Central Limit Theorem Skill 31.

Slides:

Advertisements

Similar presentations

Sampling Distributions

Advertisements

For a Normal probability distribution, let x be a random variable with a normal distribution whose mean is µ and whose standard deviation is σ. Let be.

AP Statistics Thursday, 22 January 2015 OBJECTIVE TSW investigate sampling distributions. –Everyone needs a calculator. TESTS are not graded. REMINDERS.

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,

Sampling Distributions

5.3 The Central Limit Theorem. Roll a die 5 times and record the value of each roll. Find the mean of the values of the 5 rolls. Repeat this 250 times.

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

Standard Normal Distribution

The Central Limit Theorem

Modular 13 Ch 8.1 to 8.2.

Chapter 11: Random Sampling and Sampling Distributions

Sample Distribution Models for Means and Proportions

Review of normal distribution. Exercise Solution.

Sampling Distributions. Parameter A number that describes the population Symbols we will use for parameters include  - mean  – standard deviation.

Chapter 8 Sampling Distributions Notes: Page 155, 156.

Sampling Distributions. Parameter A number that describes the population Symbols we will use for parameters include  - mean  – standard deviation.

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.

Sampling Distribution of a sample Means

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

Sampling Distribution of a Sample Mean Lecture 30 Section 8.4 Mon, Mar 19, 2007.

Statistics 300: Elementary Statistics Section 6-5.

Chapter 6.3 The central limit theorem. Sampling distribution of sample means A sampling distribution of sample means is a distribution using the means.

Chapter 7: Introduction to Sampling Distributions Section 2: The Central Limit Theorem.

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.

Areej Jouhar & Hafsa El-Zain Biostatistics BIOS 101 Foundation year.

6.3 THE CENTRAL LIMIT THEOREM. DISTRIBUTION OF SAMPLE MEANS  A sampling distribution of sample means is a distribution using the means computed from.

Section 7.2 Central Limit Theorem with Population Means HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant.

Chapter 9 found online and modified slightly! Sampling Distributions.

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.

7.2 Notes Central Limit Theorem. Theorem 7.1 – Let x be a normal distribution with mean μ and standard deviation σ. Let be a sample mean corresponding.

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:

Hypothesis Testing. Suppose we believe the average systolic blood pressure of healthy adults is normally distributed with mean μ = 120 and variance σ.

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 8 Sampling Distributions. Parameter A number that describes the population Symbols we will use for parameters include  - mean  – standard.

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

Sampling Distributions

7 Normal Curves and Sampling Distributions

Ch5.4 Central Limit Theorem

6.39 Day 2: The Central Limit Theorem

Sampling Distribution Estimation Hypothesis Testing

6-3The Central Limit Theorem.

6 Normal Curves and Sampling Distributions

6 Normal Curves and Sampling Distributions

PROBABILITY AND STATISTICS

Standard Units and the Areas Under the Standard Normal Distribution

An Example of {AND, OR, Given that} Using a Normal Distribution

Estimating p in the Binomial Distribution

Estimating µ When σ is Unknown

Sampling Distribution of the Sample Mean

Introduction to Probability & Statistics The Central Limit Theorem

Sampling Distributions

Consider the following problem

Sampling Distributions

Estimating µ When σ is Known

6 Normal Curves and Sampling Distributions

Sampling Distributions

Areas Under Any Normal Curve

Standard Units and the Areas Under the Standard Normal Distribution

The Poisson Probability Distribution

SAMPLING DISTRIBUTIONS

AGENDA: DG minutes Begin Part 2 Unit 1 Lesson 11.

Estimating µ When σ is Unknown

Estimating p in the Binomial Distribution

Estimating µ When σ is Known

Estimating 1 – 2 and p1 – p2 Skill 35.

Sample Means Section 9.3.

Day 13 AGENDA: DG minutes Begin Part 2 Unit 1 Lesson 11.

Presentation transcript:

Central Limit Theorem Skill 31

The 𝒙 Distribution, Given x is normal. Let x represent the length of a single trout taken at random from the pond. A group of biologists has determined that x has a normal distribution with mean  = 10.2 inches and standard deviation  = 1.4 inches. What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long?

Example 1 – Solution So  = 10.2 and  = 1.4 Therefore, the probability is about 0.8433 that a single trout taken at random is between 8 and 12 inches long.

Example– Probability regarding x and x What is the probability that the mean length of 5 trout taken at random is between 8 and 12 inches long?

Example–Solution

Example–Solution

The 𝒙 Distribution, Given x Follows any Distribution

Example–Central Limit Theorem A certain strain of bacteria occurs in a all raw milk. Let x be milliliter of milk the health department has found that if the milk is not contaminated then x has a distribution that basically normal. The mean of the x distribution is 2500, and the standard deviation is 300. In a large commercial dairy, the health inspector takes 42 random samples of the milk produced every day. At the end of the day, the bacteria count in each of the 42 samples is averaged to obtain the sample mean bacteria count .

Example–Central Limit Theorem Assuming the milk is not contaminated, what is the distribution of . Sample size: n=42 (Greater than 30)

Example–Central Limit Theorem Assuming the milk is not contaminated, what is the probability that the average bacteria count x-bar is between 2350 and 2650 bacteria per milliliter. Convert 2350 2650 to z values

Example–Solution

Example–Central Limit Theorem . The mean of the x distribution is 450, and the standard deviation is 30. Suppose we have a random sample of 35. Find the distribution of . Sample size: n=35 (Greater than 30) =450

Example–Central Limit Theorem . Find the probability that the average is between 400 and 500.

31 Central Limit Theorem Summarize Notes Homework Worksheet Quiz

The 𝒙 Distribution, Given x Follows any Distribution