2. Copyright © 2009 Pearson Education, Inc. Chapter 18 Sampling Distribution Models

3. Copyright © 2009 Pearson Education, Inc. Slide 1- 3 Objectives: The student will be able to: State and apply the conditions and uses of the Central Limit Theorem. Determine the mean and standard deviation (standard error) for a sampling distribution of proportions or means. Apply the sampling distribution of a proportion or a mean to application problems.

4. Copyright © 2009 Pearson Education, Inc. Slide 1- 4 Sample Proportions and Sampling Distributions The Harris poll found that of 889 U.S. adults, 40% said they believe in ghosts. CBS News found that of 808 U.S. adults, 48% said they believe in ghosts. Why are these two sample proportions different? What is the true population proportion (of ALL U.S. adults)? We’ll denote the population proportion p, and the sample proportion p ^ Consider all possible samples of size 808… if we made a histogram of the number of samples having a given p ^ what might that look like?

5. Copyright © 2009 Pearson Education, Inc. Slide 1- 5 Rather than showing real repeated samples, imagine what would happen if we were to actually draw many samples and look at their proportions. The histogram we’d get if we could see all the proportions from all possible samples is called the sampling distribution of the proportions. What would the histogram of all the sample proportions look like? The Central Limit Theorem for Sample Proportions

6. Copyright © 2009 Pearson Education, Inc. Slide 1- 6 We would expect the histogram of the sample proportions to center at the true proportion, p, in the population. 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. The Central Limit Theorem for Sample Proportions (cont)

7. Copyright © 2009 Pearson Education, Inc. Slide 1- 7 A sampling distribution model for how a sample proportion varies from sample to sample allows us to quantify that variation and how likely it is that we’d observe particular sample proportions Since sampling distribution is normally distributed we can use the full power of the Normal model! To use a Normal model, we need to specify its mean and standard deviation. We’ll put µ, the mean of the Normal, at p. The Central Limit Theorem for Sample Proportions (cont)

8. Copyright © 2009 Pearson Education, Inc. Slide 1- 8 When working with proportions, the standard deviation we will use is So, The Central Limit Theorem for Sample Proportions says: the distribution of the sample proportions is modeled with a probability model that is The Central Limit Theorem for Sample Proportions (cont)

9. Copyright © 2009 Pearson Education, Inc. Slide 1- 9 A picture of what we just discussed is as follows: The Central Limit Theorem for Sample Proportions (cont)

10. Copyright © 2009 Pearson Education, Inc. Another way of saying this…Sampling Distributions Sample statistics are random variables themselves Sample proportion (for categorical data) Sample mean (for quantitative data) They have a probability distribution, mean, standard deviation, etc.

11. Copyright © 2009 Pearson Education, Inc. Slide 1- 11 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 Independence Assumption: The sampled values must be independent of each other. 2. The Sample Size Assumption: The sample size, n, must be large enough.

12. Copyright © 2009 Pearson Education, Inc. Slide 1- 12 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 Randomization Condition,10% Condition and the Success/Failure Condition.

13. Copyright © 2009 Pearson Education, Inc. Slide 1- 13 Assumptions and Conditions (cont.) 1.Randomization Condition: The sample should be a simple random sample of the population. 2.10% Condition: If sampling has not been made with replacement, then the sample size, n, must be no larger than 10% of the population. 3.Success/Failure Condition: The sample size has to be big enough so that both np and nq are at least 10.

14. Copyright © 2009 Pearson Education, Inc. Slide 1- 14 Because we have a Normal model, for example, we know that 95% of Normally distributed values fall within two standard deviations of the mean. So we should not be surprised if 95% of various polls gave results that were near the mean but varied above and below that by no more than two standard deviations. This is what we mean by sampling error. It’s not really an error at all, but just variability you’d expect to see from one sample to another. The Central Limit Theorem for Sample Proportions (cont)

15. Copyright © 2009 Pearson Education, Inc. Worked examples 12) Public Health statistics indicate that 26.4% of American adults smoke cigarettes. Describe the sampling distribution model for the proportion of smokers among a randomly selected group of 50 adults. What are your assumptions and conditions? 15) Based on past experience, a bank believes that 7% of the people who receive loans will not make payments on time. The bank has recently approved 200 loans. What are the mean and standard deviation of the proportion of clients in this group who may not make timely payments? What assumptions underlie your model? Are the conditions met? What is the probability that over 10% of these clients will not make timely payments? Slide 1- 15

16. Copyright © 2009 Pearson Education, Inc. Practice 16) Assume that 30% of students at a university wear contact lenses. We randomly pick 100 students. Let p^ represent the proportion of students who wear contact lenses. What’s the appropriate model for the distribution of p^? Specify the name of the distribution, the mean, and the standard deviation. Be sure the verify that the conditions are met. What’s the approximate probability that more than one third of this sample wear contacts? Slide 1- 16

17. Copyright © 2009 Pearson Education, Inc. Slide 1- 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 © 2009 Pearson Education, Inc. Slide 1- 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 © 2009 Pearson Education, Inc. Slide 1- 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 © 2009 Pearson Education, Inc. Slide 1- 20 Means – Averaging More Dice Looking at the average (mean) of two dice after a simulation of 10,000 tosses: The average (mean) of three dice after a simulation of 10,000 tosses looks like:

21. Copyright © 2009 Pearson Education, Inc. Slide 1- 21 Means – Averaging Still More Dice The average (mean) of 5 dice after a simulation of 10,000 tosses looks like: The average (mean) of 20 dice after a simulation of 10,000 tosses looks like:

22. Copyright © 2009 Pearson Education, Inc. Slide 1- 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 © 2009 Pearson Education, Inc. Slide 1- 23 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.

24. Copyright © 2009 Pearson Education, Inc. Slide 1- 24 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. For example – the result of rolling a die is Uniformly distributed (not normal!) but the sampling distribution is still normal The CLT works better (and faster) the closer the population model is to a Normal itself. It also works better for larger samples. The Central Limit Theorem: The Fundamental Theorem of Statistics (cont.)

25. Copyright © 2009 Pearson Education, Inc. Slide 1- 25 Assumptions and Conditions The CLT requires essentially the same assumptions we saw for modeling proportions:  Independence Assumption: The sampled values must be independent of each other.  Sample Size Assumption: The sample size must be sufficiently large.

26. Copyright © 2009 Pearson Education, Inc. Slide 1- 26 Assumptions and Conditions (cont.) We can’t check these directly, but we can think about whether the Independence Assumption is plausible. We can also check some related conditions: Randomization Condition: The data values must be sampled randomly. 10% Condition: When the sample is drawn without replacement, the sample size, n, should be no more than 10% of the population. Large Enough Sample Condition: The CLT doesn’t tell us how large a sample we need. For now, you need to think about your sample size in the context of what you know about the population.

27. Copyright © 2009 Pearson Education, Inc. Slide 1- 27 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?

28. Copyright © 2009 Pearson Education, Inc. Slide 1- 28 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. So the sampling distribution of means can be modeled by N( μ, σ/√n )

29. Copyright © 2009 Pearson Education, Inc. 38) Statistics indicate that Ithaca, NY gets an average rainfall of 35.4” of rain each year, with a standard deviation of 4.2”. Assume that a Normal model applies During what percentage of years does Ithaca get more than 40” of rain? Less than how much rain falls in the driest 20% of all years? A Cornell student is in Ithaca for 4 years. Let y(bar) represent the mean amount of rain for those 4 years. Describe the sampling distribution model of this sample mean y(bar). What’s the probability that those 4 years average less than 30” of rain? Slide 1- 29

30. Copyright © 2009 Pearson Education, Inc. A restaurateur anticipates serving 180 people on a Friday evening and believes that about 20% of the patrons will order the steak special. How many of those specials should he plan on ordering in order to be 95% sure (i.e. only a 5% chance of running out of food) of having enough steaks on hand to meet customer demand? This yields a proportion of 0.249 (or 45 steaks). Slide 1- 30

31. Copyright © 2009 Pearson Education, Inc. 43) The College Board reported the score distribution shown in the table for all students who took the 2006 AP Statistics Exam: Find the mean and standard deviation of the scores If we select a random sample of 40 AP students would we expect their scores to follow a Normal Model? Consider the mean scores of random samples of 40 AP stats students. Describe the sampling model for these means An AP stats teacher had 63 students preparing to take the AP exam. He considers his students to be “typical” of all the national students. What’s the probability that his students will achieve an average score of at least 3? Slide 1- 31 ScorePercent of Students 512.6 422.2 325.3 218.3 121.6

32. Copyright © 2009 Pearson Education, Inc. 48) The weight of potato chips in a bag is stated to be 10 ounces. The amount that the machine puts in these bags is believed to have a normal model with mean 10.2 oz and standard deviation of 0.12 oz. What fraction of all bags are underweight? Some of the chips are sold in “bargain packs” of 3 bags. What is the probability that none of the 3 is underweight? What’s the probability that the mean weight of the 3 bags is below 10 oz. What’s the probability that the mean weight of a 24-bag case is below 10 oz? Slide 1- 32