Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-1 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.Chap 7-2 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.Chap 7-3 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.Chap 7-4 Types of Samples Quota Samples Non-Probability Samples JudgmentChunk Probability Samples Simple Random Systematic Stratified Cluster Convenience

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-5 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.Chap 7-6 Types of Samples  In a probability sample, items in the sample are chosen on the basis of known probabilities. Probability Samples Simple Random SystematicStratifiedCluster

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-7 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.

Random Number Table

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-9 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 8 th 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.Chap 7-10 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.Chap 7-11 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.Chap 7-12 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.Chap 7-13 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.Chap 7-14 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.Chap 7-15 Measurement Error  A person with one watch always knows what time it is;  A person with two watches always searches to identify the correct one;  A person with ten watches is always reminded of the difficulty in measuring time

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-16 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.

Population (Size of population = N) Sample number 1 Sample number 2 Sample number 3 Sample number N C n Each sample size = n

Point Estimators Sample Population

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-19 Sampling Distributions Sample Mean Example  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.Chap 7-20 Sampling Distributions Sample Mean Example Summary Measures for the Population Distribution: A B C D P(x) x Uniform Distribution

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-21 Sampling Distributions Sample Mean Example 1 st Obs. 2 nd Observation ,1818,2018,2218, ,1829,2020,2220, ,1822,2022,2222, ,1824,2024,2224,24 Now consider all possible samples of size n=2 1 st Obs. 2 nd Observation Sample Means 16 possible samples (sampling with replacement)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-22 Sampling Distributions Sample Mean Example Sampling Distribution of All Sample Means 1 st Obs 2 nd Observation Sample Means P(X) X (no longer uniform) Sample Means Distribution _

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-23 Sampling Distributions Sample Mean Example Summary Measures of this Sampling Distribution:

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-24 Sampling Distributions Sample Mean Example Population N = 4 Sample Means Distribution n = A B C D P(X) X P(X) X _ _

Example: Let N = {2,6,10,11,21} Find µ, median and σ µ = 10 median = 10 σ = 6.36 How many samples of size 3 are possible? navemedSxSx σ 1 2,6, ,6, ,6, ,10, ,10, ,11, ,10, ,10, ,11, ,11, C 3 = 10 med P(x).3.4.3

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-26 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.Chap 7-27 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.Chap 7-28 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.Chap 7-29 Sampling Distributions Properties: Normal Pop. (i.e. is unbiased ) Normal Population Distribution Normal Sampling Distribution (has the same mean)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-30 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.Chap 7-31 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.Chap 7-32 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.Chap 7-33 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.Chap 7-34 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.Chap 7-35 Sampling Distributions Example = 2( ) = 2(.1915) = Z Standardized Normal Distribution Sampling Distribution Sample Population Distribution x

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-36 Sampling Distributions Example First, compute Z values for both 7.75 and Now, use the cumulative normal table to compute the correct probability.

As the sample size increases, the sampling distribution of sample means approaches a normal distribution.

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, a.) if one woman is randomly selected, find the probability that her weight is greater than 150 lb. b.) if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lb.

a.) if one woman is randomly selected, find the probability that her weight is greater than 150 lb.  =  = = z = =

a.) if one woman is randomly selected, the probability that her weight is greater than 150 lb. is  =  = =

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, b.) if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lb.

b.) if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lb.  x =  x  = 29 =

b.) if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lb.  x =  x  = z = =

b.) if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lb.  x =  x  = = z = =

b.) if 36 different women are randomly selected, the probability that their mean weight is greater than 150 lb is  x =  x  = = z = =

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, a.) if one woman is randomly selected, find the probability that her weight is greater than 150 lb. P(x > 150) =

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, a.) if one woman is randomly selected, find the probability that her weight is greater than 150 lb. P(x > 150) = b.) if 36 different women are randomly selected, their mean weight is greater than 150 lb. P(x  > 150) =

Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, a.) if one woman is randomly selected, find the probability that her weight is greater than 150 lb. P(x > 150) = b.) if 36 different women are randomly selected, their mean weight is greater than 150 lb. P(x > 150) = It is much easier for an individual to deviate from the mean than it is for a group of 36 to deviate from the mean.

Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, Find an interval that will include 90% of the sample means   x  =

Central Limit Theorem 1. Rainfall in the Northeast has a mean pH level of 4.8 and a standard deviation of.5. What is the probability that the average pH level of 36 randomly selected future rainstorms in the area will have a pH value greater than 5.0?  x = 4.8 n = 36  =.5

Central Limit Theorem 2. Average amount spent by a customer at a particular store averages $ with a standard deviation of $ If 50 customers are chosen at random, find the following probabilities, a. The average amount spent exceeds $  x = n = 50  = 17.10

Central Limit Theorem 2. Average amount spent by a customer at a particular store averages $ with a standard deviation of $ If 50 customers are chosen at random, find the following probabilities, b. The average amount spent is between $50.00 and $  x = n = 50  = 17.10

Central Limit Theorem 3. A production for steel rods have an average of 6 meters and a variance of 64 cm. These rods are tied into bundles of 40. a.What is the probability that the average length of a randomly selected bundle is less than 598 cm?  x = 6 n = 40  = 8

Central Limit Theorem 3. A production for steel rods have an average of 6 meters and a variance of 64 cm. These rods are tied into bundles of 40. b. What is the probability that the average length of a bundle is less than 598 cm or more than 601 cm?  x = 6 n = 40  = 8

Central Limit Theorem 3. A production for steel rods have an average of 6 meters and a variance of 64 cm. These rods are tied into bundles of 40. c. Suppose that the bundle size is 25. What is the probability of selecting a bundle that has an average length less than 598 cm?  x = 6 n = 25  = 8

Sample Proportion Standard Error of the Proportion

Finding Z for the Sampling Distribution of the Proportion

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-58 Sampling Distributions The Proportion  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.Chap 7-59 Sampling Distributions The Proportion  Standard error for the proportion:  Z value for the proportion:

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-60 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.Chap 7-61 Sampling Distributions The Proportion: Example Find : Convert to standardized normal:

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-62 Sampling Distributions The Proportion: Example Use cumulative normal table: P(0 ≤ Z ≤ 1.44) = P(Z ≤ 1.44) – 0.5 =.4251 Z Standardize Sampling Distribution Standardized Normal Distribution.400 p

Central Limit Theorem 1. If 100 consumers are randomly selected regarding their brand preference, what is the probability of observing a sample proportion preferring Brand A as small as.15 or less, if in fact, 25% of the population prefer Brand A?

Central Limit Theorem 2. In a random survey, 40% of all U.S. employees participate in self-insurance health plans. a. In a random sample of 100 employees, what is the probability that at least half of these employees participate in such a plan?

Central Limit Theorem 2. In a random survey, 40% of all U.S. employees participate in self-insurance health plans. b. What is the probability that out of these 100 employees that between 20% and 30% participate?

Central Limit Theorem 3. Tests show that 25% of the population eats breakfast. In a random sample of 500 people, what is the probability that less than 15% chosen eat breakfast?

Central Limit Theorem 3. Tests show that 25% of the population eats breakfast. In a random sample of 500 people, what is the probability that between 20% and 50% eat breakfast?

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 7-68 Chapter Summary  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 In this chapter, we have