bias and variability lecture 28 section 8.3 tue, oct 25, 2005

Bias and Bias and VariabilityVariability

Lecture 28Lecture 28

Section 8.3Section 8.3

Tue, Oct 25, 2005Tue, Oct 25, 2005

Unbiased StatisticsUnbiased Statistics

Unbiased statisticUnbiased statistic – A statistic whose – A statistic whose average value equals the parameter average value equals the parameter that it is estimating.that it is estimating.

We have already seen that We have already seen that pp^̂ is an is an unbiased estimator of unbiased estimator of pp, because , because pp^̂ = = pp..

Would the sample range be an Would the sample range be an unbiased estimator of the population unbiased estimator of the population range?range?

Variability of a StatisticVariability of a Statistic

The The variabilityvariability of a statistic is a of a statistic is a measure of how spread out the measure of how spread out the sampling distribution of that statistic sampling distribution of that statistic is.is.

All estimators exhibit some All estimators exhibit some variability.variability.

The less variability, the better.The less variability, the better.

The ParameterThe Parameter

The parameter

Unbiased, Low VariabilityUnbiased, Low Variability

The parameter

The sampling distribution

Unbiased, High Unbiased, High VariabilityVariability

The parameter

Biased, High VariabilityBiased, High Variability

The parameter

Biased, Low VariabilityBiased, Low Variability

The parameter

Accuracy and PrecisionAccuracy and Precision

An unbiased statistic allows us to An unbiased statistic allows us to make make accurateaccurate estimates. estimates.

A low variability statistic allows us to A low variability statistic allows us to make make preciseprecise estimates. estimates.

The best estimator is one that is The best estimator is one that is unbiased and with low variability.unbiased and with low variability. Then we can make estimates that are Then we can make estimates that are

both accurate both accurate andand precise. precise.

The Sampling The Sampling Distribution of Distribution of pp^̂

Since the mean of Since the mean of pp^̂ equals equals pp, then , then pp^̂ is is an an unbiasedunbiased estimator of estimator of pp..

Because Because nn appears in the denominator of appears in the denominator of the standard deviation,the standard deviation, The standard deviation of The standard deviation of pp^̂ decreasesdecreases as as nn increasesincreases..

Therefore, for large samples (large Therefore, for large samples (large nn), ), pp^̂ has a has a lower variabilitylower variability than it does for than it does for small samples.small samples.

In that respect, larger samples are better.In that respect, larger samples are better.

ExperimentExperiment

I will use randBin(50, .1, 200) to simulate I will use randBin(50, .1, 200) to simulate selecting 50 people from a population that is selecting 50 people from a population that is 10% male 200 times, and counting the males.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.

ExperimentExperiment

Divide the list by 50 to get proportions.Divide the list by 50 to get proportions. Store the results in list LStore the results in list L11..

STO LSTO L11.. Compute the statistics for LCompute the statistics for L11..

1-Var Stats L1-Var Stats L11.. What are the means and standard What are the means and standard

deviations?deviations? Do they seem to change, depending on Do they seem to change, depending on

the population proportion?the population proportion?

Sampling Distributions and Sampling Distributions and Hypothesis TestingHypothesis Testing

Suppose we choose a sample of Suppose we choose a sample of nn students from an unknown population.students from an unknown population.

However, we know that the population However, we know that the population consists of either 1/3 freshmen or 2/3 consists of either 1/3 freshmen or 2/3 freshmen.freshmen.

Our purpose is to test the following Our purpose is to test the following hypotheses:hypotheses: HH00: : pp = 1/3. = 1/3.

HH11: : pp = 2/3. = 2/3.


Under Under HH00, the sampling distribution of , the sampling distribution of pp^̂ should be should be normal,normal, pp^̂ = 1/3, = 1/3, pp^̂ = = ((1/3)(2/3)/((1/3)(2/3)/nn) = 0.4714/) = 0.4714/nn..

Under Under HH11, the sampling distribution of , the sampling distribution of pp^̂ should be should be normal,normal, pp^̂ = 2/3, = 2/3, pp^̂ = = ((2/3)(1/3)/((2/3)(1/3)/nn) = 0.4714/) = 0.4714/nn..


The likelihood of being able to tell The likelihood of being able to tell the difference based on the difference based on pp^̂ will will depend on the sample size.depend on the sample size.

The larger the sample, the more The larger the sample, the more likely it is that we will be able to likely it is that we will be able to distinguish between the two distinguish between the two hypothetical populations.hypothetical populations.

PDFs of PDFs of pp^̂ for for nn = 5 = 5

H0 H1


H0 H1

Let’s Do It!Let’s Do It!

Let’s do it! 8.5, p. 521 – Probabilities Let’s do it! 8.5, p. 521 – Probabilities about the Proportion of People with about the Proportion of People with Type B Blood.Type B Blood.

Let’s do it! 8.6, p. 523 – Estimating the Let’s do it! 8.6, p. 523 – Estimating the Proportion of Patients with Side Effects.Proportion of Patients with Side Effects.

Let’s do it! 8.7, p. 525 – Testing Let’s do it! 8.7, p. 525 – Testing hypotheses about Smoking Habits.hypotheses about Smoking Habits. See Example 8.5, p. 524 – Testing See Example 8.5, p. 524 – Testing

Hypotheses about the Proportion of Hypotheses about the Proportion of Cracked Bottles.Cracked Bottles.

bias and variability lecture 28 section 8.3 tue, oct 25, 2005

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