1. Stat 512 – Day 7 Probability Models, Inference (Ch. 4)

2. Last Time – Random Sampling Need an unbiased method for selecting samples from a larger population so you believe sample obtained is representative of the population  Simple random sample = every sample of size n has the same chance of being selected  Other probability methods: systematic, multistage, cluster, and stratified sampling Goal: Allows us to generalize sample results back to the population of interest

3. Last Time - Sampling Compare results Randomized? Getting the observational units in the first place! Population Random? sample parameter statistic

4. Scope of Conclusions Subjects randomly assigned to groups? YesNo Subjects randomly selected fromYes population?No Can generalize to population Can draw cause-and- effect conclusions

5. But: Nonsampling Errors Even if sample is properly selected, are other sources of bias and misleading results

6. Example Should Cal Poly allow speeches on campus that might incite violence?  13 Yes, 10 No Should Cal Poly forbid speeches on campus that might incite violence  4 Yes, 18 No Lost ticket, would you buy another?  8 Yes, 13 No Lost $20, would you buy another?  20 Yes, 3 No 200 lives saved?  12 Program A, 11 Program B 400 lives lost?  4 Program A, 18 Program B Prediction: more likely if lost ticket Prediction: Program A more likely when in terms of lives saved

7. Nonsampling Errors March 6-8, 2004 Wall Street Journal/NBC poll of 1,018 adults GAY MARRIAGE opinions depend on how the question is asked. To one poll question, a 52%-43% majority opposes a constitutional amendment "making it illegal for gay couples to marry." A 54%-42% majority responds favorably to a second query that omits the word "illegal" and more benignly asks about an amendment "that defined marriage as a union only between a man and a woman."

8. Sources of Nonsampling Errors Sensitive questions  Social acceptability Wording of question  Collided vs. made contact Appearance of interviewer Delivery of questions, instructions Order of choices Unsure response, change mind, faulty memory

9. Second Benefit of Random Sampling Sample statistics follow a predictable, probabilistic pattern Will always be some random sampling error but we will be able to estimate its size…

10. Second Benefit of Random Sampling Population sample random?

11. Second Benefit of Random Sampling Population sample pop mean

12. Second Benefit of Random Sampling Population sample =sample mean pop mean

13. Second Benefit of Random Sampling Population sample pop mean Sampling distribution

14. Last Time parameter Sampling method is unbiased

15. Which is “better?” parameter

16. Last Time

17. Population size…

18. Population sample pop mean

19. Example 1: Sampling Penny Ages Suppose I take repeated samples of n=5 pennies and calculate the date on the pennies  Expected shape of the population?  Behavior of sampling distribution?

20. Sampling Penny Ages http://statweb.calpoly.edu/chance/applets/ SampleData/SampleData.html

21. Sampling Penny Ages Distribution of sample means will be reasonably symmetric if  Sample size is large Convention: n > 30 OR  Population distribution is normal Standard deviation of the sample means follows the formula

22. Normal Distribution N(  )

23. Example 2: Body Temperatures The probability of temp < 98.2 is.2919 About 29% of randomly selected adults would have a temperature at most 98.2 0 F.

24. Example 2: Body Temperatures There is only a 3% chance that a random person will have a temperature in excess of 100 0 F.

25. Example 2: Body Temperatures Less than.0001 probability that a sample would have an average temperature below 98.2 0 F

26. Example 2: Body Temperature We have strong evidence that a random sample of 130 adults from a population with  =98.6 0 F would give a sample mean of 98.2 or less by chance alone So after observing a sample mean of 98.2, we have strong evidence that the statement that  =98.6 0 F is wrong

27. Example 3: Reese’s Pieces

28. Distribution of Sample Simple Bar Graph

29. Sampling Distribution About 6% of random samples (with 25 candies) will have a sample proportion of.60 or higher when  =.45.

30. Sampling Distribution About.4% of random samples (with 75 candies) will have a sample proportion of.60 or higher when  =.45.

31. When n=3 Can’t use the CLT! Would need other methods to help your friend

32. For Thursday Submit PP 7 (3 pts) Read Ch. 6  Lots of new terminology but same basic reasoning… HW 4 (by Friday)