1. Today Today: Chapter 8, start Chapter 9 Assignment: Recommended Questions: 9.1, 9.8, 9.20, 9.23, 9.25

2. Properties of Sample Means and Proportions If X=(X 1, X 2,…,X n ) represents random sample then the expected value of the sample mean is: The variance of the sample mean is:

3. Properties of Sample Means and Proportions If X=(X 1, X 2,…,X n ) represents random sample from a Bernoulli population, then the expected value of the sample proportion (sample mean) is: The variance of the sample proportion (sample mean) is:

4. Example A population of males has mean height of 70 inches and standard deviation of 3 inches For a random sample of size 4, what is the mean and variance of the sample mean For a random sample of size 40, what is the mean and variance of the sample mean

5. Properties of Sample Means from Normal Populations Suppose X=(X 1, X 2,…,X n ) represents random sample from a Normal population The distribution of the sample mean is:

6. Example A population of males has mean height of 70 inches and standard deviation of 3 inches and approximately follows a normal distribution For a random sample of size 4, what is the probability that the observed sample mean is less than 69 For a random sample of size 40, what is the probability that the observed sample mean is less than 69

7. The Central Limit Theorem Finding the sampling distribution of a statistic (e.g., sample mean or sample variance) can be hard Turns out that for many common statistics, when the sample size is large, the distribution of the sample statistic will be approximately normal

8. The Central Limit Theorem Suppose X=(X 1, X 2,…,X n ) represents random sample from a distribution with mean μ and finite variance σ 2, then That is,

9. The Central Limit Theorem What does this mean in practice? What is a large sample?

10. Example A population of males has mean height of 70 inches and standard deviation of 3 inches For a random sample of size 4, what is the probability that the observed sample mean is less than 69 For a random sample of size 40, what is the probability that the observed sample mean is less than 69

11. Example Given that X had pdf: The mean and variance of X are: Find the mean and variance of

12. Example What is the asymptotic distribution of T?

13. CLT for Bernoulli Distributions We saw that the mean and variance of the sample proportion are: For large samples, the distribution of the sample proportion is: What is a large sample in this case?

14. Example (similar to 8-R8) Consider a population in which the percentage of Hispanics is 10% Find the probability that in a random sample of 200, the percentage of Hispanics is less than 10% Find the probability that 30 or fewer Hispanics are in the sample

15. Chapter 9 - Estimation When given a model (e.g., N(μ, σ 2 )), would like to draw a sample and estimate the parameter in the model Example –The Neilson Corp. draws samples of TV viewers to estimate the proportion of viewers watching various TV shows –If a sample of size n=1000 is taken on Thursday at 8:00 pm, what is the distribution of the number of people watching Friends –How do we estimate the population proportion of people watching Friends?

16. Errors in Estimation (9.1) Idea is to use samples of data to estimate parameters from models The estimators are sample statistics Not all estimators are necessarily good estimators This section looks at errors in estimators

17. Errors in Estimation (9.1) Suppose have a random sample (X 1, X 2,…,X n ) from some population and wish to estimate a parameter θ Let T be the statistic used to estimate θ Although T will vary from sample to sample, would like to be close to (equal to) θ on average T will be viewed as a good estimator of θ if its sampling distribution is centered at θ

18. Errors in Estimation (9.1) Unbiased estimator: Bias MSE:

19. Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a distribution with mean μ and finite variance σ 2, then Show that the sample mean is unbiased What is the MSE of T