Lesson 6 4 The Central Limit Theorem Based
Lesson 6. 4: The Central Limit Theorem Based on slides provided by Pearson to accompany Triola’s Essentials of Statistics text, 6 th edition, and modified by Ms. Townsend for our course. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Elementary Statistics Sixth Edition Chapter 6: Normal Probability Distributions 6. 1 The Standard Normal Distribution 6. 2 Real Applications of Normal Distributions 6. 3 Sampling Distributions and Estimators 6. 4 The Central Limit Theorem Based on slides provided by Pearson, and modified by Ms. Townsend Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Key Concept of Lesson 6. 4 We introduce and apply the Central Limit Theorem. We want to understand • what the Central Limit Theorem says, • when and how we can apply it, and • how it can be used with the Rare Event Rule for Inferential Statistics to compute and then interpret probabilities. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Lesson 6. 4 Learning Outcomes 1. State the Central Limit Theorem in your own words. 2. Determine whether the distribution of sample means may be approximated by a normal distribution. Then, compute probabilities associated with a range of values of sample means. 3. Apply the Central Limit Theorem and the Rare Event Rule for Inferential Statistics to compute and interpret low probabilities. 4. Discuss the finite population correction factor, and when we might need to use that correction factor. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Lesson 6. 4 Learning Outcome 1 State the Central Limit Theorem in your own words. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
The Central Limit Theorem • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Sampling Distribution of the Sample Means • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
• Example – Normal Distribution of Sample Means Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
What The Central Limit Theorem Does Not Say • The Central Limit Theorem does not say that the distribution of sample data will approach a normal distribution as the sample size increases. • The Central Limit Theorem says that – The sample means approach a normal distribution as the sample size increases, and – The mean and standard deviation of the sample means are related to the mean and standard deviation of the distribution of the original population in a very specific way. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Lesson 6. 4 Learning Outcome 2 Determine whether the distribution of sample means may be approximated by a normal distribution. Then, compute probabilities associated with a range of values of sample means. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Requirements and Formulas for Problems Involving Sample Means • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Procedure for Practical Problem Solving (1 of 2) 1. Check Requirements. – When working with sample means, verify that the normal distribution can be used by confirming that § The original population has a normal distribution, or § The sample size is n > 30. – Remember, if the requirements aren’t met, the methods of this section don’t apply. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Procedure for Practical Problem Solving (2 of 2) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Pulse Rates of Females (1 of 7) Assume that females have pulse rates that are normally distributed with a mean of 74. 0 beats per minute (bpm) and a standard deviation of 12. 5 bpm. a. Find the probability that 1 randomly selected adult female has a pulse rate less than 80 bpm. b. Find the probability that a sample of 16 randomly selected adult females has a mean pulse rate less than 80 bpm. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Pulse Rates of Females (2 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Pulse Rates of Females (3 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Pulse Rates of Females (4 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Pulse Rates of Females (5 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Pulse Rates of Females (6 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Pulse Rates of Females (7 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Lesson 6. 4 Learning Outcome 3 Apply the Central Limit Theorem and the Rare Event Rule for Inferential Statistics to compute and interpret low probabilities. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
The Rare Event Rule for Inferential Statistics Revisited If, under a given assumption, the probability of a particular observed event is very small and the observed event occurs significantly less often or significantly more often than what we typically expect with that assumption, we conclude that the assumption is probably not correct. • One of the most important rules of statistics! • Plays an essential role in hypothesis testing. • If we find that a probability is very low, the rare event rule influences our interpretation of that probability. We may choose to reconsider our original assumptions. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
The Rare Event Rule for Inferential Statistics Revisited If, under a given assumption, the probability of a particular observed event is very small and the observed event occurs significantly less often or significantly more often than what we typically expect with that assumption, we conclude that the assumption is probably not correct. Statisticians reject explanations based on very low probabilities. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Elevator Safety (1 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Elevator Safety (2 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Elevator Safety (3 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Elevator Safety (4 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Elevator Safety (5 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Elevator Safety (6 of 7) • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Example: Elevator Safety (7 of 7) If an older elevator were designed taking the outdated mean of of the weight of men (174 lb) into account, the designer may have felt comfortable with the relatively rare event that the elevator was overloaded approximately 7% of those times in which 27 men rode the elevator. But, if years later, they found that the elevator overloaded much more often than expected (because the true mean weight of men had shifted to 189 lb), we might revisit the initial assumptions. Since the probability was small, but actual events contradict that probability, we should assume that our assumptions are incorrect. That’s the Rare Event Rule in action. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Lesson 6. 4 Learning Outcome 4 Discuss the finite population correction factor, and when we might need to use that correction factor. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
• Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
Correction for a Finite Population • Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
- Slides: 33