The Central Limit Theorem Section 9.3.2. Starter Assume I have 1000 pennies in a jar Let X = the age of a penny in years –If the date is 2007, X = 0 –If.

Slides:

Advertisements

Similar presentations

Sampling distributions. Example Take random sample of students. Ask “how many courses did you study for this past weekend?” Calculate a statistic, say,

Advertisements

Sampling distributions. Example Take random sample of 1 hour periods in an ER. Ask “how many patients arrived in that one hour period ?” Calculate statistic,

Central Limit Theorem.

Chapter Six Sampling Distributions McGraw-Hill/Irwin Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

PROBABILITY AND SAMPLES: THE DISTRIBUTION OF SAMPLE MEANS.

The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers CHAPTER 7 Sampling Distributions 7.3 Sample Means.

Chapter 7 The Normal Probability Distribution 7.5 Sampling Distributions; The Central Limit Theorem.

Chapter 11: Random Sampling and Sampling Distributions

The Central Limit Theorem For simple random samples from any population with finite mean and variance, as n becomes increasingly large, the sampling distribution.

Normal and Sampling Distributions A normal distribution is uniquely determined by its mean, , and variance,  2 The random variable Z = (X-  /  is.

M31- Dist of X-bars 1  Department of ISM, University of Alabama, Sampling Distributions Our Goal? To make a decisions. What are they? Why do.

The Central Limit Theorem. 1. The random variable x has a distribution (which may or may not be normal) with mean and standard deviation. 2. Simple random.

The Distribution of Sample Proportions Section

Each child born to a particular set of parents has probability of 0.25 having blood type O. Suppose these parents have 5 children. Let X = number of children.

Sampling Distribution of a sample Means

Correct Answer: A AP1.14. Boxplots of two data sets are shown.

Section 9.3 Distribution of Sample Means AP Statistics February 5, 2010 Berkley High School.

Sampling Distribution of the Sample Mean. Example a Let X denote the lifetime of a battery Suppose the distribution of battery battery lifetimes has 

Chapter 10 – Sampling Distributions Math 22 Introductory Statistics.

Section 5.2 The Sampling Distribution of the Sample Mean.

Estimation This is our introduction to the field of inferential statistics. We already know why we want to study samples instead of entire populations,

1 Chapter 10: Introduction to Inference. 2 Inference Inference is the statistical process by which we use information collected from a sample to infer.

TobiasEcon 472 Law of Large Numbers (LLN) and Central Limit Theorem (CLT)

Statistics 300: Elementary Statistics Section 6-5.

Distributions of the Sample Mean

Biostatistics Unit 5 – Samples. Sampling distributions Sampling distributions are important in the understanding of statistical inference. Probability.

1 Chapter 7 Sampling Distributions. 2 Chapter Outline  Selecting A Sample  Point Estimation  Introduction to Sampling Distributions  Sampling Distribution.

The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers CHAPTER 7 Sampling Distributions 7.3 Sample Means.

Sample Variability Consider the small population of integers {0, 2, 4, 6, 8} It is clear that the mean, μ = 4. Suppose we did not know the population mean.

AP Statistics Section 9.3B The Central Limit Theorem.

Section 6-5 The Central Limit Theorem. THE CENTRAL LIMIT THEOREM Given: 1.The random variable x has a distribution (which may or may not be normal) with.

8 Sampling Distribution of the Mean Chapter8 p Sampling Distributions Population mean and standard deviation,  and   unknown Maximal Likelihood.

Sampling Distributions. Essential Question: How is the mean of a sampling distribution related to the population mean or proportion?

Estimation Chapter 8. Estimating µ When σ Is Known.

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 7 Sampling and Sampling Distributions.

Chapter 18: Sampling Distribution Models

Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.

Chapter 18 - Part 2 Sampling Distribution Models for.

© 2010 Pearson Prentice Hall. All rights reserved Chapter Sampling Distributions 8.

STA 2023 Section 5.4 Sampling Distributions and the Central Limit Theorem.

Distributions of Sample Means. z-scores for Samples  What do I mean by a “z-score” for a sample? This score would describe how a specific sample is.

Statistical Analysis – Chapter 5 “Central Limit Theorem” Dr. Roderick Graham Fashion Institute of Technology.

Unit 6 Section : The Central Limit Theorem  Sampling Distribution – the probability distribution of a sample statistic that is formed when samples.

Sec 6.3 Bluman, Chapter Review: Find the z values; the graph is symmetrical. Bluman, Chapter 63.

Sampling Distribution, Chp Know the difference between a parameter and a statistic.

Central Limit Theorem Let X 1, X 2, …, X n be n independent, identically distributed random variables with mean  and standard deviation . For large n:

Chapter 7: Sampling Distributions Section 7.2 Sample Proportions.

Review Normal Distributions –Draw a picture. –Convert to standard normal (if necessary) –Use the binomial tables to look up the value. –In the case of.

Sampling Distributions Chapter 18. Sampling Distributions A parameter is a number that describes the population. In statistical practice, the value of.

THE CENTRAL LIMIT THEOREM. Sampling Distribution of Sample Means Definition: A distribution obtained by using the means computed from random samples of.

CHAPTER 7 Sampling Distributions

Ch5.4 Central Limit Theorem

Sampling Distribution of a sample Means

Chapter 7 Sampling and Sampling Distributions

Sec. 7-5: Central Limit Theorem

Chapter 7: Sampling Distributions

Sampling Distribution of the Sample Mean

The Normal Probability Distribution Summary

Chapter 18 – Central Limit Theorem

MATH 2311 Section 4.4.

Sampling Distributions

CHAPTER 15 SUMMARY Chapter Specifics

9.3 Sample Means.

CHAPTER 7 Sampling Distributions

CHAPTER 7 Sampling Distributions

CHAPTER 7 Sampling Distributions

Sample Means Section 9.3.

Warmup Which of the distributions is an unbiased estimator?

Review of Hypothesis Testing

CHAPTER 7 Sampling Distributions

Presentation transcript:

The Central Limit Theorem Section 9.3.2

Starter Assume I have 1000 pennies in a jar Let X = the age of a penny in years –If the date is 2007, X = 0 –If the date is 2006, X = 1, etc Draw a smooth curve that is your best guess of the shape of the distribution of X Write a verbal description of your guess of the distribution

Today’s Objectives Perform an activity that demonstrates the effect of the Central Limit Theorem Write a statement of The Central Limit Theorem California Standard 9.0 Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially.

How Old Are Pennies in Circulation? That’s a hard question to answer. Let’s try an easier one: What’s the average age of 1000 randomly collected pennies? –We could agree that the answer to this question is a reasonable approximation to the answer to the main question. We could now actually record the ages of all 1000 pennies and calculate the mean, or we could approach the problem by sampling. I will give you 25 pennies in a cup that are a sample of the 1000 pennies. We will take samples of increasing size and see what happens.

Sample the population n = 2 Shake up your pennies in the cup and select a sample of 2 pennies Calculate the mean age of the 2 pennies –Record the sample mean you found Replace the pennies, shake again, and choose another sample of 2 pennies –Record the new sample mean Plot your sample means on the whiteboard –You should be plotting two points Draw a sketch and write a description of the sampling distribution

Sample the population n = 10 Shake up your pennies in the cup and select a sample of 10 pennies Calculate the mean age of the 10 pennies –Record the sample mean you found Replace the pennies, shake again, and choose another sample of 10 pennies –Record the new sample mean Plot your sample means on the whiteboard –You should be plotting two points Draw a sketch and write a description of the sampling distribution

Sample the population n = 25 Combine your pennies with one or more partners and draw a new sample of 25. Record the ages of all 25 of your pennies and calculate the sample mean. Plot your sample mean on the whiteboard. Draw a sketch and write a description of the sampling distribution. –NOTE: KEEP YOUR SAMPLE MEAN IN YOUR NOTES. WE WILL USE IT AGAIN IN CHAPTER 10.

The Central Limit Theorem If an SRS of size n is drawn from a population of any shape with mean μ and standard deviation σ, and if n is large, then the sampling distribution of the sample means will be approximately normal with mean μ and standard deviation σ/ √n –How large is large? For most situations, n=30 will be large enough. –The more “non-normal” the population, the larger n should be to get good results

Today’s Objectives Perform an activity that demonstrates the effect of the Central Limit Theorem Write a statement of The Central Limit Theorem California Standard 9.0 Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially.

Homework Read pages Do problems