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**Statistics for Managers Using Microsoft® Excel 5th Edition**

Chapter 7 Sampling and Sampling Distributions Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Learning Objectives In this chapter, you will learn:**

To distinguish between different survey sampling methods 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 Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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Why Sample? Selecting a sample is less time-consuming than selecting every item in the population (census). Selecting a sample is less costly than selecting every item in the population. An analysis of a sample is less cumbersome and more practical than an analysis of the entire population. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Non-Probability Samples**

Types of Samples Samples Non-Probability Samples Probability Samples Simple Random Stratified Judgment Chunk Cluster Systematic Quota Convenience Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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Types of Samples In a nonprobability sample, items included are chosen without regard to their probability of occurrence. In convenience sampling, items are selected based only on the fact that they are easy, inexpensive, or convenient to sample. In a judgment sample, you get the opinions of pre-selected experts in the subject matter. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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Types of Samples In a probability sample, items in the sample are chosen on the basis of known probabilities. Probability Samples Simple Random Systematic Stratified Cluster Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Simple Random Sampling**

Every individual or item from the frame has an equal chance of being selected Selection may be with replacement (selected individual is returned to frame for possible reselection) or without replacement (selected individual isn’t returned to the frame). Samples obtained from table of random numbers or computer random number generators. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Systematic Sampling Decide on sample size: n**

Divide frame of N individuals into groups of k individuals: k=N/n Randomly select one individual from the 1st group Select every kth individual thereafter For example, suppose you were sampling n = 9 individuals from a population of N = 72. So, the population would be divided into k = 72/9 = 8 groups. Randomly select a member from group 1, say individual 3. Then, select every 8th individual thereafter (i.e. 3, 11, 19, 27, 35, 43, 51, 59, 67) Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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Stratified Sampling Divide population into two or more subgroups (called strata) according to some common characteristic. A simple random sample is selected from each subgroup, with sample sizes proportional to strata sizes. Samples from subgroups are combined into one. This is a common technique when sampling population of voters, stratifying across racial or socio-economic lines. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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Cluster Sampling Population is divided into several “clusters,” each representative of the population. A simple random sample of clusters is selected. All items in the selected clusters can be used, or items can be chosen from a cluster using another probability sampling technique. A common application of cluster sampling involves election exit polls, where certain election districts are selected and sampled. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Comparing Sampling Methods**

Simple random sample and Systematic sample Simple to use May not be a good representation of the population’s underlying characteristics Stratified sample Ensures representation of individuals across the entire population Cluster sample More cost effective Less efficient (need larger sample to acquire the same level of precision) Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Evaluating Survey Worthiness**

What is the purpose of the survey? Were data collected using a non-probability sample or a probability sample? Coverage error – appropriate frame? Nonresponse error – follow up Measurement error – good questions elicit good responses Sampling error – always exists Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Types of Survey Errors Coverage error or selection bias**

Exists if some groups are excluded from the frame and have no chance of being selected Non response error or bias People who do not respond may be different from those who do respond Sampling error Chance (luck of the draw) variation from sample to sample. Measurement error Due to weaknesses in question design, respondent error, and interviewer’s impact on the respondent Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions**

A sampling distribution is a distribution of all of the possible values of a 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. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Numerical Data**

Suppose your population (simplified) was four people at your institution. Population size N=4 Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years) Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Sample Mean Example**

Summary Measures for the Population Distribution: P(x) .3 .2 .1 x A B C D Uniform Distribution Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Sample Mean Example**

Now consider all possible samples of size n=2 16 Sample Means 1st Obs. 2nd Observation 18 20 22 24 18,18 18,20 18,22 18,24 20,18 29,20 20,22 20,24 22,18 22,20 22,22 22,24 24,18 24,20 24,22 24,24 1st Obs. 2nd Observation 18 20 22 24 19 21 23 16 possible samples (sampling with replacement) Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Sample Mean Example**

Sampling Distribution of All Sample Means 16 Sample Means Sample Means Distribution P(X) 1st Obs 2nd Observation 18 20 22 24 19 21 23 .3 .2 .1 _ X (no longer uniform) Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Sample Mean Example**

Summary Measures of this Sampling Distribution: Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Sample Mean Example**

Population N = 4 Sample Means Distribution n = 2 _ P(X) P(X) .3 .3 .2 .2 .1 .1 _ X A B C D X Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Standard Error**

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: Note that the standard error of the mean decreases as the sample size increases. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Standard Error: Normal Pop.**

If a population is normal with mean μ and standard deviation σ, the sampling distribution of the mean is also normally distributed with and (This assumes that sampling is with replacement or sampling is without replacement from an infinite population) Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Z Value: Normal Pop.**

Z-value for the sampling distribution of the sample mean: where: = sample mean = population mean = population standard deviation n = sample size Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Properties: Normal Pop.**

Normal Population Distribution (i.e is unbiased ) Normal Sampling Distribution (has the same mean) Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Properties: Normal Pop.**

For sampling with replacement: As n increases, decreases Larger sample size Smaller sample size Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Non-Normal Population**

The Central Limit Theorem states that as the sample size (that is, the number of values in each sample) gets large enough, the sampling distribution of the mean is approximately normally distributed. This is true regardless of the shape of the distribution of the individual values in the population. Measures of the sampling distribution: Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Non-Normal Population**

Population Distribution Sampling Distribution (becomes normal as n increases) Larger sample size Smaller sample size Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Non-Normal Population**

For most distributions, n > 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n > 15 will give a sampling distribution that is nearly normal For normal population distributions, the sampling distribution of the mean is always normally distributed Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions 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 7.75 and 8.25? Even if the population is not normally distributed, the central limit theorem can be used (n > 30). So, the distribution of the sample mean is approximately normal with Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions Example (Numerical data)**

So, the distribution of the sample mean is approximately normal (with n > 30) Use PhStats Normal distribution calculator with the revised parameters Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions The Proportion (Categorical Data)**

The proportion of the population having some characteristic is denoted π. Sample proportion ( p ) provides an estimate of π: 0 ≤ p ≤ 1 p has a binomial distribution (assuming sampling with replacement from a finite population or without replacement from an infinite population) Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distribution of p**

Approximated by a normal distribution if: nπ 5 & n(1- π) 5 Mean for the proportion: Standard error for the proportion: Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions The Proportion: 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? In other words, if π = .4 and n = 200, what is P(.40 ≤ p ≤ .45) ? Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Sampling Distributions The Proportion: Example**

Find : p ; Use PhStats Normal distribution calculator with the revised parameters P(.40 ≤ p ≤ .45)= .4251 If from a finite population, check the fpc box and input N Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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**Chapter Summary In this chapter, we have**

Described different types of samples Examined survey worthiness and types of survey errors 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 Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

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