# Chapter 18. Sampling Distribution Models. Copyright 2010, 2007, 2004 Pearson Education, Inc.

Size: px
Start display at page:

Transcription

1

2 Chapter 18 Sampling Distribution Models Copyright 2010, 2007, 2004 Pearson Education, Inc.

3 Normal Model When we talk about one data value and the Normal model we used the notation: N(μ, σ) Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-3

4 Scenario Suppose everyone in AP Stats at Geneva High School took a survey which entailed the question Do you believe in ghosts? Are the answers received going to be categorical or quantitative? Would everyone get the same proportion of students that said yes and no? Why or why not? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-4

5 Categorical Data Deals with sample proportions Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-5

6 The Central Limit Theorem for Sample Proportions Rather than showing real repeated samples, imagine what would happen if we were to actually draw many samples. Now imagine what would happen if we looked at the sample proportions for these samples. 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? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-6

7 Modeling the Distribution of Sample Proportions (cont.) We would expect the histogram of the sample proportions to center at the true proportion, p, in the population. As far as the shape of the histogram goes, we can simulate a bunch of random samples that we didn t really draw. 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. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-7

8 Modeling the Distribution of Sample Proportions (cont.) Modeling how sample proportions vary from sample to sample is one of the most powerful ideas we ll see in this course. 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 a sample proportion in any particular interval. 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. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-8

9 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 N p, pq n pq n Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-9

10 Modeling the Distribution of Sample Proportions (cont.) A picture of what we just discussed is as follows: Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-10

11 The Central Limit Theorem for Sample Proportions (cont) Because we have a Normal model, for example, we know that 95% of Normally distributed values are 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. A better term would be sampling variability. A reasonable variance/error is ±2σ Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-11

12 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 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-12

13 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. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-13

14 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, the 10% Condition and the Success/Failure Condition. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-14

15 Assumptions and Conditions (cont.) 1. Randomization Condition: The sample should be a simple random sample of the population % Condition: 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 (number of successes) and nq (number of failures) are at least 10. So, we need a large enough sample that is not too large. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-15

16 A Sampling Distribution Model for a Proportion A proportion is no longer just a computation from a set of data. It is now a random variable quantity that has a probability distribution. This distribution is called the sampling distribution model for proportions. Even though we depend on sampling distribution models, we never actually get to see them. We never actually take repeated samples from the same population and make a histogram. We only imagine or simulate them. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-16

17 A Sampling Distribution Model for a Proportion (cont.) Still, sampling distribution models are important because they act as a bridge from the real world of data to the imaginary world of the statistic and enable us to say something about the population when all we have is data from the real world. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-17

18 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 ˆp (sample proportion of success) is modeled by a Normal model with Mean: ( ˆp) p Standard deviation: SD( ˆp) pq n Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-18

19 Example p. 434 #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. a) What are the mean and standard deviation of the proportion of clients in this group who may not make timely payments? b) What assumptions underlie you model? Are the conditions met? Explain. c) What s the probability that over 10% of these clients will not make timely payments? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-19

20 Ch 18 Homework p. 432 #3, 5, 7, 9, 16, 22 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-20

21 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. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-21

22 Quantitative Data Deals with sample means Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-22

23 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 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-23

24 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: Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-24

25 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: Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-25

26 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: Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-26

27 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. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-27

28 The Fundamental Theorem of Statistics The sampling distribution of any mean becomes more nearly 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). Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-28

29 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 more skewed, the larger the sample size we need. The CLT works better (and faster) the closer the population model is to a Normal itself. It also works better for larger samples. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-29

30 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. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-31

31 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. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-32

32 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? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-33

33 But Which Normal? (cont.) The Normal model for the sampling distribution of the mean has a standard deviation equal to SD y n where σ is the population standard deviation. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-34

34 Central Limit Theorem So the normal model dealing with sample means is written like N(, n ) Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-36

35 About Variation The standard deviation of the sampling distribution declines only with the square root of the sample size (the denominator contains the square root of n). Therefore, the variability decreases as the sample size increases. 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.) Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-37

36 Example p. 437 #49 A waiter believes the distribution of his tips has a model that is slightly skewed to the right, with a mean of \$9.60 and a standard deviation of \$5.40. a) Explain why you cannot determine the probability that a given party will tip him at least \$20. b) Can you estimate the probability that the next 4 parties will tip an average of at least \$15? Explain. c) Is it likely that his 10 parties today will tip an average of at least \$15? Explain. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-38

37 k Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-39

38 Homework Chapter 18 Homework: p. 432 #31, 33, 37, 50, Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-40

39 The Real World and the Model World Be careful! Now we have two distributions to deal with. The first is the real world distribution of the sample, which we might display with a histogram. The second is the math world sampling distribution of the statistic, which we model with a Normal model based on the Central Limit Theorem. Don t confuse the two! Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-41

40 Sampling Distribution Models Always remember that the statistic itself is a random quantity. We can t know what our statistic will be because it comes from a random sample. Fortunately, for the mean and proportion, the CLT tells us that we can model their sampling distribution directly with a Normal model. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-42

41 Sampling Distribution Models (cont.) There are two basic truths about sampling distributions: 1. Sampling distributions arise because samples vary. Each random sample will have different cases and, so, a different value of the statistic. 2. Although we can always simulate a sampling distribution, the Central Limit Theorem saves us the trouble for means and proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-43

42 What Can Go Wrong? Don t confuse the sampling distribution with the distribution of the sample. When you take a sample, you look at the distribution of the values, usually with a histogram, and you may calculate summary statistics. The sampling distribution is an imaginary collection of the values that a statistic might have taken for all random samples the one you got and the ones you didn t get. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-45

43 What Can Go Wrong? (cont.) Beware of observations that are not independent. The CLT depends crucially on the assumption of independence. You can t check this with your data you have to think about how the data were gathered. Watch out for small samples from skewed populations. The more skewed the distribution, the larger the sample size we need for the CLT to work. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-46

44 What have we learned? Sample proportions and means will vary from sample to sample that s sampling error (sampling variability). Sampling variability may be unavoidable, but it is also predictable! Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-47

45 What have we learned? (cont.) We ve learned to describe the behavior of sample proportions when our sample is random and large enough to expect at least 10 successes and failures. We ve also learned to describe the behavior of sample means (thanks to the CLT!) when our sample is random (and larger if our data come from a population that s not roughly unimodal and symmetric). Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-48

46 Homework Chapter 18 Homework: p. 432 #2(1), 4(3), 6(5), 8(7), 10(9), 16, 22, 30(29), 34, 38, 50, 52 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-49

47 Example p. 432 #1 See book Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-50

48 Example p. 432 #3 The philanthropic organization in Exercise 1 expects about a 5% success rate when they send fundraising letters to the people on their mailing list. In Exercise 1 you looked at the histograms showing distributions of sample proportions from 1000 simulated mailings for samples of size 20, 50, 100, and 200. The sample statistics from each simulation were as follows: a) According to the CLT, what should the theoretical mean and standard deviation be for these sample sizes? n Mean St. Dev Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-51

49 Example p. 432 #3 The philanthropic organization in Exercise 1 expects about a 5% success rate when they send fundraising letters to the people on their mailing list. In Exercise 1 you looked at the histograms showing distributions of sample proportions from 1000 simulated mailings for samples of size 20, 50, 100, and 200. The sample statistics from each simulation were as follows: b) How close are those n Mean St. Dev theoretical values to what was observed in these simulations? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-52

50 Example p. 432 #3 The philanthropic organization in Exercise 1 expects about a 5% success rate when they send fundraising letters to the people on their mailing list. In Exercise 1 you looked at the histograms showing distributions of sample proportions from 1000 simulated mailings for samples of size 20, 50, 100, and 200. The sample statistics from each simulation were as follows: c) Looking at the histograms n Mean St. Dev in Exercise 1, at what sample size would you be comfortable using the Normal model as an approximation for the sampling distribution? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-53

51 Example p. 432 #3 The philanthropic organization in Exercise 1 expects about a 5% success rate when they send fundraising letters to the people on their mailing list. In Exercise 1 you looked at the histograms showing distributions of sample proportions from 1000 simulated mailings for samples of size 20, 50, 100, and 200. The sample statistics from each simulation were as follows: d) What does the n Mean St. Dev Success/Failure Condition say about the choice you made in part c? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-54

52 Example p. 433 #5 In a large class of introductory Statistics students, the professor has each person toss a coin 16 times and calculate the proportion of his or her tosses that were heads. The students then report their results, and the professor plots a histogram of these several proportions. d) Explain why a Normal model should not be used here. a) What shape would you expect this histogram to be? Why? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-55

53 Example p. 433 #5 In a large class of introductory Statistics students, the professor has each person toss a coin 16 times and calculate the proportion of his or her tosses that were heads. The students then report their results, and the professor plots a histogram of these several proportions. b) Where do you expect the histogram to be centered? c) How much variability would you expect among these proportions? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-56

54 Example p. 433 #7 Suppose the class in Exercise 5 repeats the cointossing experiment. a) The students toss the coins 25 times each. Use the Rule to describe the sampling distribution. b) Confirm that you can use the Normal model here. c) They increase the number of tosses to 64 each. Draw and label the appropriate sampling distribution model. Check the appropriate conditions to justify your model. d) Explain how the sampling distribution model changes as the number of tosses increases. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-57

55 Example p. 433 #9 One of the students in the introductory Statistics class in Exercise 7 claims to have tossed her coin 200 times and found only 42% heads. What do you think of this claim? Explain. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-58

56 Example p. 434 #21 Just before a referendum on a school budget, a local newspaper polls 400 random voters in an attempt to predict whether the budget will pass. Suppose that the budget actually has the support of 52% of the voters. What s the probability the newspaper s sample will lead them to predict defeat (less than 50%)? Be sure to verify that the assumptions and conditions necessary for our analysis are met. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-59

57 Example p. 434 #29 See book Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-60

58 Example p. 432 #31 Researchers measured the Waist Sizes of 250 men in a study on body fat. The true mean and standard deviation of the Waist Sizes for the 250 men are in and in, respectively. In Exercise 29 you looked at the histograms of simulations that drew samples of sizes 2, 5, 10, and 20 (with replacement). The summary statistics for these simulations were as follows: n Mean St. Dev a) According to the CLT, what should the theoretical mean and standard deviation be for each of these sample sizes? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-61

59 Example p. 432 #31 Researchers measured the Waist Sizes of 250 men in a study on body fat. The true mean and standard deviation of the Waist Sizes for the 250 men are in and in, respectively. In Exercise 29 you looked at the histograms of simulations that drew samples of sizes 2, 5, 10, and 20 (with replacement). The summary statistics for these simulations were as follows: n Mean St. Dev b) How close are those theoretical values to what was observed in these simulations? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-62

60 Example p. 432 #31 Researchers measured the Waist Sizes of 250 men in a study on body fat. The true mean and standard deviation of the Waist Sizes for the 250 men are in and in, respectively. In Exercise 29 you looked at the histograms of simulations that drew samples of sizes 2, 5, 10, and 20 (with replacement). The summary statistics for these simulations were as follows: n Mean St. Dev c) Looking at the histograms in Exercise 29, at what sample size would you be comfortable using the Normal model as an approximation for the sampling distribution? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-63

61 Example p. 432 #31 Researchers measured the Waist Sizes of 250 men in a study on body fat. The true mean and standard deviation of the Waist Sizes for the 250 men are in and in, respectively. In Exercise 29 you looked at the histograms of simulations that drew samples of sizes 2, 5, 10, and 20 (with replacement). The summary statistics for these simulations were as follows: d) What about the shape of n Mean St. Dev the distribution of Waist Size explains your choice of sample size in part c? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-64

62 Example p. 436 #33 A college s data about the incoming freshman indicates that the mean of their high school GPAs was 3.4, with a standard deviation of 0.35; the distribution was roughly mound-shaped and only slightly skewed. The students are randomly assigned to freshman writing seminars in groups of 25. What might the mean GPA of one of these seminar groups be? Describe the appropriate sampling distribution model shape, center, spread with attention to assumptions and conditions. Make a sketch using the Rule. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-65

63 Example p. 437 #49 A waiter believes the distribution of his tips has a model that is slightly skewed to the right, with a mean of \$9.60 and a standard deviation of \$5.40. a) Explain why you cannot determine the probability that a given party will tip him at least \$20. b) Can you estimate the probability that the next 4 parties will tip an average of at least \$15? Explain. c) Is it likely that his 10 parties today will tip an average of at least \$15? Explain. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-66

64 k Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-67

65 Example p. 438 #51 A waiter believes the distribution of his tips has a model that is slightly skewed to the right, with a mean of \$9.60 and a standard deviation of \$5.40. The waiter usually waits on about 40 parties over a weekend of work. a) Estimate the probability that he will earn at least \$500 in tips. b) How much does he earn on the best 10% of such weekends? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-68

66 Homework Chapter 18 Homework: p. 432 #2(1), 4(3), 6(5), 8(7), 10(9), 16, 22, 30(29), 32(31), 34, 38, 50(49), 52(51) Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-69

67 Practice 1 Groovy M&M s are supposed to make up 30% of the candies sold. In a large bag of 250 M&M s, what is the probability that we get at least 25% groovy candies? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-70

68 Practice 2 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? Explain. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 18-71

### STA Why Sampling? Module 6 The Sampling Distributions. Module Objectives

STA 2023 Module 6 The Sampling Distributions Module Objectives In this module, we will learn the following: 1. Define sampling error and explain the need for sampling distributions. 2. Recognize that sampling

### Sampling Distribution Models. Chapter 17

Sampling Distribution Models Chapter 17 Objectives: 1. Sampling Distribution Model 2. Sampling Variability (sampling error) 3. Sampling Distribution Model for a Proportion 4. Central Limit Theorem 5. Sampling

### Chapter 18. Sampling Distribution Models /51

Chapter 18 Sampling Distribution Models 1 /51 Homework p432 2, 4, 6, 8, 10, 16, 17, 20, 30, 36, 41 2 /51 3 /51 Objective Students calculate values of central 4 /51 The Central Limit Theorem for Sample

### Sampling Distribution Models. Central Limit Theorem

Sampling Distribution Models Central Limit Theorem Thought Questions 1. 40% of large population disagree with new law. In parts a and b, think about role of sample size. a. If randomly sample 10 people,

### STA Module 8 The Sampling Distribution of the Sample Mean. Rev.F08 1

STA 2023 Module 8 The Sampling Distribution of the Sample Mean Rev.F08 1 Module Objectives 1. Define sampling error and explain the need for sampling distributions. 2. Find the mean and standard deviation

### Chapter 18 Sampling Distribution Models

Chapter 18 Sampling Distribution Models The histogram above is a simulation of what we'd get if we could see all the proportions from all possible samples. The distribution has a special name. It's called

### Chapter 15 Sampling Distribution Models

Chapter 15 Sampling Distribution Models 1 15.1 Sampling Distribution of a Proportion 2 Sampling About Evolution According to a Gallup poll, 43% believe in evolution. Assume this is true of all Americans.

### ACMS Statistics for Life Sciences. Chapter 13: Sampling Distributions

ACMS 20340 Statistics for Life Sciences Chapter 13: Sampling Distributions Sampling We use information from a sample to infer something about a population. When using random samples and randomized experiments,

### MAT Mathematics in Today's World

MAT 1000 Mathematics in Today's World Last Time We discussed the four rules that govern probabilities: 1. Probabilities are numbers between 0 and 1 2. The probability an event does not occur is 1 minus

### Chapter 7. Scatterplots, Association, and Correlation. Copyright 2010 Pearson Education, Inc.

Chapter 7 Scatterplots, Association, and Correlation Copyright 2010 Pearson Education, Inc. Looking at Scatterplots Scatterplots may be the most common and most effective display for data. In a scatterplot,

### Chapter 8. Linear Regression. Copyright 2010 Pearson Education, Inc.

Chapter 8 Linear Regression Copyright 2010 Pearson Education, Inc. Fat Versus Protein: An Example The following is a scatterplot of total fat versus protein for 30 items on the Burger King menu: Copyright

### Chapter 6. September 17, Please pick up a calculator and take out paper and something to write with. Association and Correlation.

Please pick up a calculator and take out paper and something to write with. Sep 17 8:08 AM Chapter 6 Scatterplots, Association and Correlation Copyright 2015, 2010, 2007 Pearson Education, Inc. Chapter

### Chapter 18 Summary Sampling Distribution Models

Uit 5 Itroductio to Iferece Chapter 18 Summary Samplig Distributio Models What have we leared? Sample proportios ad meas will vary from sample to sample that s samplig error (samplig variability). Samplig

### Chapter 14. From Randomness to Probability. Copyright 2012, 2008, 2005 Pearson Education, Inc.

Chapter 14 From Randomness to Probability Copyright 2012, 2008, 2005 Pearson Education, Inc. Dealing with Random Phenomena A random phenomenon is a situation in which we know what outcomes could happen,

### Chapter 24. Comparing Means. Copyright 2010 Pearson Education, Inc.

Chapter 24 Comparing Means Copyright 2010 Pearson Education, Inc. Plot the Data The natural display for comparing two groups is boxplots of the data for the two groups, placed side-by-side. For example:

### Chapter 6 The Standard Deviation as a Ruler and the Normal Model

Chapter 6 The Standard Deviation as a Ruler and the Normal Model Overview Key Concepts Understand how adding (subtracting) a constant or multiplying (dividing) by a constant changes the center and/or spread

### From the Data at Hand to the World at Large

From the Data at Hand to the World at Large PART V Chapter 18 Sampling Distribution Models Chapter 19 Confidence Intervals for Proportions Chapter 20 Testing Hypotheses About Proportions Chapter 21 More

### Linear Regression. Linear Regression. Linear Regression. Did You Mean Association Or Correlation?

Did You Mean Association Or Correlation? AP Statistics Chapter 8 Be careful not to use the word correlation when you really mean association. Often times people will incorrectly use the word correlation

### appstats8.notebook October 11, 2016

Chapter 8 Linear Regression Objective: Students will construct and analyze a linear model for a given set of data. Fat Versus Protein: An Example pg 168 The following is a scatterplot of total fat versus

### Chapter 8. Linear Regression. The Linear Model. Fat Versus Protein: An Example. The Linear Model (cont.) Residuals

Chapter 8 Linear Regression Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8-1 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Fat Versus

### 1. Create a scatterplot of this data. 2. Find the correlation coefficient.

How Fast Foods Compare Company Entree Total Calories Fat (grams) McDonald s Big Mac 540 29 Filet o Fish 380 18 Burger King Whopper 670 40 Big Fish Sandwich 640 32 Wendy s Single Burger 470 21 1. Create

### Chapter 7 Summary Scatterplots, Association, and Correlation

Chapter 7 Summary Scatterplots, Association, and Correlation What have we learned? We examine scatterplots for direction, form, strength, and unusual features. Although not every relationship is linear,

### Chapter 27 Summary Inferences for Regression

Chapter 7 Summary Inferences for Regression What have we learned? We have now applied inference to regression models. Like in all inference situations, there are conditions that we must check. We can test

### appstats27.notebook April 06, 2017

Chapter 27 Objective Students will conduct inference on regression and analyze data to write a conclusion. Inferences for Regression An Example: Body Fat and Waist Size pg 634 Our chapter example revolves

### Chapter 15. Probability Rules! Copyright 2012, 2008, 2005 Pearson Education, Inc.

Chapter 15 Probability Rules! Copyright 2012, 2008, 2005 Pearson Education, Inc. The General Addition Rule When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter

### Lesson 19: Understanding Variability When Estimating a Population Proportion

Lesson 19: Understanding Variability When Estimating a Population Proportion Student Outcomes Students understand the term sampling variability in the context of estimating a population proportion. Students

### AP Statistics. Chapter 6 Scatterplots, Association, and Correlation

AP Statistics Chapter 6 Scatterplots, Association, and Correlation Objectives: Scatterplots Association Outliers Response Variable Explanatory Variable Correlation Correlation Coefficient Lurking Variables

### Statistic: a that can be from a sample without making use of any unknown. In practice we will use to establish unknown parameters.

Chapter 9: Sampling Distributions 9.1: Sampling Distributions IDEA: How often would a given method of sampling give a correct answer if it was repeated many times? That is, if you took repeated samples

### CHAPTER 18 SAMPLING DISTRIBUTION MODELS STAT 203

1 CHAPTER 18 SAMPLING DISTRIBUTION MODELS STAT 203 Outline 2 Sampling Distribution for Proportions Sample Proportions The mean The standard deviation The Distribution Model Assumptions and Conditions Sampling

### STA Module 4 Probability Concepts. Rev.F08 1

STA 2023 Module 4 Probability Concepts Rev.F08 1 Learning Objectives Upon completing this module, you should be able to: 1. Compute probabilities for experiments having equally likely outcomes. 2. Interpret

### Two-sample Categorical data: Testing

Two-sample Categorical data: Testing Patrick Breheny April 1 Patrick Breheny Introduction to Biostatistics (171:161) 1/28 Separate vs. paired samples Despite the fact that paired samples usually offer

### MA 1125 Lecture 15 - The Standard Normal Distribution. Friday, October 6, Objectives: Introduce the standard normal distribution and table.

MA 1125 Lecture 15 - The Standard Normal Distribution Friday, October 6, 2017. Objectives: Introduce the standard normal distribution and table. 1. The Standard Normal Distribution We ve been looking at

### *Karle Laska s Sections: There is no class tomorrow and Friday! Have a good weekend! Scores will be posted in Compass early Friday morning

STATISTICS 100 EXAM 3 Spring 2016 PRINT NAME (Last name) (First name) *NETID CIRCLE SECTION: Laska MWF L1 Laska Tues/Thurs L2 Robin Tu Write answers in appropriate blanks. When no blanks are provided CIRCLE

### Probability Distributions

CONDENSED LESSON 13.1 Probability Distributions In this lesson, you Sketch the graph of the probability distribution for a continuous random variable Find probabilities by finding or approximating areas

### THE SAMPLING DISTRIBUTION OF THE MEAN

THE SAMPLING DISTRIBUTION OF THE MEAN COGS 14B JANUARY 26, 2017 TODAY Sampling Distributions Sampling Distribution of the Mean Central Limit Theorem INFERENTIAL STATISTICS Inferential statistics: allows

### We're in interested in Pr{three sixes when throwing a single dice 8 times}. => Y has a binomial distribution, or in official notation, Y ~ BIN(n,p).

Sampling distributions and estimation. 1) A brief review of distributions: We're in interested in Pr{three sixes when throwing a single dice 8 times}. => Y has a binomial distribution, or in official notation,

### LECTURE 15: SIMPLE LINEAR REGRESSION I

David Youngberg BSAD 20 Montgomery College LECTURE 5: SIMPLE LINEAR REGRESSION I I. From Correlation to Regression a. Recall last class when we discussed two basic types of correlation (positive and negative).

### ( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of

Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they

### Chapter 26: Comparing Counts (Chi Square)

Chapter 6: Comparing Counts (Chi Square) We ve seen that you can turn a qualitative variable into a quantitative one (by counting the number of successes and failures), but that s a compromise it forces

### We're in interested in Pr{three sixes when throwing a single dice 8 times}. => Y has a binomial distribution, or in official notation, Y ~ BIN(n,p).

Sampling distributions and estimation. 1) A brief review of distributions: We're in interested in Pr{three sixes when throwing a single dice 8 times}. => Y has a binomial distribution, or in official notation,

### The Central Limit Theorem

The Central Limit Theorem Patrick Breheny March 1 Patrick Breheny STA 580: Biostatistics I 1/23 Kerrich s experiment A South African mathematician named John Kerrich was visiting Copenhagen in 1940 when

### STA Module 5 Regression and Correlation. Learning Objectives. Learning Objectives (Cont.) Upon completing this module, you should be able to:

STA 2023 Module 5 Regression and Correlation Learning Objectives Upon completing this module, you should be able to: 1. Define and apply the concepts related to linear equations with one independent variable.

### The Central Limit Theorem

The Central Limit Theorem Patrick Breheny September 27 Patrick Breheny University of Iowa Biostatistical Methods I (BIOS 5710) 1 / 31 Kerrich s experiment Introduction 10,000 coin flips Expectation and

### Lecture 8 Sampling Theory

Lecture 8 Sampling Theory Thais Paiva STA 111 - Summer 2013 Term II July 11, 2013 1 / 25 Thais Paiva STA 111 - Summer 2013 Term II Lecture 8, 07/11/2013 Lecture Plan 1 Sampling Distributions 2 Law of Large

### Warm-up Using the given data Create a scatterplot Find the regression line

Time at the lunch table Caloric intake 21.4 472 30.8 498 37.7 335 32.8 423 39.5 437 22.8 508 34.1 431 33.9 479 43.8 454 42.4 450 43.1 410 29.2 504 31.3 437 28.6 489 32.9 436 30.6 480 35.1 439 33.0 444

### COMP6053 lecture: Sampling and the central limit theorem. Jason Noble,

COMP6053 lecture: Sampling and the central limit theorem Jason Noble, jn2@ecs.soton.ac.uk Populations: long-run distributions Two kinds of distributions: populations and samples. A population is the set

### Descriptive Statistics (And a little bit on rounding and significant digits)

Descriptive Statistics (And a little bit on rounding and significant digits) Now that we know what our data look like, we d like to be able to describe it numerically. In other words, how can we represent

### Ch. 7 Statistical Intervals Based on a Single Sample

Ch. 7 Statistical Intervals Based on a Single Sample Before discussing the topics in Ch. 7, we need to cover one important concept from Ch. 6. Standard error The standard error is the standard deviation

### Discrete Mathematics for CS Spring 2006 Vazirani Lecture 22

CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 22 Random Variables and Expectation Question: The homeworks of 20 students are collected in, randomly shuffled and returned to the students.

### STAT 201 Chapter 5. Probability

STAT 201 Chapter 5 Probability 1 2 Introduction to Probability Probability The way we quantify uncertainty. Subjective Probability A probability derived from an individual's personal judgment about whether

### Using Dice to Introduce Sampling Distributions Written by: Mary Richardson Grand Valley State University

Using Dice to Introduce Sampling Distributions Written by: Mary Richardson Grand Valley State University richamar@gvsu.edu Overview of Lesson In this activity students explore the properties of the distribution

### STA Module 10 Comparing Two Proportions

STA 2023 Module 10 Comparing Two Proportions Learning Objectives Upon completing this module, you should be able to: 1. Perform large-sample inferences (hypothesis test and confidence intervals) to compare

### Lecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 9.1-1

Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola Copyright 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 9.1-1 Chapter 9 Inferences

### Review of the Normal Distribution

Sampling and s Normal Distribution Aims of Sampling Basic Principles of Probability Types of Random Samples s of the Mean Standard Error of the Mean The Central Limit Theorem Review of the Normal Distribution

### COMP6053 lecture: Sampling and the central limit theorem. Markus Brede,

COMP6053 lecture: Sampling and the central limit theorem Markus Brede, mb8@ecs.soton.ac.uk Populations: long-run distributions Two kinds of distributions: populations and samples. A population is the set

### Discrete Mathematics and Probability Theory Fall 2014 Anant Sahai Note 15. Random Variables: Distributions, Independence, and Expectations

EECS 70 Discrete Mathematics and Probability Theory Fall 204 Anant Sahai Note 5 Random Variables: Distributions, Independence, and Expectations In the last note, we saw how useful it is to have a way of

### value of the sum standard units

Stat 1001 Winter 1998 Geyer Homework 7 Problem 18.1 20 and 25. Problem 18.2 (a) Average of the box. (1+3+5+7)=4=4. SD of the box. The deviations from the average are,3,,1, 1, 3. The squared deviations

### Chapter 4: An Introduction to Probability and Statistics

Chapter 4: An Introduction to Probability and Statistics 4. Probability The simplest kinds of probabilities to understand are reflected in everyday ideas like these: (i) if you toss a coin, the probability

### the probability of getting either heads or tails must be 1 (excluding the remote possibility of getting it to land on its edge).

Probability One of the most useful and intriguing aspects of quantum mechanics is the Heisenberg Uncertainty Principle. Before I get to it however, we need some initial comments on probability. Let s first

### Part 13: The Central Limit Theorem

Part 13: The Central Limit Theorem As discussed in Part 12, the normal distribution is a very good model for a wide variety of real world data. And in this section we will give even further evidence of

### Chapter 5 Simplifying Formulas and Solving Equations

Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify

### What is a parameter? What is a statistic? How is one related to the other?

Chapter Seven: SAMPLING DISTRIBUTIONS 7.1 Sampling Distributions Read 424 425 What is a parameter? What is a statistic? How is one related to the other? Example: Identify the population, the parameter,

### CHAPTER 7. Parameters are numerical descriptive measures for populations.

CHAPTER 7 Introduction Parameters are numerical descriptive measures for populations. For the normal distribution, the location and shape are described by µ and σ. For a binomial distribution consisting

### Chapter 23. Inferences About Means. Monday, May 6, 13. Copyright 2009 Pearson Education, Inc.

Chapter 23 Inferences About Means Sampling Distributions of Means Now that we know how to create confidence intervals and test hypotheses about proportions, we do the same for means. Just as we did before,

### Chapter 18: Sampling Distribution Models

Chapter 18: Sampling Distribution Models Suppose I randomly select 100 seniors in Scott County and record each one s GPA. 1.95 1.98 1.86 2.04 2.75 2.72 2.06 3.36 2.09 2.06 2.33 2.56 2.17 1.67 2.75 3.95

### Chapter 24. Comparing Means

Chapter 4 Comparing Means!1 /34 Homework p579, 5, 7, 8, 10, 11, 17, 31, 3! /34 !3 /34 Objective Students test null and alternate hypothesis about two!4 /34 Plot the Data The intuitive display for comparing

### Senior Math Circles November 19, 2008 Probability II

University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles November 9, 2008 Probability II Probability Counting There are many situations where

### MA 1125 Lecture 33 - The Sign Test. Monday, December 4, Objectives: Introduce an example of a non-parametric test.

MA 1125 Lecture 33 - The Sign Test Monday, December 4, 2017 Objectives: Introduce an example of a non-parametric test. For the last topic of the semester we ll look at an example of a non-parametric test.

### Do students sleep the recommended 8 hours a night on average?

BIEB100. Professor Rifkin. Notes on Section 2.2, lecture of 27 January 2014. Do students sleep the recommended 8 hours a night on average? We first set up our null and alternative hypotheses: H0: μ= 8

### - a value calculated or derived from the data.

Descriptive statistics: Note: I'm assuming you know some basics. If you don't, please read chapter 1 on your own. It's pretty easy material, and it gives you a good background as to why we need statistics.

### Chapter 8: An Introduction to Probability and Statistics

Course S3, 200 07 Chapter 8: An Introduction to Probability and Statistics This material is covered in the book: Erwin Kreyszig, Advanced Engineering Mathematics (9th edition) Chapter 24 (not including

### Part 3: Parametric Models

Part 3: Parametric Models Matthew Sperrin and Juhyun Park April 3, 2009 1 Introduction Is the coin fair or not? In part one of the course we introduced the idea of separating sampling variation from a

### Business Statistics. Lecture 5: Confidence Intervals

Business Statistics Lecture 5: Confidence Intervals Goals for this Lecture Confidence intervals The t distribution 2 Welcome to Interval Estimation! Moments Mean 815.0340 Std Dev 0.8923 Std Error Mean

### Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 12. Random Variables: Distribution and Expectation

CS 70 Discrete Mathematics and Probability Theory Fall 203 Vazirani Note 2 Random Variables: Distribution and Expectation We will now return once again to the question of how many heads in a typical sequence

### DISTRIBUTIONS USED IN STATISTICAL WORK

DISTRIBUTIONS USED IN STATISTICAL WORK In one of the classic introductory statistics books used in Education and Psychology (Glass and Stanley, 1970, Prentice-Hall) there was an excellent chapter on different

### What Is a Sampling Distribution? DISTINGUISH between a parameter and a statistic

Section 8.1A What Is a Sampling Distribution? Learning Objectives After this section, you should be able to DISTINGUISH between a parameter and a statistic DEFINE sampling distribution DISTINGUISH between

### 4/19/2009. Probability Distributions. Inference. Example 1. Example 2. Parameter versus statistic. Normal Probability Distribution N

Probability Distributions Normal Probability Distribution N Chapter 6 Inference It was reported that the 2008 Super Bowl was watched by 97.5 million people. But how does anyone know that? They certainly

### Problems from Probability and Statistical Inference (9th ed.) by Hogg, Tanis and Zimmerman.

Math 224 Fall 2017 Homework 1 Drew Armstrong Problems from Probability and Statistical Inference (9th ed.) by Hogg, Tanis and Zimmerman. Section 1.1, Exercises 4,5,6,7,9,12. Solutions to Book Problems.

### The PROMYS Math Circle Problem of the Week #3 February 3, 2017

The PROMYS Math Circle Problem of the Week #3 February 3, 2017 You can use rods of positive integer lengths to build trains that all have a common length. For instance, a train of length 12 is a row of

### Lecture 8: Conditional probability I: definition, independence, the tree method, sampling, chain rule for independent events

Lecture 8: Conditional probability I: definition, independence, the tree method, sampling, chain rule for independent events Discrete Structures II (Summer 2018) Rutgers University Instructor: Abhishek

### Statistics Primer. A Brief Overview of Basic Statistical and Probability Principles. Essential Statistics for Data Analysts Using Excel

Statistics Primer A Brief Overview of Basic Statistical and Probability Principles Liberty J. Munson, PhD 9/19/16 Essential Statistics for Data Analysts Using Excel Table of Contents What is a Variable?...

### Essentials of Statistics and Probability

May 22, 2007 Department of Statistics, NC State University dbsharma@ncsu.edu SAMSI Undergrad Workshop Overview Practical Statistical Thinking Introduction Data and Distributions Variables and Distributions

### Solving with Absolute Value

Solving with Absolute Value Who knew two little lines could cause so much trouble? Ask someone to solve the equation 3x 2 = 7 and they ll say No problem! Add just two little lines, and ask them to solve

### Exam #2 Results (as percentages)

Oct. 30 Assignment: Read Chapter 19 Try exercises 1, 2, and 4 on p. 424 Exam #2 Results (as percentages) Mean: 71.4 Median: 73.3 Soda attitudes 2015 In a Gallup poll conducted Jul. 8 12, 2015, 1009 adult

### Special Theory Of Relativity Prof. Shiva Prasad Department of Physics Indian Institute of Technology, Bombay

Special Theory Of Relativity Prof. Shiva Prasad Department of Physics Indian Institute of Technology, Bombay Lecture - 6 Length Contraction and Time Dilation (Refer Slide Time: 00:29) In our last lecture,

### AP Statistics S C A T T E R P L O T S, A S S O C I A T I O N, A N D C O R R E L A T I O N C H A P 6

AP Statistics 1 S C A T T E R P L O T S, A S S O C I A T I O N, A N D C O R R E L A T I O N C H A P 6 The invalid assumption that correlation implies cause is probably among the two or three most serious

### 3 PROBABILITY TOPICS

Chapter 3 Probability Topics 135 3 PROBABILITY TOPICS Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr) Introduction It is often necessary

### ( ) P A B : Probability of A given B. Probability that A happens

A B A or B One or the other or both occurs At least one of A or B occurs Probability Review A B A and B Both A and B occur ( ) P A B : Probability of A given B. Probability that A happens given that B

### Steve Smith Tuition: Maths Notes

Maths Notes : Discrete Random Variables Version. Steve Smith Tuition: Maths Notes e iπ + = 0 a + b = c z n+ = z n + c V E + F = Discrete Random Variables Contents Intro The Distribution of Probabilities

### Hypothesis testing I. - In particular, we are talking about statistical hypotheses. [get everyone s finger length!] n =

Hypothesis testing I I. What is hypothesis testing? [Note we re temporarily bouncing around in the book a lot! Things will settle down again in a week or so] - Exactly what it says. We develop a hypothesis,

### Chapter 6. The Standard Deviation as a Ruler and the Normal Model 1 /67

Chapter 6 The Standard Deviation as a Ruler and the Normal Model 1 /67 Homework Read Chpt 6 Complete Reading Notes Do P129 1, 3, 5, 7, 15, 17, 23, 27, 29, 31, 37, 39, 43 2 /67 Objective Students calculate

### Chapter 1: Introduction. Material from Devore s book (Ed 8), and Cengagebrain.com

1 Chapter 1: Introduction Material from Devore s book (Ed 8), and Cengagebrain.com Populations and Samples An investigation of some characteristic of a population of interest. Example: Say you want to

### Chapter 1 Review of Equations and Inequalities

Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve

### 1 Probability Distributions

1 Probability Distributions In the chapter about descriptive statistics sample data were discussed, and tools introduced for describing the samples with numbers as well as with graphs. In this chapter

### 3.2 Probability Rules

3.2 Probability Rules The idea of probability rests on the fact that chance behavior is predictable in the long run. In the last section, we used simulation to imitate chance behavior. Do we always need

### Lesson 6-1: Relations and Functions

I ll bet you think numbers are pretty boring, don t you? I ll bet you think numbers have no life. For instance, numbers don t have relationships do they? And if you had no relationships, life would be

### Functions. If x 2 D, then g(x) 2 T is the object that g assigns to x. Writing the symbols. g : D! T

Functions This is the second of three chapters that are meant primarily as a review of Math 1050. We will move quickly through this material. A function is a way of describing a relationship between two

### 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Tutorial:A Random Number of Coin Flips

6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Tutorial:A Random Number of Coin Flips Hey, everyone. Welcome back. Today, we're going to do another fun problem that