1. Lecture 28 Section 8.3 Fri, Mar 4, 2005
Bias and Variability
Lecture 28
Section 8.3
Fri, Mar 4, 2005

2. 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?

3. 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.

4. The Parameter
The parameter

5. Unbiased, Low Variability
The sampling
distribution
The parameter

6. Unbiased, High Variability
The parameter

7. Biased, High Variability
The parameter

8. Biased, Low Variability
The parameter

9. 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.

10. 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.

11. 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.

12. Experiment Divide the list by 50 to get proportions.
Store the results in list L1.
STO L1.
Compute the statistics for L1.
1-Var Stats L1.
What are the means and standard deviations?
Do they seem to change, depending on the population proportion?

13. 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.

14. Sampling Distributions and Hypothesis Testing
Under H0, the sampling distribution of p^ should be
normal,
p^ = 1/3,
p^ = ((1/3)(2/3)/n) = /n.
Under H1, the sampling distribution of p^ should be
p^ = 2/3,
p^ = ((2/3)(1/3)/n) = /n.

15. 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.

16. PDFs of p^ for n = 5 H0 H1

17. PDFs of p^ for n = 10 H0 H1

18. PDFs of p^ for n = 20 H0 H1

19. PDFs of p^ for n = 50 H0 H1

20. PDFs of p^ for n = 100 H0 H1

21. 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.