Chapter 7 Introduction to Sampling Distributions Business Statistics: QMIS 220, by Dr. M. Zainal

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 _ __ _ QMIS 220, by Dr. M. Zainal Chap 7-2

Inferential statistics Drawing conclusions and/or making decisions concerning a population based only on sample data Consists of methods that use sample results to help make decisions or predictions about a population. Elections Review: Inferential Statistics QMIS 220, by Dr. M. Zainal Chap 7-3

Sample statistics Population parameters (known) Inference (unknown, but can be estimated from sample evidence) Review: Inferential Statistics QMIS 220, by Dr. M. Zainal Chap 7-4

Review: Inferential Statistics Estimation e.g., Estimate the population mean weight using the sample mean weight Hypothesis Testing e.g., Use sample evidence to test the claim that the population mean weight is 120 pounds Drawing conclusions and/or making decisions concerning a population based on sample results. QMIS 220, by Dr. M. Zainal Chap 7-5

Review: Key Definitions A population is the entire collection of things under consideration A parameter is a summary measure computed to describe a characteristic of the population A sample is a portion of the population selected for analysis A statistic is a summary measure computed to describe a characteristic of the sample QMIS 220, by Dr. M. Zainal Chap 7-6

Review: Population vs. Sample a b c d ef gh i jk l m n o p q rs t u v w x y z PopulationSample b c g i n o r u y QMIS 220, by Dr. M. Zainal Chap 7-7

Review: Why Sample? Less time consuming than a census Less costly to administer than a census It is possible to obtain statistical results of a sufficiently high precision based on samples. QMIS 220, by Dr. M. Zainal Chap 7-8

Review: Sampling Techniques Convenience Sampling Techniques Nonstatistical Sampling Judgment Statistical Sampling Simple Random Systematic Stratified Cluster QMIS 220, by Dr. M. Zainal Chap 7-9

Review: Statistical Sampling Items of the sample are chosen based on known or calculable probabilities Statistical Sampling (Probability Sampling) SystematicStratifiedClusterSimple Random QMIS 220, by Dr. M. Zainal Chap 7-10

Simple Random Sampling Every possible sample of a given size has an equal chance of being selected Selection may be with replacement or without replacement The sample can be obtained using a table of random numbers or computer random number generator QMIS 220, by Dr. M. Zainal Chap 7-11

Stratified Random Sampling Divide population into subgroups (called strata) according to some common characteristic Select a simple random sample from each subgroup Combine samples from subgroups into one Population Divided into 4 strata Sample QMIS 220, by Dr. M. Zainal Chap 7-12

Decide on sample size: n Divide frame of N individuals into groups of k individuals: k=N/n Randomly select one individual from the 1 st group Select every k th individual thereafter Systematic Random Sampling N = 64 n = 8 k = 8 First Group QMIS 220, by Dr. M. Zainal Chap 7-13

Cluster Sampling Divide population into several “clusters,” each representative of the population Select a simple random sample of clusters All items in the selected clusters can be used, or items can be chosen from a cluster using another probability sampling technique Population divided into 16 clusters. Randomly selected clusters for sample QMIS 220, by Dr. M. Zainal Chap 7-14

Examples of poor samplings The technique of sampling has been widely used, both properly and improperly, in the area of politics. During the 1936 presidential race where the Literary Digest predicted Alf Landon to win the election over Franklin D. Roosevelt. QMIS 220, by Dr. M. Zainal Chap 7-15

Sampling Error So far, we have stressed the benefits of drawing a sample from a population. However, in statistics, as in life, there's no such thing as a free lunch. By sampling, we expose ourselves to errors that can lead to inaccurate conclusions about the population. The type of error that a statistician is most concerned about is called sampling error. QMIS 220, by Dr. M. Zainal Chap 7-16

Sampling Error 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 QMIS 220, by Dr. M. Zainal Chap 7-17

Sampling Error As the entire population is rarely measured, the sampling error cannot be directly calculated. With inferential statistics, we'll be able to assign probabilities to certain amounts of sampling error later. It occurs when we select a sample that is not a perfect match to the entire population. Sampling errors are a small price to pay to avoid measuring an entire population. QMIS 220, by Dr. M. Zainal Chap 7-18

Sampling Error One way to reduce the sampling error of a statistical study is to increase the size of the sample. In general, the larger the sample size, the smaller the sampling error. If you increase the sample size until it reaches the size of the population, then the sampling error will be reduced to 0. But in doing so, we lose the benefits of sampling. QMIS 220, by Dr. M. Zainal Chap 7-19

Calculating Sampling Error 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: QMIS 220, by Dr. M. Zainal Chap 7-20

Review Population mean:Sample Mean: where: μ = Population mean x = sample mean x i = Values in the population or sample N = Population size n = sample size QMIS 220, by Dr. M. Zainal Chap 7-21

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 QMIS 220, by Dr. M. Zainal Chap 7-22

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 QMIS 220, by Dr. M. Zainal Chap 7-23

Sampling Distribution A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population QMIS 220, by Dr. M. Zainal Chap 7-24

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) A B C D QMIS 220, by Dr. M. Zainal Chap 7-25

A B C D Uniform Distribution P(x) x (continued) Summary Measures for the Population Distribution: Developing a Sampling Distribution QMIS 220, by Dr. M. Zainal Chap 7-26

16 possible samples (sampling with replacement) Now consider all possible samples of size n=2 (continued) 16 Sample Means QMIS 220, by Dr. M. Zainal Chap 7-27 Developing a Sampling Distribution

Sampling Distribution of All Sample Means P(x) x Sample Means Distribution 16 Sample Means _ (continued) (no longer uniform) QMIS 220, by Dr. M. Zainal Chap 7-28 Developing a Sampling Distribution

Summary Measures of this Sampling Distribution: (continued) QMIS 220, by Dr. M. Zainal Chap 7-29 Developing a Sampling Distribution

Comparing the Population with its Sampling Distribution P(x) x A B C D Population N = 4 P(x) x _ Sample Means Distribution n = 2 QMIS 220, by Dr. M. Zainal Chap 7-30

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: Properties of a Sampling Distribution Theorem 1 Theorem 2 QMIS 220, by Dr. M. Zainal Chap 7-31

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 QMIS 220, by Dr. M. Zainal Chap 7-32

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 QMIS 220, by Dr. M. Zainal Chap 7-33

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 QMIS 220, by Dr. M. Zainal Chap 7-34

Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution Properties The sample mean is an unbiased estimator QMIS 220, by Dr. M. Zainal Chap 7-35

The sample mean is a consistent estimator (the value of x becomes closer to μ as n increases) : Sampling Distribution Properties Larger sample size Small sample size (continued) x Population As n increases, decreases QMIS 220, by Dr. M. Zainal Chap 7-36

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 QMIS 220, by Dr. M. Zainal Chap 7-37

n↑ Central Limit Theorem As the sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population QMIS 220, by Dr. M. Zainal Chap 7-38

Population Distribution Sampling Distribution (becomes normal as n increases) Central Tendency Variation (Sampling with replacement) Larger sample size Smaller sample size If the Population is not Normal (continued) Sampling distribution properties: QMIS 220, by Dr. M. Zainal Chap 7-39

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 QMIS 220, by Dr. M. Zainal Chap 7-40

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? QMIS 220, by Dr. M. Zainal Chap 7-41

Example 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 (continued) QMIS 220, by Dr. M. Zainal Chap 7-42

Example Solution (continued) -- find z-scores: (continued) z Sampling Distribution Standard Normal Distribution Population Distribution ? ? ? ? ? ? ?? ? ? ? ? SampleStandardize x QMIS 220, by Dr. M. Zainal Chap 7-43

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 QMIS 220, by Dr. M. Zainal Chap 7-44

Sampling Distribution of p Approximated by a normal distribution if: where and (where π = population proportion) Sampling Distribution P( p ) p QMIS 220, by Dr. M. Zainal Chap 7-45

z-Value for Proportions If sampling is without replacement and n is greater than 5% of the population size, then must use the finite population correction factor: Standardize p to a z value with the formula: QMIS 220, by Dr. M. Zainal Chap 7-46

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) ? QMIS 220, by Dr. M. Zainal Chap 7-47

Example if π =.4 and n = 200, what is P(.40 ≤ p ≤.45) ? (continued) Find : Convert to standard normal: QMIS 220, by Dr. M. Zainal Chap 7-48

Example z Standardize Sampling Distribution Standardized Normal Distribution if π =.4 and n = 200, what is P(.40 ≤ p ≤.45) ? (continued) Use standard normal table: P(0 ≤ z ≤ 1.44) = p QMIS 220, by Dr. M. Zainal Chap 7-49

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 QMIS 220, by Dr. M. Zainal Chap 7-50

Copyright The materials of this presentation were mostly taken from the PowerPoint files accompanied Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. QMIS 220, by Dr. M. Zainal Chap 7-51