Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.. Chap 7-1 Chapter 7 Sampling Distributions Basic Business Statistics 10 th Edition

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.. Chap 7-1

Chapter 7

Sampling Distributions

Basic Business Statistics10th Edition

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 7-2

Learning Objectives

In this chapter, you learn: The concept of the sampling distributionThe concept of the sampling distribution

To compute probabilities related to the sample To compute probabilities related to the sample mean and the sample proportionmean and the sample proportion

The importance of the Central Limit TheoremThe importance of the Central Limit Theorem

To distinguish between different survey sampling methods

To evaluate survey worthiness and survey errors


Chapter 7 On-Line Problem Assignment (using MathXL)

From Textbook Chapter 7

Problem 7.17

Problem 7.25

Problem 7.21

Problem 7.29

Problem 7.33

Login Web Address for MathXL/MyStatLab On-Line

Problems


http://www.mathxl.com




Sampling Distribution of

the Mean

Sampling Distribution of the Proportion



A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population

It is the probability distribution of a statistic


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

X

μμX

n

σσ

X

),(2

nNX



A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population

It is the probability distributionprobability distribution of a statistic


Sampling Distribution of the Mean




Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.


• When the population is normally distributed

Shape: Regardless of sample size, the distribution of sample means will be normally distributed.

Center: The mean of the distribution of sample means is the mean of the population. Sample size does not affect the center of the distribution.

Spread: The standard deviation of the distribution of sample means, or the standard error, is


Z-value for Sampling Distributionof the Mean

Z-value for the sampling distribution of :

where: = sample mean

= population mean

= population standard deviation

n = sample size

Xμσ

n

σμ)X(

σ

)μX(Z

X

X

X


Normal Population Distribution

Normal Sampling Distribution (has the same mean)

Sampling Distribution Properties

(i.e. is unbiased )xx

x

μμx

μ

xμ


Sampling Distribution

Properties

As n increases,

decreasesLarger sample size

Smaller sample size

x

(continued)

xσ

μ

(i.e. is consistent )x


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.

Properties of the sampling distribution:

andμμx n

σσx


n↑

Central Limit Theorem

As the sample size gets large enough…

the sampling distribution becomes almost normal regardless of shape of population

x


Central Limit Theorem

• According to the Central Limit Theorem (CLT), the larger the sample size (n 30), the more normal the distribution of sample means becomes. The CLT is central to the concept of statistical inference because it permits us to draw conclusions about the population based strictly on sample data without having knowledge about the distribution of the underlying population.



• When the population is not normally distributed

Shape: When the sample size taken from such a population is sufficiently large, the distribution of its sample means will be approximately normally distributed regardless of the shape of the underlying population those samples are taken from. According to the Central Limit Theorem, the larger the sample size, the more normal the distribution of sample means becomes.

© 2002 The Wadsworth Group


Population Distribution

Sampling Distribution (becomes normal as n increases)

Central Tendency

Variation

x

x

Larger sample size

Smaller sample size

If the Population is not Normal(continued)

Sampling distribution properties:

μμx

n

σσx

xμ

μ


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

For normal population distributions, the sampling distribution of the mean is always normally distributed


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?


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 = 8

…and standard deviation

(continued)

x

xμ

0.536

3

n

σσx


Example

Solution (continued):(continued)

0.31080.4)ZP(-0.4

363

8-8.2

nσ

μ- X

363

8-7.8P 8.2) X P(7.8

Z7.8 8.2 -0.4 0.4


Standard Normal Distribution .1554

+.1554

Population Distribution

??

??

?????

??? Sample Standardize

8μ 8μX

0μz xX


Table E.14 on page 835 of Text


Sample ProblemSample ProblemSample ProblemSample Problem

You’re an operations analyst for Sprint-Nextel. Long-distance telephone calls are normally distributed with = 8 min. & = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes?

© 1984-1994 T/Maker Co.


Example:

8 =2 25

7.8 8.2 ?

n

P X


Standardized Normal

Distribution2

.425

X 1Z

8X 8.2 Z0Z

0.5

7.8 8 8.2 87.8 8.2

2 / 25 2 / 25

.5 .5 .3830

X

X

XP X P

P Z

7.8 0.5

.1915

X


Another ExampleAnother Example

• Example: When a production machine is properly calibrated, it requires an average of 25 seconds per unit produced, with a standard deviation of 3 seconds. For a simple random sample of n = 36 units, the sample mean is found to be 26.2 seconds per unit. When the machine is properly calibrated, what is the probability that the mean for a simple random sample of this size will be at least 26.2 seconds? Standardized sample mean:

3,25,2.26 x

40.2

36

3252.26

z

0082.04918.5.)40.2()2.26( zPxP




Sampling Distribution of

the Mean



Population Proportion

Categorical Variable

E.g., Gender, Voted for Bush, College Degree

Proportion of Population Having a Certain

Characteristic ( Sample Proportion Provides an Estimate of

If Two Outcomes, X Has a Binomial Distribution

Possess or do not possess characteristic

p

)(


Population Proportions

π = the proportion of the population having some characteristic

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)

size sample

interest ofstic characteri the having sample the in itemsofnumber

n

Xp


Sampling Distribution of p

Approximated by a

normal distribution if:

where

and

(where π = population proportion)

Sampling DistributionP( ps)

.3

.2

.1 0

0 . 2 .4 .6 8 1 p

πpμn

)(1σp

ππ

5p)n(1

5np

and

30n ifor



• When the sample proportion of successes in a sample of n trials is p,

Center: The center of the distribution of sample proportions is the center of the population, .

Spread: The standard deviation of the distribution of sample proportions, or the standard error, is

n

)(1σp

ππ

))1(

,(n

Np


Z-Value for Proportions

n)(1

p

σ

pZ

p

Standardize p to a Z value with the formula:

))1(

,(n

Np


Example

If the true proportion of voters who support

Proposition A is π = 0.4, what is the probability

that a sample of size 200 yields a sample

proportion between 0.40 and 0.45?

i.e.: if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?


Example

if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

(continued)

0.03464200

0.4)0.4(1

n

)(1σp

1.44)ZP(0

0.03464

0.400.45Z

0.03464

0.400.40P0.45)pP(0.40

Find :

Convert to standard normal:

pσ


Example

Z0.45 1.44

0.4251

Standardize

Sampling DistributionStandardized

Normal Distribution

if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

(continued)

Use standard normal table: P(0 ≤ Z ≤ 1.44) = 0.4251

0.40 0p


Another Proportion Example Another Proportion Example

• Example: The campaign manager for a political candidate claims that 55% of registered voters favor the candidate over her strongest opponent. Assuming that this claim is true, what is the probability that in a simple random sample of 300 voters, at least 60% would favor the candidate over her strongest opponent? = 0.55, p = 0.60, n = 300 Standardized sample proportion:

74.10287.0

05.0

300)55.01(55.0

55.060.0

z

0409.04591.5.)74.1()60.0( zPpP


Sample ProblemSample ProblemSample ProblemSample Problem

You’re manager of a bank. 40% of depositors have multiple accounts. You select a random sample of 200 customers. What is the probability that the sample proportion of depositors with multiple accounts would be between 40% & 43% ?

© 1984-1994 T/Maker Co.


Sampling Distribution of Proportion Solution Solution P(.40 P(.40 pp .43) .43)

Sampling Distribution Normal Distribution

Standardized

Z p -

-

= .43 - .40= 0.87

n

)1(

200

40140 ).(.

p = .0346

p

= 1

= 0 .87 Z

.3078

p = .40 .43

?30n


Example: 200 .4 .43 ?Sn p P p

.43 .4.43 .87 .8078

.4 1 .4

200

S

S

S pS

p

pP p P P Z


Standardized Normal

DistributionSp

1Z

Sp

Sp Z0.43 .87



• When the sample proportion of successes in a sample of n trials is p,

Center: The center of the distribution of sample proportions is the center of the population, .

Spread: The standard deviation of the distribution of sample proportions, or the standard error, is

n

)(1σp

ππ

))1(

,(n

Np


Reasons for Drawing a Sample

Less time consuming than a census

Less costly to administer than a census

Less cumbersome and more practical to administer than a census of the targeted population


Nonprobability Sample Items included are chosen without regard to

their probability of occurrence

Probability Sample Items in the sample are chosen on the basis

of known probabilities

Types of Samples Used


Types of Samples Used

Quota

Samples

Non-Probability Samples

Judgement Chunk

Probability Samples

Simple Random

Systematic

Stratified

ClusterConvenience

(continued)


Probability Sampling

Items in the sample are chosen based on known probabilities

Probability Samples

Simple Random

Systematic Stratified Cluster


Simple Random Samples

Every individual or item from the frame has an equal chance of being selected

Selection may be with replacement or without replacement

Samples obtained from table of random numbers or computer random number generators


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

Systematic Samples

N = 64

n = 8

k = 8

First Group


Stratified Samples

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

Population

Divided

into 4

strata

Sample


Cluster Samples

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

Population divided into 16 clusters. Randomly selected

clusters for sample


Advantages and Disadvantages

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)


Evaluating Survey Worthiness

What is the purpose of the survey? Is the survey based on a probability sample? Coverage error – appropriate frame? Nonresponse error – follow up Measurement error – good questions elicit good

responses Sampling error – always exists


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

Nonresponse error or bias People who do not respond may be different from those

who do respond

Sampling error Variation from sample to sample will always exist

Measurement error Due to weaknesses in question design, respondent

error, and interviewer’s effects on the respondent


Types of Survey Errors

Coverage error

Non response error

Sampling error

Measurement error

Excluded from frame

Follow up on nonresponses

Random differences from sample to sample

Bad or leading question

(continued)


Standard Error of the Mean 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 (i.e. the standard deviation):

(This assumes that sampling is with replacement or sampling is without replacement from an infinite population)

Note that the standard error of the mean decreases as the sample size increases

n

σσ

X


Chapter Summary

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 Described different types of samples and sampling

techniques Examined survey worthiness and types of survey

errors

Documents

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.. Chap 7-1 Chapter 7 Sampling Distributions Basic Business Statistics 10 th Edition