Transcript Section X.X
Section 7.1 Introduction to the Central Limit Theorem HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Use the Central Limit Theorem to identify characteristics of a sampling distribution. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Theorem The Central Limit Theorem (CLT) For any given population with mean, , and standard deviation, , a sampling distribution of sample means will have the following three characteristics if either the sample size, n, is at least 30 or the population is normally distributed. 1. The mean of a sampling distribution of sample means, x , equals the mean of the population, . mx m HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Theorem The Central Limit Theorem (CLT) (cont.) 2. The standard deviation of a sampling distribution of sample means, x , equals the standard deviation of the population, , divided by the square root of the sample size, n. s sx n HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Theorem The Central Limit Theorem (CLT) (cont.) 3. The shape of a sampling distribution of sample means will approach that of a normal distribution, regardless of the shape of the population distribution. The larger the sample size, the better the normal distribution approximation will be. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.1: Calculating the Standard Deviation for a Sampling Distribution of Sample Means Using the Central Limit Theorem Suppose that the standard deviation of movie ticket prices in the United States is $0.72. Suppose sample means are calculated for samples of size 52; that is, ticket prices are recorded for different samples of 52 theaters and the sample means are calculated. What would be the standard deviation of the sampling distribution of the sample means, that is, the standard error of the mean? HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.1: Calculating the Standard Deviation for a Sampling Distribution of Sample Means Using the Central Limit Theorem (cont.) Solution x n 0.72 52 0.099846 0.100 So, the sampling distribution’s standard deviation would be x $0.10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.2: Applying the Central Limit Theorem The following histogram represents the population distribution of the weights of 600 horse jockeys. The mean of the population is 116.2 pounds and the standard deviation is 3.9 pounds. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.2: Applying the Central Limit Theorem (cont.) Let’s now consider the sampling distribution of the sample means for samples of size n = 45. a. Can the Central Limit Theorem be applied to this sampling distribution? If so, explain how the conditions are met. b. What is the sampling distribution’s mean? c. What is the sampling distribution’s standard deviation? d. What is the sampling distribution’s shape? HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.2: Applying the Central Limit Theorem (cont.) Solution a. Yes, the Central Limit Theorem is applicable in this scenario because the sample size, n = 45, satisfies the criteria n ≥ 30. b. According to the CLT, the sampling distribution’s mean is equal to the mean of the population. Therefore, x 116.2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.2: Applying the Central Limit Theorem (cont.) c. To find the standard deviation of the sampling distribution we will have to do a small calculation. . Hence, we have The CLT says that x n 3.9 x 0.58. 45 d. Although the original distribution was skewed to the right, the CLT tells us that the sampling distribution will be approximately normal, so it will be bellshaped. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.2: Applying the Central Limit Theorem (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.3: Applying the Central Limit Theorem The following graph displays the starting salaries for law school graduates entering the workforce in 2007. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.3: Applying the Central Limit Theorem Note: The graph is based on 23,337 salaries. A few salaries above $200,000 are excluded for clarity. Reproduced with permission of the National Association for Law Placement, Inc. (NALP). Source: NALP: The Association for Legal Career Professionals. “Another Picture Worth 1,000 Words.” July 2008. http://www.nalp.org/anotherpicture?s=Another%20 Picture%20Worth%201%2C000%20Words (19 Dec. 2011). HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.3: Applying the Central Limit Theorem (cont.) a. Describe the shape of this distribution. b. Consider the sampling distribution of the sample means created from this population for samples of size 25. Can the Central Limit Theorem be applied in this situation? c. Consider the sampling distribution of the sample means created from this population for samples of size 50. Can the Central Limit Theorem be applied in this situation? HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.3: Applying the Central Limit Theorem (cont.) d. Consider the sampling distribution of the sample means created from this population for samples of size 200. What would you expect the shape of this distribution to be? How would it compare to the sampling distribution described in part c.? Solution a. The distribution is best classified as bimodal. b. No, the Central Limit Theorem is not applicable in this situation because neither condition is met. The sample size is not large enough (25 < 30), and the population data are not normally distributed. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7.3: Applying the Central Limit Theorem (cont.) c. Yes, the Central Limit Theorem is applicable here because the sample size is sufficiently large (50 ≥ 30). d. We would expect the sampling distribution created from samples of size 200 to resemble a normal distribution. In fact, the larger the value of the sample size, n, the more “normal” the graph will appear. So then, it follows that the sampling distribution described in part d. will more closely resemble a true normal distribution than the sampling distribution described in part c. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.