Chapter 18 -- Part 1

1. Chapter 18 -- Part 1 Sampling Distribution Models for

2. Sampling Distribution Models Population Parameter? Population Inference Sample Statistic Sample

3. Objectives • Describe the sampling distribution of a sample proportion • Understand that the variability of a statistic depends on the size of the sample • Statistics based on larger samples are less variable

4. Review • Chapter 12 – Sample Surveys • Parameter (Population Characteristics) • m (mean) • p (proportion) • Statistic (Sample Characteristics) • (sample mean) • (sample proportion)

5. Review • Chapter 12 • “Statistics will be different for each sample. These differences obey certain laws of probability (but only for random samples).” • Chapter 14 • Taking a sample from a population is a random phenomena. That means: • The outcome is unknown before the event occurs • The long term behavior is predictable

6. Example • Who? Stat 101 students in Sections G and H. • What? Number of siblings. • When? Today. • Where? In class. • Why? To find out what proportion of students’ have exactly one sibling.

7. Example • Population • Stat 101 students in sections G and H. • Population Parameter • Proportion of all Stat 101 students in sections G and H who have exactly one sibling.

8. Example • Sample • 4 randomly selected students. • Sample Statistic • The proportion of the 4 students who have exactly one sibling.

9. Example • Sample 1 • Sample 2 • Sample 3

10. What Have We Learned • Different samples produce different sample proportions. • There is variation among sample proportions. • Can we model this variation?

11. Example • Senators Population Characteristics • p = proportion of Democratic Senators • Take SRS of size n = 10 • Calculate Sample Characteristics • = sample proportion of Democratic Senators

12. Sample 1 0.2 2 0.5 3 0.6 4 0.3 5 0.7 Example

13. SRS characteristics • Values of and are random • Change from sample to sample • Different from population characteristics • p = 0.50

14. Imagine • Repeat taking SRS of size n = 10 • Collection of values for and ARE DATA • Summarize data – make a histogram • Shape, Center and Spread • Sampling distribution for

15. Sampling Distribution for • Mean (Center) • We would expect on average to get p. • Say is unbiased for p.

16. Sampling Distribution for • Standard deviation (Spread) • As sample size n gets larger, gets smaller • Larger samples are more accurate

17. n=2 n=5 n=10 n=25 Example • 50% of people on campus favor current academic calendar. • 1. Select n people. • 2. Find sample proportion of people favoring current academic calendar. • 3. Repeat sampling. • 4. What does sampling distribution of sample proportion look like?

19. n=2 n=10 n=50 n=100 Example • 10% of all people are left handed. • 1. Select n people. • 2. Find sample proportion of left handed people. • 3. Repeat sampling. • 4. What does sampling distribution of sample proportion look like?

21. Sampling Distribution for • Shape • Normal Distribution • Two assumptions must hold in order for us to be able to use the normal distribution • The sampled values must be independent of each other • The sample size, n, must be large enough

22. Sampling Distribution for • It is hard to check that these assumptions hold, so we will settle for checking the following conditions • 10% Condition – the sample size, n, is less than 10% of the population size • Success/Failure Condition – np > 10, n(1-p) > 10 • These conditions seem to contradict one another, but they don't!

23. Sampling Distribution for • Assuming the two conditions are true (must be checked for each problem), then the sampling distribution for is

24. Sampling Distribution for • But the sampling distribution has a center (mean) of p (a population proportion) often times we don’t know p. • Let be the center.

25. Example • Senators • Check assumptions (p = 0.50) • 10(0.50) = 5 and 10(0.50) = 5 • n = 10 is 10% of the population size. • Assumption 1 does not hold. • Sampling Distribution of ????

26. Example #1 • Public health statistics indicate that 26.4% of the U.S. adult population smoked cigarettes in 2002. Use the 68-95-99.7 Rule to describe the sampling distribution for the sample proportion of smokers among 50 adults.

27. Example #1 • Check assumptions: • np = (50)(0.264) = 13.2 > 10 nq = (50)(0.736) = 36.8 > 10 • n = 50, less than 10% of population • Therefore, the sampling distribution for the proportion of smokers is

28. Example # 1 • About 68% of samples have a sample proportion between 20.2% and 32.6% • About 95% of samples have a sample proportion between 14% and 38.8% • About 99.7% of samples have a sample proportion between 7.8% and 45%

29. Example #2 • Information on a packet of seeds claims that the germination rate is 92%. What's the probability that more than 95% of the 160 seeds in the packet will germinate? • Check assumptions:1. np = (160)(0.92) = 147.2 > 10 nq = (160)(0.08) = 12.8 > 10 2. n = 160, less than 10% of all seeds?

30. Review - Standardizing • You can standardize using the formula

31. Review • Chapter 6 – The Normal Distribution • Y~ N(70,3) • Do you remember the 68-95-99.7 Rule?

32. Example #2 • Therefore, the sampling distribution for the proportion of seeds that will germinate is

33. Big Picture Population Parameter? Population Inference Sample Statistic Sample

34. Big Picture • Before we would take one random sample and compute our sample statistic. Presently we are focusing on: This is an estimate of the population parameter p. • But we realized that if we took a second random sample that from sample 1 could possibly be different from the we would get from sample 2. But from sample 2 is also an estimate of the population parameter p. • If we take a third sample then the for third sample could possibly be different from the first and seconds. Etc.

35. Big Picture • So there is variability in the sample statistic . • If we randomized correctly we can consider as random (like rolling a die) so even though the variability is unavoidable it is understandable and predictable!!! (This is the absolutely amazing part).

36. Big Picture • So for a sufficiently large sample size (n) we can model the variability in with a normal model so:

37. Big Picture • The hard part is trying to visualize what is going on behind the scenes. The sampling distribution of is what a histogram would look like if we had every possible sample available to us. (This is very abstract because we will never see these other samples). • So lets just focus on two things:

38. Take Home Message • 1. Check to see that • A. the sample size, n, is less than 10% of the population size • B. np > 10, n(1-p) > 10 • 2. If these hold then can be modeled with a normal distribution that is:

39. Example #3 • When a truckload of apples arrives at a packing plant, a random sample of 150 apples is selected and examined for bruises, discoloration, and other defects. The whole truckload will be rejected if more than 5% of the sample is unsatisfactory (i.e. damaged). Suppose that actually 8% of the apples in the truck do not meet the desired standard. What is the probability of accepting the truck anyway?

40. Example #3 • What is the sampling distribution? • np = (150)(0.08) = 12>10nq = (150)(0.92) = 138>10 • n = 150 > 10% of all applesSo, the sampling distribution is N(0.08,0.022).What is the probability of accepting the truck anyway?