Chapter 7 Introduction to Sampling Distributions Business Statistics: A Decision-Making Approach 7th Edition Chapter 7 Introduction to Sampling Distributions Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Chapter Goals After completing this chapter, you should be able to: Define the concept of sampling error Determine the mean and standard deviation for the sampling distribution of the sample mean, x Determine the mean and standard deviation for the sampling distribution of the sample proportion, p Describe the Central Limit Theorem and its importance Apply sampling distributions for both x and p _ _ _ _ Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Sampling Error Problems: Sample Statistics are used to estimate Population Parameters ex: X is an estimate of the population mean, μ Problems: Different samples provide different estimates of the population parameter Sample results have potential variability, thus sampling error exits Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Calculating Sampling Error The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population Example: (for the mean) where: Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Review Population mean: Sample Mean: where: μ = Population mean x = sample mean xi = Values in the population or sample N = Population size n = sample size Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Example If the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = 99.2 degrees, then the sampling error is Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Sampling Errors Different samples will yield different sampling errors The sampling error may be positive or negative ( may be greater than or less than μ) The expected sampling error decreases as the sample size increases Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Sampling Distribution A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, 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 Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, 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 Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, 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) Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, 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) Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Summary Measures of this Sampling Distribution: Developing a Sampling Distribution (continued) Summary Measures of this Sampling Distribution: Business Statistics: A Decision-Making Approach, 7e © 2008 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, 7e © 2008 Prentice-Hall, Inc.

Properties of a Sampling Distribution For any population, the average value of all possible sample means computed from all possible random samples of a given size from the population is equal to the population mean: The standard deviation of the possible sample means computed from all random samples of size n is equal to the population standard deviation divided by the square root of the sample size: Theorem 1 Theorem 2 Business Statistics: A Decision-Making Approach, 7e © 2008 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 Theorem 3 Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

z-value for Sampling Distribution of x Z-value for the sampling distribution of : where: = sample mean = population mean = population standard deviation n = sample size Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Finite Population Correction Apply the Finite Population Correction if: the sample is large relative to the population (n is greater than 5% of N) and… Sampling is without replacement Then Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Sampling Distribution Properties The sample mean is an unbiased estimator Normal Population Distribution Normal Sampling Distribution (has the same mean) Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Sampling Distribution Properties (continued) The sample mean is a consistent estimator (the value of x becomes closer to μ as n increases): Population x Small sample size As n increases, decreases Larger sample size Business Statistics: A Decision-Making Approach, 7e © 2008 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 and Theorem 4 Business Statistics: A Decision-Making Approach, 7e © 2008 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, 7e © 2008 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, 7e © 2008 Prentice-Hall, Inc.

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 is sufficient For normal population distributions, the sampling distribution of the mean is always normally distributed Business Statistics: A Decision-Making Approach, 7e © 2008 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, 7e © 2008 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 = μ = 8 …and standard deviation Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Example Solution (continued) -- find z-scores: x (continued) z Population Distribution Sampling Distribution Standard Normal Distribution .1554 +.1554 ? ? ? ? ? ? ? ? ? ? Sample Standardize ? ? -0.4 0.4 x 7.8 8.2 z Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Population Proportions, π π = the proportion of the population having some characteristic Sample proportion ( p ) provides an estimate of π : If two outcomes, p has a binomial distribution Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Sampling Distribution of p Approximated by a normal distribution if: where and Sampling Distribution P( p ) .3 .2 .1 p 0 . 2 .4 .6 8 1 (where π = population proportion) Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

z-Value for Proportions Standardize p to a z value with the formula: If sampling is without replacement and n is greater than 5% of the population size, then must use the finite population correction factor: Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Example If the true proportion of voters who support Proposition A is π = .4, what is the probability that a sample of size 200 yields a sample proportion between .40 and .45? i.e.: if π = .4 and n = 200, what is P(.40 ≤ p ≤ .45) ? Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Example if π = .4 and n = 200, what is P(.40 ≤ p ≤ .45) ? Find : (continued) if π = .4 and n = 200, what is P(.40 ≤ p ≤ .45) ? Find : Convert to standard normal: Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Standardized Normal Distribution Example (continued) if π = .4 and n = 200, what is P(.40 ≤ p ≤ .45) ? Use standard normal table: P(0 ≤ z ≤ 1.44) = .4251 Standardized Normal Distribution Sampling Distribution .4251 Standardize .40 .45 1.44 p z Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Chapter Summary Discussed sampling error 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 Discussed sampling from finite populations Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.