Sampling Distributions Chapter 6 Sampling Distributions Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Random Sampling Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Definitions A population: Consist of the totality of observations with which we are concerned A sample is a subset of a population Let X1 , X2 , … , Xn be n independent random variables, each having the same probability distribution f(x). We then define X1 , X2 , … , Xn to be random sample of size n from the population f(x). Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Random Samples The rv’s X1,…,Xn are said to form a (simple random sample of size n if The Xi’s are independent rv’s. Every Xi has the same probability distribution. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Some Important Statistics Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Statistic A statistic is any quantity whose value can be calculated from sample data. Prior to obtaining data, there is uncertainty as to what value of any particular statistic will result. or Any function of the random variables constituting of random sample called a statistic. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. If X1 , X2 , … , Xn represent a random sample of size n, then the sample mean is defined by the statistic If X1 , X2 , … , Xn represent a random sample of size n, then the sample variance is defined by the statistic Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Sampling Distribution of the Mean Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Using the Sample Mean Let X1,…, Xn be a random sample from a distribution with mean value and standard deviation Then In addition, with To = X1 +…+ Xn, Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Normal Population Distribution Let X1,…, Xn be a random sample from a normal distribution with mean value and standard deviation Then for any n, is normally distributed with mean and standard deviation Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Central Limit Theorem Let X1,…, Xn be a random sample from a distribution with mean value and variance Then if n sufficiently large, has approximately a normal distribution with and To also has approximately a normal distribution with The larger the value of n, the better the approximation. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Central Limit Theorem small to moderate n large n Population distribution Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Rule of Thumb If n > 30, the Central Limit Theorem can be used. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Developing the Distribution Of the Sample Mean Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Developing a Sampling Distribution Assume there is a population … Population size N=4 Random variable, x, is age of individuals Values of x: 18, 20, 22, 24 (years) D A C B Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Developing a Sampling Distribution (continued) Summary Measures for the Population Distribution: P(x) .3 .2 .1 x 18 20 22 24 A B C D Uniform Distribution Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Now consider all possible samples of size n=2 Developing a Sampling Distribution (continued) Now consider all possible samples of size n=2 16 Sample Means 16 possible samples (sampling with replacement) Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Sampling Distribution of All Sample Means Developing a Sampling Distribution (continued) Sampling Distribution of All Sample Means Sample Means Distribution 16 Sample Means P(x) .3 .2 .1 _ 18 19 20 21 22 23 24 x (no longer uniform) Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Summary Measures of this Sampling Distribution: Developing a Sampling Distribution (continued) Summary Measures of this Sampling Distribution: Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Comparing the Population with its Sampling Distribution Sample Means Distribution n = 2 P(x) P(x) .3 .3 .2 .2 .1 .1 _ x 18 20 22 24 A B C D 18 19 20 21 22 23 24 x Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

If the Population is Normal If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

z-value for Sampling Distribution of x Z value for the sampling distribution of Where the sample mean the population mean population standard deviation n sample size Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Sampling Distribution Properties Normal Population Distribution (i.e. is unbiased ) Normal Sampling Distribution (has the same mean) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Sampling Distribution Properties (continued) For sampling with replacement Decreases As n increases Smaller sample size Larger sample size Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

If the Population is not Normal We can apply the Central Limit Theorem: Even if the population is not normal, …sample means from the population will be approximately normal as long as the sample size is large enough …and the sampling distribution will have Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Central Limit Theorem the sampling distribution becomes almost normal regardless of shape of population As the sample size gets large enough… n↑ Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

If the Population is not Normal (continued) Population Distribution Sampling distribution properties: Central Tendency Sampling Distribution (becomes normal as n increases) Variation Larger sample size Smaller sample size (Sampling with replacement) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Example Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2? Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Example (continued) Solution: Even if the population is not normally distributed, the central limit theorem can be used (n > 30) … so the sampling distribution of is approximately normal … with mean = 8 …and standard deviation Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Example Solution (continued): (continued) z Population Distribution Sampling Distribution Standard Normal Distribution .1554 +.1554 ? ? ? ? ? ? ? ? ? ? Sample Standardize ? ? -0.4 0.4 7.8 8.2 z Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Population Proportions, P P = the proportion of population having some characteristic Sample proportion ( ) provides an estimate of P is: Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Sampling Distribution of Approximated by a normal distribution if: Sampling Distribution P( ) .3 .2 .1 0 . 2 .4 .6 8 1 where and (where p = population proportion) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

z-Value for Proportions Standardize to a z value with the formula: Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Example If the true proportion of voters who support Proposition A is p = .4, what is the probability that a sample of size 200 yields a sample proportion between .40 and .45? i.e.: if p = .4 and n = 200, what is P(.40 ≤ ≤ .45) ? Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Example if p = .4 and n = 200, what is P(.40 ≤ ≤ .45) ? Find : (continued) if p = .4 and n = 200, what is P(.40 ≤ ≤ .45) ? Find : Convert to standard normal: Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Standardized Normal Distribution Example (continued) if p = .4 and n = 200, what is P(.40 ≤ ≤ .45) ? Use standard normal table: P(0 ≤ z ≤ 1.44) = .4251 Standardized Normal Distribution Sampling Distribution .4251 Standardize .40 .45 1.44 z Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

The Distribution of a Linear Combination Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Linear Combination Given a collection of n random variables X1,…, Xn and n numerical constants a1,…,an, the rv is called a linear combination of the Xi’s. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Expected Value of a Linear Combination Let X1,…, Xn have mean values and variances of respectively Whether or not the Xi’s are independent, Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Variance of a Linear Combination If X1,…, Xn are independent, and Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Difference Between Two Random Variables and, if X1 and X2 are independent, Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Difference Between Normal Random Variables If X1, X2,…Xn are independent, normally distributed rv’s, then any linear combination of the Xi’s also has a normal distribution. The difference X1 – X2 between two independent, normally distributed variables is itself normally distributed. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.