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Chapter 7 7-1 Sampling Distributions Learning Objectives In this chapter, you learn: The concept of the sampling distribution To compute probabilities related to the sample mean and the sample proportion The importance of the Central Limit Theorem Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-1 Sampling Distributions A sampling distribution is a distribution of all of the possible values of a sample statistic for a given size sample selected from a population. For example, suppose you sample 50 students from your college regarding their mean GPA. If you obtained many different samples of 50, you will compute a different mean for each sample. We are interested in the distribution of all potential mean GPA we might calculate for any given sample of 50 students. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Basic Business Statistics, 10/e Chap 7-2 © 2006 Prentice Hall, Inc. Chapter 7 7-2 Developing a Sampling Distribution Assume there is a population … A Population size N=4 B C D Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years) Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-3 Developing a Sampling Distribution (continued) Summary Measures for the Population Distribution: ∑X µ= = P(x) i N .3 18 + 20 + 22 + 24 = 21 4 σ= ∑ (X − µ) i N .2 .1 0 2 = 2.236 18 20 22 24 A B C D x Uniform Distribution Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Basic Business Statistics, 10/e Chap 7-4 © 2006 Prentice Hall, Inc. Chapter 7 7-3 Developing a Sampling Distribution (continued) Now consider all possible samples of size n=2 1st Obs 16 Sample Means 2nd Observation 18 20 22 24 18 18,18 18,20 18,22 18,24 20 20,18 20,20 20,22 20,24 1st 2nd Observation Obs 18 20 22 24 22 22,18 22,20 22,22 22,24 18 18 19 20 21 24 24,18 24,20 24,22 24,24 20 19 20 21 22 22 20 21 22 23 16 possible samples (sampling with replacement) 24 21 22 23 24 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-5 Developing a Sampling Distribution (continued) Sampling Distribution of All Sample Means Sample Means Distribution 16 Sample Means 1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 _ P(X) .3 .2 .1 0 18 19 20 21 22 23 24 _ X (no longer uniform) Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Basic Business Statistics, 10/e Chap 7-6 © 2006 Prentice Hall, Inc. Chapter 7 7-4 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-7 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-8 Basic Business Statistics, 10/e © 2006 Prentice Hall, Inc. Chapter 7 7-5 Developing a Sampling Distribution (continued) Summary Measures of this Sampling Distribution: µX = σX = = ∑X i = N 18 + 19 + 21 + L + 24 = 21 16 ∑(X − µ i X )2 N (18 - 21)2 + (19 - 21)2 + L + (24 - 21)2 = 1.58 16 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-9 Comparing the Population with its Sampling Distribution Population N=4 µ = 21 Sample Means Distribution n=2 µ X = 21 σ = 2.236 _ P(X) .3 P(X) .3 .2 .2 .1 .1 0 18 20 22 24 A B C D Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Basic Business Statistics, 10/e σ X = 1.58 X 0 18 19 20 21 22 23 24 _ X Chap 7-10 © 2006 Prentice Hall, Inc. Chapter 7 7-6 Sampling Distribution of the Mean or Proportion Sampling Distributions Sampling Distribution of the Mean Sampling Distribution of the Proportion Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-11 Sample Mean Sampling Distribution: Standard Error of the Mean Different samples of the same size from the same population will yield different sample means A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean: (This assumes that sampling is with replacement or sampling is without replacement from an infinite population) σX = σ n Note that the standard error of the mean decreases as the sample size increases Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Basic Business Statistics, 10/e Chap 7-12 © 2006 Prentice Hall, Inc. Chapter 7 7-7 Sample Mean Sampling Distribution: If the Population is Normal If a population is normal with mean µ and standard deviation σ, the sampling distribution of X is also normally distributed with µX = µ and σX = σ n Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-13 Z-value for Sampling Distribution of the Mean Z-value for the sampling distribution of X : Z= where: (X − µX ) σX = ( X − µ) σ n X = sample mean µ = population mean σ = population standard deviation n = sample size Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Basic Business Statistics, 10/e Chap 7-14 © 2006 Prentice Hall, Inc. Chapter 7 7-8 Sampling Distribution Properties µx = µ (i.e. x is unbiased ) Normal Population Distribution µ x µx x Normal Sampling Distribution (has the same mean) Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-15 Sampling Distribution Properties (continued) As n increases, Larger sample size σ x decreases Smaller sample size µ Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Basic Business Statistics, 10/e x Chap 7-16 © 2006 Prentice Hall, Inc. Chapter 7 7-9 Determining An Interval Including A Fixed Proportion of the Sample Means Find a symmetrically distributed interval around µ that will include 95% of the sample means when µ = 368, σ = 15, and n = 25. Since the interval contains 95% of the sample means 5% of the sample means will be outside the interval Since the interval is symmetric 2.5% will be above the upper limit and 2.5% will be below the lower limit. From the standardized normal table, the Z score with 2.5% (0.0250) below it is -1.96 and the Z score with 2.5% (0.0250) above it is 1.96. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-17 Determining An Interval Including A Fixed Proportion of the Sample Means (continued) Calculating the lower limit of the interval XL = µ+Z σ 15 = 368 + ( −1.96) = 362.12 n 25 Calculating the upper limit of the interval σ 15 = 368 + (1.96) = 373.88 n 25 95% of all sample means of sample size 25 are between 362.12 and 373.88 XU = µ + Z Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Basic Business Statistics, 10/e Chap 7-18 © 2006 Prentice Hall, Inc. Chapter 7 7-10 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. Properties of the sampling distribution: µx = µ and σx = σ n Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-19 Central Limit Theorem As the sample size gets large enough… n↑ the sampling distribution becomes almost normal regardless of shape of population x Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Basic Business Statistics, 10/e Chap 7-20 © 2006 Prentice Hall, Inc. Chapter 7 7-11 Central Limit Theorem Three major elements: Population distribution Sample mean (x-bar) is normal distributed Sample size (n) is large enough How large of (n) is large enough? n > 30 for _____________population n > 15 for ______________population n is any for _____________population Practical implications: E(x-bar) = µ(x-bar) = µ and σ(x-bar) = σ/√n E(p-bar) = p0 and σ(p-bar) = sqrt[p*(1-p)/sqrt(n)] Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-21 If the Population is not Normal (continued) Sampling distribution properties: Population Distribution Central Tendency µx = µ Variation σx = σ n Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Basic Business Statistics, 10/e µ x Sampling Distribution (becomes normal as n increases) Larger sample size Smaller sample size µx x Chap 7-22 © 2006 Prentice Hall, Inc. Chapter 7 7-12 How Large is Large Enough? For most distributions, n > 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n > 15 For normal population distributions, the sampling distribution of the mean is always normally distributed Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-23 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? Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Basic Business Statistics, 10/e Chap 7-24 © 2006 Prentice Hall, Inc. Chapter 7 7-13 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 approximately normal … with mean µx = 8 …and standard deviation σ x = x is = 3 σ n 36 = 0.5 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-25 Example (continued) Solution (continued): 7.8 - 8 X -µ 8.2 - 8 P(7.8 < X < 8.2) = P < < 3 σ 3 36 n 36 = P(-0.4 < Z < 0.4) = 0.3108 Population Distribution ??? ? ?? ? ? ? ?? µ=8 Sampling Distribution ? Sample X Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Basic Business Statistics, 10/e Standard Normal Distribution .1554 +.1554 Standardize 7.8 µX = 8 8.2 x -0.4 µz = 0 0.4 Z Chap 7-26 © 2006 Prentice Hall, Inc. Chapter 7 7-14 Sampling Distribution of the Proportion Sampling Distributions Sampling Distribution of the Mean Sampling Distribution of the Proportion Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-27 Population Proportions π = the proportion of the population having some characteristic Sample proportion ( p ) provides an estimate of π: p= X number of items in the sample having the characteristic of interest = n sample size 0≤ p≤1 p is approximately distributed as a normal distribution when n is large (assuming sampling with replacement from a finite population or without replacement from an infinite population) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Basic Business Statistics, 10/e Chap 7-28 © 2006 Prentice Hall, Inc. Chapter 7 7-15 Sampling Distribution of p Approximated by a normal distribution if: P(ps) Sampling Distribution .3 .2 .1 0 nπ ≥ 5 and 0 .2 .4 .6 8 1 p n(1 − π ) ≥ 5 where µp = π and σp = π(1− π ) n (where π = population proportion) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-29 Z-Value for Proportions Standardize p to a Z value with the formula: Z= Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Basic Business Statistics, 10/e p −π = σp p −π π (1− π ) n Chap 7-30 © 2006 Prentice Hall, Inc. Chapter 7 7-16 Example If the true proportion of voters who support Proposition A is π = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45? i.e.: if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-31 Example (continued) if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? Find σ p : σ p = π (1− π ) n = 0.4(1 − 0.4) = 0.03464 200 0.45 − 0.40 0.40 − 0.40 Convert to P(0.40 ≤ p ≤ 0.45) = P ≤Z≤ standardized 0.03464 0.03464 normal: = P(0 ≤ Z ≤ 1.44) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Basic Business Statistics, 10/e Chap 7-32 © 2006 Prentice Hall, Inc. Chapter 7 7-17 Example Central Limit Thoerem (CLT) online (continued) if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? Use standardized normal table: P(0 ≤ Z ≤ 1.44) = 0.4251 Standardized Normal Distribution Sampling Distribution 0.4251 Standardize 0.40 0.45 p 0 Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. 1.44 Z Chap 7-33 Chapter Summary Introduced sampling distributions Described the sampling distribution of the mean For normal populations Using the Central Limit Theorem Described the sampling distribution of a proportion Calculated probabilities using sampling distributions Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Basic Business Statistics, 10/e Chap 7-34 © 2006 Prentice Hall, Inc.