The Central Limit Theorem

Slides:



Advertisements
Similar presentations
Sampling Distribution Models Sampling distributions of Proportions and Means C.L.T.
Advertisements

Chapter 7 Hypothesis Testing
Central Limit Theorem Given:
Distributions of the Sample Mean
THE STANDARD NORMAL DISTRIBUTION Individual normal distributions all have means and standard deviations that relate to them specifically. In order to compare.
One sample means Testing a sample mean against a population mean.
Normal Distributions: Finding Probabilities 1 Section 5.2.
Chapter 8 Hypothesis Testing
Sampling Distributions
Reminders: Parameter – number that describes the population Statistic – number that is computed from the sample data Mean of population = µ Mean of sample.
Sampling distributions. Example Take random sample of students. Ask “how many courses did you study for this past weekend?” Calculate a statistic, say,
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,
Statistical Significance What is Statistical Significance? What is Statistical Significance? How Do We Know Whether a Result is Statistically Significant?
5.2 The Sampling Distribution of a Sample Mean
Slide Slide 1 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Section 6-4 Sampling Distributions and Estimators Created by.
Statistical Significance What is Statistical Significance? How Do We Know Whether a Result is Statistically Significant? How Do We Know Whether a Result.
Fall 2006 – Fundamentals of Business Statistics 1 Chapter 6 Introduction to Sampling Distributions.
PROBABILITY AND SAMPLES: THE DISTRIBUTION OF SAMPLE MEANS.
6-5 The Central Limit Theorem
Chapter 11: Random Sampling and Sampling Distributions
The Central Limit Theorem. A water taxi sank in Baltimore’s Inner Harbor. Assume the weights of men is are normally distributed with a mean of 172.
From Last week.
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.
AP Statistics Chapter 9 Notes.
Statistics Sampling Distributions
Probability and Samples
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by.
Chapter 6 Lecture 3 Sections: 6.4 – 6.5.
Chapter 10 – Sampling Distributions Math 22 Introductory Statistics.
Statistics 300: Elementary Statistics Section 6-5.
Probabilistic & Statistical Techniques Eng. Tamer Eshtawi First Semester Eng. Tamer Eshtawi First Semester
Slide Slide 1 Section 6-5 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.
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.
§ 5.3 Normal Distributions: Finding Values. Probability and Normal Distributions If a random variable, x, is normally distributed, you can find the probability.
1 Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY.
Chapter 6 Lecture 3 Sections: 6.4 – 6.5. Sampling Distributions and Estimators What we want to do is find out the sampling distribution of a statistic.
Chapter 7: The Distribution of Sample Means. Frequency of Scores Scores Frequency.
Chapter 18: The Central Limit Theorem Objective: To apply the Central Limit Theorem to the Normal Model CHS Statistics.
STA 2023 Section 5.4 Sampling Distributions and the Central Limit Theorem.
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:
An Example of {AND, OR, Given that} Using a Normal Distribution By Henry Mesa.
MATH Section 4.4.
{ Chapter 3 Lesson 9 Z-Scores  Z-Score- The value z when you take an x value in the data set, subtract the mean from it, then divide by the standard.
And distribution of sample means
Lecture Slides Elementary Statistics Twelfth Edition
Ch5.4 Central Limit Theorem
The Central Limit Theorem
Sampling Distributions and Estimators
Sec. 7-5: Central Limit Theorem
Lecture Slides Elementary Statistics Twelfth Edition
SAMPLING DISTRIBUTION. Probability and Samples Sampling Distributions Central Limit Theorem Standard Error Probability of Sample Means.
Sampling Distributions
Elementary Statistics
Overview and Basics of Hypothesis Testing
Applications of the Normal Distribution
ANATOMY OF THE EMPIRICAL RULE
An Example of {AND, OR, Given that} Using a Normal Distribution
Consider the following problem
Lecture Slides Elementary Statistics Twelfth Edition
Use the graph of the given normal distribution to identify μ and σ.
Applications of the Normal Distribution
Chapter 7: The Distribution of Sample Means
Central Limit Theorem Accelerated Math 3.
The Central Limit Theorem
The Normal Curve Section 7.1 & 7.2.
An Example of {AND, OR, Given that} Using a Normal Distribution
Lecture Slides Essentials of Statistics 5th Edition
Z-Scores 10/13/2015 Statistics Mr. DeOms
Consider the following problem
Presentation transcript:

The Central Limit Theorem The Central Limit Theorem tells us that for a population with any distribution, the distribution of the sample mean approaches a normal distribution as the sample size increases. Furthermore, if the original distribution has mean 𝜇 and standard deviation 𝜎, the mean of the sample means will be 𝜇 and the standard deviation of the sample means will be 𝜎 𝑛 , where 𝑛 is the sample size.

The Central Limit Theorem Principles to use the Central Limit Theorem For a population with any distribution, if 𝑛>30, then the sample means will have a distribution that can be approximated by a normal distribution with mean 𝜇 and standard deviation 𝜎 𝑛 . If 𝑛≤30 and the original population has a normal distribution, then the sample means have a normal distribution with mean 𝜇 and standard deviation 𝜎 𝑛 . If 𝑛≤30 and the original population does not have a normal distribution, then we cannot apply the central limit theorem! There is a cool Chart on page 288 summarizing how to use the Central Limit Theorem.

The Central Limit Theorem Notation for the Sampling Distribution of 𝒙 If all possible random samples of size n are selected from a population with mean 𝜇 and standard deviation 𝜎, the sample means is denoted by 𝜇 𝑥 , so 𝝁 𝒙 =𝝁

The Central Limit Theorem Notation for the Sampling Distribution of 𝒙 If all possible random samples of size n are selected from a population with mean 𝜇 and standard deviation 𝜎, the sample means is denoted by 𝝁 𝒙 , so 𝝁 𝒙 =𝝁 Also the standard deviation of the sample means is denoted by 𝝈 𝒙 , so 𝝈 𝒙 = 𝝈 𝒏 𝜎 𝑥 is called the standard error of the mean.

The Central Limit Theorem Lets Look at example 1.

The Central Limit Theorem Lets Look at example 1. Note: Individual value: When working with individual values from a normally distributed population, use the methods from last class. Use 𝒛= 𝒙−𝝁 𝝈 Sample of values: When working with a mean for some sample (or group), be sure to use the value of 𝜎/ 𝑛 for the standard deviation of the sample means. Use 𝒛= 𝒙 −𝝁 𝝈 𝒏 .

The Central Limit Theorem Lets Look at example 1. Note: Individual value: When working with individual values from a normally distributed population, use the methods from last class. Use 𝒛= 𝒙−𝝁 𝝈 or normalcdf(lower, upper, mean, stdev) Sample of values: When working with a mean for some sample (or group), be sure to use the value of 𝜎/ 𝑛 for the standard deviation of the sample means. Use 𝒛= 𝒙 −𝝁 𝝈 𝒏 or normalcdf(lower, upper, mean, 𝝈 𝒏 ). Now Lets do example 2 on page 290.

The Central Limit Theorem A water taxi sank in Baltimore’s Inner Harbor. Assume the weights of men is are normally distributed with a mean of 172 lb. and a standard deviation of 29 lb. Find the probability that if an individual man is randomly selected, his weight will be greater than 175 lb.

The Central Limit Theorem A water taxi sank in Baltimore’s Inner Harbor. Assume the weights of men is are normally distributed with a mean of 172 lb. and a standard deviation of 29 lb. Find the probability that if an individual man is randomly selected, his weight will be greater than 175 lb. Find the probability that 20 randomly selected men will have a mean weight that is greater than 175 lb.

The Central Limit Theorem Recall the Rare Event rule for inferential Statistics If under a given assumption, the probability of a particular observed event is exceptionally small (such as less than 0.05), we conclude that the assumption is probably not correct.

The Central Limit Theorem The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. If 1 pregnant woman is randomly selected, find the probability that her length of pregnancy is less than 260 days.

The Central Limit Theorem The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. If 1 pregnant woman is randomly selected, find the probability that her length of pregnancy is less than 260 days. If 25 randomly selected women are put on a special diet just before they become pregnant, find the probability that their lengths of pregnancy have a mean that is less than 260 days.

The Central Limit Theorem The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. If 1 pregnant woman is randomly selected, find the probability that her length of pregnancy is less than 260 days. If 25 randomly selected women are put on a special diet just before they become pregnant, find the probability that their lengths of pregnancy have a mean that is less than 260 days. If the 25 women do have a mean of less that 260 days, does it appear that the, does it appear that the diet has an effect on the length of pregnancy?

The Central Limit Theorem Membership in Mensa requires and IQ score of above 131.5. Nine candidates take IQ tests, and their summary results indicated that their mean IQ score is 133. (IQ scores are normally distributed with a mean of 100 and a standard deviation of 15). If 1 person is randomly selected from the general population, find the probability of getting someone with an IQ score of at least 133.

The Central Limit Theorem Membership in Mensa requires and IQ score of above 131.5. Nine candidates take IQ tests, and their summary results indicated that their mean IQ score is 133. (IQ scores are normally distributed with a mean of 100 and a standard deviation of 15). If 1 person is randomly selected from the general population, find the probability of getting someone with an IQ score of at least 133. If 9 people are randomly selected, find the probability that their mean IQ is at least 133.

The Central Limit Theorem Membership in Mensa requires and IQ score of above 131.5. Nine candidates take IQ tests, and their summary results indicated that their mean IQ score is 133. (IQ scores are normally distributed with a mean of 100 and a standard deviation of 15). If 1 person is randomly selected from the general population, find the probability of getting someone with an IQ score of at least 133. If 9 people are randomly selected, find the probability that their mean IQ is at least 133. Although the summary results are available, the individual scores have been lost. Can is be concluded that all 9 candidates have IQ scores above 131.5 so that they are all eligible for Mensa membership?

Homework!! 6-5:1-9, 11 – 19 odd.