7-1
The Excel NORMDIST Function

Computes the cumulative probability to the
value X
NORMDIST
X
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
7-2
The Excel NORMDIST Function

Computes the cumulative probability to the
value X

No need to convert into a Z value

Needs the
mean, and
standard
distribution
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
7-3
The Excel NORMINV Function

To compute an X value that has a cumulative
probability of a certain amount

If mean is 100 and the standard deviation is 20,
the X value with a cumulative probability of .95
is 132.9.
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
Business Statistics:
A First Course
5th Edition
Chapter 7
Sampling and Sampling Distributions
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.
7-4
7-5
Why Sample?

Why not use the entire population?

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.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-6
A Sampling Process Begins
With A Sampling Frame


The sampling frame is a listing of items that
make up the population
Sampling Frame For
Population With 850
Items
Frames are data sources
such as population lists,
Item Name
directories, or maps
Bev R.
Ulan X.
.
.
Joan T.
Paul F.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Item #
001
002
.
.
849
850
A Sampling Process Begins
With A Sampling Frame

The sampling frame is a listing of items that
make up the population

Inaccurate or biased results can result if a frame
excludes certain portions of the population

Using different frames to generate data can lead to
dissimilar conclusions
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-7
7-8
Types of Samples
Samples
Non-Probability
Samples
Judgment
Convenience
Probability Samples
Simple
Random
Stratified
Systematic
Cluster
In nonprobability sampling, items not are selected by
probability.
Statistical inference methods cannot be used.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-9
Types of Samples
Samples
In judgment sampling, experts are selected.
Non-Probability
Probability Samples
May Samples
not be able to generalize to the general public.
Judgment
Convenience
Simple
Random
Stratified
Systematic
Cluster
In convenience sampling, items are easy, quickly,
convenient, or convenient to sample.
Surveys of users visiting a website.
Representative of what population?
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-10
Types of Samples
Samples
Non-Probability
In probability
sampling,
Samples
you select
frame items
based on probabilities.
To be used whenever
Judgment
Convenience
possible to get unbiased
inferences about the
population of interest.
Probability Samples
Simple
Random
Stratified
Systematic
Cluster
There are four types of
probability sampling methods.
We will look at two.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Probability Sample:
Simple Random Sample
7-11

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.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Probability Sample:
Simple Random Sample

The book describes tables of random numbers


7-12
This is prehistoric.
The Excel RAND() function

Randomly generates a number between 0 and 1

If you have a sampling frame numbered 1 to 850,
create a table each cell having the equation
=850*RAND()

A random number between 0 and 500 will appear in
each cell
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Probability Sample:
Simple Random Sample

The Excel RAND() function

If you make any changes to the worksheet, the
numbers will be recalculated.

To prevent this,

Select the range holding the random numbers

Give the command to copy them

Select the range again

Give the command to paste special
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-13
Probability Sample:
Stratified Sample
7-14

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
Population
Divided
into 4
strata
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Probability Sample:
Stratified Sample
7-15

Samples from subgroups are combined into one

This is a common technique when sampling
population of voters, stratifying across racial or
socio-economic lines.

Sampling employees of different levels in an
organization.
Population
Divided
into 4
strata
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-16
Evaluating a Survey Design

What is the purpose of the survey?

Is the survey based on a probability sample?

How will various errors be controlled?
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-17
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

Example: Calling households during the daytime will
exclude many working people

Always consider who is being undercounted
Excluded from
frame
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-18
Types of Survey Errors

Non response error or bias

People who do not respond may be different from
those who do respond

Income level

Ethnic background

Busier

Try multiple times to reach nonrespondents

Do selective follow-up to determine differences
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-19
Types of Survey Errors

Sampling error

Variation from sample to sample will always exist

Increasing sample size minimizes this
Random
differences from
sample to sample
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-20
Types of Survey Errors

Measurement error

Problems in the design of questions

Leading questions


Ambiguous questions


Hawthorne effect: Answering to please the
interviewer
Can be interpreted in different ways
Questions the respondent cannot answer
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-21
Ethics

Purposely create survey errors

Coverage errors


Nonresponse Errors


Purposely exclude certain groups to give more
favorable results
Ignore nonrespondents, whose answers you
probably will not like
Measurement Errors

Asking leading questions
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-22
Sampling Distributions

A sampling distribution is a distribution of all of the
possible values of a sample statistic for a given size
sample selected from a population.

Sample to find the mean age for college students.

If you did this for N times, you would get N sample
statistic values for the mean.

The totality of these values
would constitute the
sampling distribution.
μ
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
x
7-23
Sampling Distributions

A sampling distribution is a distribution of all of the
possible values of a sample statistic for a given size
sample selected from a population.

Most sample values
will be close to the
population mean
P(X)
.3
.2
.1
0
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
18 19
20 21 22 23
24
_
X
Sample Mean Sampling Distribution: 7-24
Standard Error of the Mean

A measure of the variability in the mean from sample to
sample is given by the Standard Error of the Mean:
(This assumes that sampling is with replacement or
sampling is without replacement from an infinite population)
σX

σ

n
Note that the standard error of the mean decreases as
the sample size increases
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Sample Mean Sampling Distribution:
If the Population is Normal

7-25
If a population is normally distributed with mean
μ and standard deviation σ, the sampling
distribution of X is also normally distributed
with
μX  μ
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
σ
σX 
n
Z-value for Sampling
Distribution of the Mean

Z-value for the sampling distribution of X :
Z
where:
( X  μX )
σX
( X  μ)

σ
n
X = sample mean
μ = population mean
σ = population standard deviation
n = sample size
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-26
7-27
Sampling Distribution Properties
μx  μ

Normal Population
Distribution
μ
(i.e.
x is unbiased )
Normal Sampling
Distribution
(has the same mean)
μx
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
x
x
7-28
Sampling Distribution Properties
(continued)
As n increases,
Larger
sample size
σ x decreases
Smaller
sample size
μ
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
x
Determining An Interval Including A
Fixed Proportion of the Sample Means
7-29
Find a symmetrically distributed interval around µ
that will include 95% of the sample means when µ
= 368, σ = 15, and n = 25.

Since the interval contains 95% of the sample means
5% of the sample means will be outside the interval

Since the interval is symmetric 2.5% will be above
the upper limit and 2.5% will be below the lower limit.

From the standardized normal table, the Z score with
2.5% (0.0250) below it is -1.96 and the Z score with
2.5% (0.0250) above it is 1.96.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
Determining An Interval Including A 7-30
Fixed Proportion of the Sample Means
(continued)

Calculating the lower limit of the interval
σ
15
XL  μ  Z
 368  (1.96)
 362.12
n
25

Calculating the upper limit of the interval
σ
15
XU  μ  Z
 368  (1.96)
 373.88
n
25

95% of all sample means of sample size 25 are
between 362.12 and 373.88
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-31
Sample Mean Sampling Distribution:
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:
μx  μ
σ
σx 
n
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-32
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 will
usually give a sampling distribution is almost
normal

For normal population distributions, the
sampling distribution of the mean is always
normally distributed
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-33
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?
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-34
Example
(continued)
Solution:


Even if the population is not normally
distributed, the central limit theorem can be
used (n > 30)
… so the sampling distribution of
approximately normal
x
is

… with mean μx = 8

σ
3

 0.5
…and standard deviation σ x 
n
36
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-35
Example
(continued)
Solution (continued):


 7.8 - 8
X -μ
8.2 - 8 
P(7.8  X  8.2)  P



3
σ
3


36
n
36


 P(-0.4  Z  0.4)  0.3108
Population
Distribution
???
?
??
?
?
?
?
?
μ8
Sampling
Distribution
Standard Normal
Distribution
Sample
.1554
+.1554
Standardize
?
X
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7.8
μX  8
8.2
x
-0.4
μz  0
0.4
Z
7-36
Population Proportions
π = the proportion of the population having
some characteristic

p
Sample proportion ( p ) provides an estimate
of π:
X
number of items in the sample having the characteristic of interest

n
sample size


0≤ p≤1
p is approximately distributed as a normal distribution
when n is large
(assuming sampling with replacement from a finite population or
without replacement from an infinite population)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-37
Sampling Distribution of p

Approximated by a
normal distribution if:

Sampling Distribution
.3
.2
.1
0
nπ  5
and
0
n(1  π )  5
where
P( ps)
μp  π
and
.2
.4
π(1 π )
σp 
n
(where π = population proportion)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
.6
8
1
p
7-38
Z-Value for Proportions
Standardize p to a Z value with the formula:
p 
Z

σp
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
p 
 (1  )
n
7-39
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) ?
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-40
Example
(continued)

if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?
Find σ p : σ p 
 (1  )
n
0.4(1 0.4)

 0.03464
200
0.45  0.40 
 0.40  0.40
Convert to
P(0.40  p  0.45)  P
Z

standardized
0.03464 
 0.03464
normal:
 P(0  Z  1.44)
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
7-41
Example
(continued)

if π = 0.4 and n = 200, what is
P(0.40 ≤ p ≤ 0.45) ?
Use standardized normal table:
P(0 ≤ Z ≤ 1.44) = 0.4251
Standardized
Normal Distribution
Sampling Distribution
0.4251
Standardize
0.40
0.45
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..
p
0
1.44
Z
7-42
Chapter Summary







Probability and nonprobability samples
Two common probability samples
Examined survey worthiness and types of errors
Introduced sampling distributions
Described the sampling distribution of the mean
 For normal populations
 Using the Central Limit Theorem for any
distribution
Described the sampling distribution of a proportion
Calculated probabilities using sampling distributions
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..