estimating the Proportion of Patients with Side Effects. Let’s do it! 8.6, p. 523 – Estimating the Proportion of Patients with Side Effects. Let’s do it! 8.7, p. 525 – Testing hypotheses about Smoking Habits. Let’s do it! 8.7, p. 525 – Testing hypotheses about Smoking Habits. See Example 8.5, p. 524 – Testing Hypotheses about the Proportion of Cracked Bottles. See Example 8.5, p. 524 – Testing Hypotheses about the Proportion of Cracked Bottles.