1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 18 Sampling Distribution Models

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 3 Experiment – Holiday Nerds Each member of the class will receive a 1 teaspoon sample of holiday Nerd candies. Find the proportion (percentage) of green Nerds in you sample. Mark your proportion on the dot plot on the board. Questions… What is the shape of this sampling model? What is its mean?

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 4 Modeling the Distribution of Sample Proportions (cont.) It turns out that the histogram is unimodal, symmetric, and centered at p. More specifically, it’s an amazing and fortunate fact that a Normal model is just the right one for the histogram of sample proportions. To use a Normal model, we need to specify its mean and standard deviation. The mean of this particular Normal is at p.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 5 Modeling the Distribution of Sample Proportions (cont.) When working with proportions, knowing the mean automatically gives us the standard deviation as well—the standard deviation we will use is So, the distribution of the sample proportions is modeled with a probability model that is

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 6 Modeling the Distribution of Sample Proportions (cont.) A picture of what we just discussed is as follows:

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 7 How Good Is the Normal Model? The Normal model gets better as a good model for the distribution of sample proportions as the sample size gets bigger. Just how big of a sample do we need? This will soon be revealed…

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 8 Assumptions and Conditions Most models are useful only when specific assumptions are true. There are two assumptions in the case of the model for the distribution of sample proportions: 1. The sampled values must be independent of each other. 2. The sample size, n, must be large enough.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 9 Assumptions and Conditions (cont.) Assumptions are hard—often impossible—to check. That’s why we assume them. Still, we need to check whether the assumptions are reasonable by checking conditions that provide information about the assumptions. The corresponding conditions to check before using the Normal to model the distribution of sample proportions are the 10% Condition and the Success/Failure Condition.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 10 Assumptions and Conditions (cont.) 1.10% condition: If sampling has not been made with replacement, then the sample size, n, must be no larger than 10% of the population. 2.Success/failure condition: The sample size has to be big enough so that both and are greater than 10. So, we need a large enough sample that is not too large.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 11 The Sampling Distribution Model for a Proportion (cont.) Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with Mean: Standard deviation:

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 12 Example 1: Cars on the Interstate Of all cars on the interstate, 80% exceed the speed limit. What proportion of speeders might we see among the next 50 cars? Check 10% condition: Check Success/Failure Condition: Find mean and standard deviation. Explain the model in context.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 13 Example continued… 1. 50 is less than 10% of all cars on the highway. 2. np = 50(.8) = 40 and nq = 50(.2) = 10 (This model is approximately Normal) 3. 4. Normal(.8,.057) is the model for p hat. 5. Discuss.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 14 Example 2: Harley-Davidson Motorcycles Harley-Davidson motorcycles make up 14% of all the motorcycles registered in the United States. You plan to interview an SRS of 500 motorcycle owners. How likely is your sample to contain 20% of more who own Harley’s?

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 15 Example 3: Lefties Suppose that about 13% of the population is left- handed. A 200 seat school auditorium has been built with 15 “lefty seats”, seats that have the built- in desk on the left rather than the right arm of the chair. In a class of 90 students, what’s the probability that there will not be enough seats for the left-handed students?

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 16 Example 3 continued…. You are trying to find the probability that more than 15 out of 90 students will be lefties. (This is 16.7%) 1. 90 is less than 10% of the student body (think BHS) 2. np = 90(.13) = 11.7 ≥ 10 nq = 90(.87) = 78.3 ≥ 10 (approximately Normal) 3. N(13,.035) 4. P(p-hat>.167) = P(z>1.06) =.1446 5. There is about a 14.5% chance that there will not be enough seats for the left-handed students in the class.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 17 What About Quantitative Data? Proportions summarize categorical variables. The Normal sampling distribution model looks like it will be very useful. Can we do something similar with quantitative data? We can indeed. Even more remarkable, not only can we use all of the same concepts, but almost the same model.

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 18 Simulating the Sampling Distribution of a Mean Like any statistic computed from a random sample, a sample mean also has a sampling distribution. We can use simulation to get a sense as to what the sampling distribution of the sample mean might look like…

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 19 Means – The “Average” of One Die Let’s start with a simulation of 10,000 tosses of a die. A histogram of the results is:

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 20 Means – Averaging More Dice Looking at the average of two dice after a simulation of 10,000 tosses: The average of three dice after a simulation of 10,000 tosses looks like:

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 21 Means – Averaging Still More Dice The average of 5 dice after a simulation of 10,000 tosses looks like: The average of 20 dice after a simulation of 10,000 tosses looks like:

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 22 Means – What the Simulations Show As the sample size (number of dice) gets larger, each sample average is more likely to be closer to the population mean. So, we see the shape continuing to tighten around 3.5 And, it probably does not shock you that the sampling distribution of a mean becomes Normal.

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 23 The Fundamental Theorem of Statistics The sampling distribution of any mean becomes Normal as the sample size grows. All we need is for the observations to be independent and collected with randomization. We don’t even care about the shape of the population distribution! The Fundamental Theorem of Statistics is called the Central Limit Theorem (CLT).

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 24 The Fundamental Theorem of Statistics (cont.) The CLT is surprising and a bit weird: Not only does the histogram of the sample means get closer and closer to the Normal model as the sample size grows, but this is true regardless of the shape of the population distribution. The CLT works better (and faster) the closer the population model is to a Normal itself. It also works better for larger samples.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 25 The Fundamental Theorem of Statistics (cont.) The Central Limit Theorem (CLT) The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better the approximation will be.

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 26 But Which Normal? The CLT says that the sampling distribution of any mean or proportion is approximately Normal. But which Normal model? For proportions, the sampling distribution is centered at the population proportion. For means, it’s centered at the population mean. But what about the standard deviations?

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 27 But Which Normal? (cont.) The Normal model for the sampling distribution of the mean has a standard deviation equal to where σ is the population standard deviation.

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 28 But Which Normal? (cont.) The Normal model for the sampling distribution of the proportion has a standard deviation equal to

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 29 Assumptions and Conditions The CLT requires remarkably few assumptions, so there are few conditions to check: 1.Random Sampling Condition: The data values must be sampled randomly or the concept of a sampling distribution makes no sense. 2.Independence Assumption: The sample values must be mutually independent. (When the sample is drawn without replacement, check the 10% condition…) 3.Large Enough Sample Condition: There is no one- size-fits-all rule.

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 30 Diminishing Returns The standard deviation of the sampling distribution declines only with the square root of the sample size. While we’d always like a larger sample, the square root limits how much we can make a sample tell about the population. (This is an example of the Law of Diminishing Returns.)

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 31 Example 4: SAT Scores SAT scores should have a mean of 500 and a standard deviation of 100. What about the mean of random samples of 20 students? The college board assures us that SAT results are Normal. Check random sampling condition. Check 10% condition. Check Independence Assumption. What does this mean in context?

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 32 Example continued… 1. The 20 students are selected randomly. 2. 20 students represents less than 10% of all students. 3. Assumption: students SAT scores are independent, as long as they weren’t all from the same university. Discuss!!!!!!! What is the probability that your sample of students will have a mean SAT score that is less than 450?

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 33 Example 5: Cars again Speeds of cars on a highway have mean 52 mph and standard deviation of 6 mph, and are likely to be skewed to the right (a few very fast drivers). Describe what we might see in random samples of 50 cars. Check random sampling condition. Check 10% condition. Check Independence Assumption. What does this mean in context?

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 34 Example continued… 1. The 50 speeds were sampled randomly. 2. 50 cars is less than 10% of all cars. 3. It’s reasonable to think that the speeds of the cars are mutually independent. 4. The central Limit Theorem applies, because n is large. What is the probability that the mean speed of your sample of cars will be no more than 45 mph?

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 35 Example 6: Babies at Birth At birth, babies average 7.8 pounds, with a standard deviation of 2.1 pounds. A random sample of 34 babies born to mothers living near a large factory that may be polluting the air and water shows a mean birth weight of only 7.2 pounds. Is that unusually low?

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 36 Example continued… 1. 34 babies were sampled randomly. 2. As long as 340 babies were born to mothers living in the vicinity of the factory, 34 is less than 10%. 3. It’s reasonable to assume that the weights of the babies are mutually independent. 4. The central limit theorem applies since the sample size is big. Is 7.2 pounds unusually low? z = -1.67. This is within 2 standard deviations from the mean. It is low, but not unusually low. 4.7% of the samples have birth weights less than 7.2 pounds.

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 37 Standard Error Both of the sampling distributions we’ve looked at are Normal. For proportions For means

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 38 Standard Error (cont.) When we don’t know p or σ, we’re stuck, right? Nope. We will use sample statistics to estimate these population parameters. Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 18- 39 Standard Error (cont.) For a sample proportion, the standard error is For the sample mean, the standard error is