Chapter07 - Sampling and Sampling Distributions

Statistics for Managers

Using Microsoft® Excel

5th Edition

Chapter 7

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

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}

Types of Samples

Samples

Non-Probability

Samples

Judgment

Chunk

Probability Samples

Simple

Random

Systematic

Stratified

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}

Types of Samples



_{In a}

_{probability sample}

_{, items in the sample}

are chosen on the basis of known probabilities.

Probability Samples

Simple

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}

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

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}

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}

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}

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

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}

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

Sampling Distributions

Sample Mean Example



_{Suppose your population (simplified) was}

four people at your institution.



_{Population size N=4}

Sampling Distributions

Sample Mean Example

Summary Measures for the

Population

Distribution:

Sampling Distributions

Sample Mean Example

1

st

Obs.

2

nd

_Observation

18

20

22

24

18

18,18 18,20 18,22 18,24

20

20,18 29,20 20,22 20,24

22

22,18 22,20 22,22 22,24

24

24,18 24,20 24,22 24,24

Now consider all possible samples of size n=2

1

st

Obs.

2

nd

_Observation

18

20

22

24

16 possible samples

(sampling with

Sampling Distributions

Sample Mean Example

Sampling Distribution of All Sample

Means

1

st

Obs

2

nd

_Observation

18

20

22

24

Sampling Distributions

Sample Mean Example

21

Sampling Distributions

Sample Mean Example

Population

Sampling Distributions

Standard Error

n

σ

σ

_X

_



_{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}

Sampling Distributions

Standard Error: Normal Pop.

μ

μ

_X

_

n

σ

σ

_X

_



_{If a population is normal with mean μ and standard}

deviation σ, the sampling distribution of the mean is also

normally distributed with

and

Sampling Distributions

Z Value: Normal Pop.

n



_{Z-value for the sampling distribution of the sample mean:}

where:

= sample mean

= population mean

= population standard deviation

n = sample size

Sampling Distributions

Properties: Normal Pop.

(i.e. is

unbiased )

Normal Population

Distribution

Normal Sampling

Distribution

(has the same mean)

μ

μ

_x

_

x

x

Sampling Distributions

Properties: Normal Pop.



_{For sampling with replacement:}



_{As n increases,}



_decreases

σ

_x

Larger sample

size

Smaller sample

size

x

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:}

μ

Sampling Distributions

Non-Normal Population

Population Distribution

Sampling Distribution

(becomes normal as n increases)

x

x

Larger

sample

size

Smaller sample

size

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 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

8

Sampling Distributions

First, compute Z values for both 7.75 and 8.25.

Sampling Distributions

Example

= 2(.5000-.3085)

= 2(.1915)

= 0.3830

Z

-0.5 0.5

Standardized Normal

Distribution

7.75 8.25

Sampling

Distribution

Sample

x

Population

Distribution

8

Sampling Distributions

The Proportion

size

sample

interest

of

X

items

p







_{The proportion of the population having some}

characteristic is denoted π.



_{Sample proportion ( p ) provides an estimate of π:}



_{0 ≤ p ≤ 1}

Sampling Distributions

The Proportion



_{Standard error for the proportion:}

n

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}

Sampling Distributions

The Proportion: Example

.03464

Convert to

standardized

normal:

p

Sampling Distributions

The Proportion: Example

Use cumulative normal table:

P(0 ≤ Z ≤ 1.44) = P(Z ≤ 1.44) – 0.5 = .4251

.4251

Standardize

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}