Transcript Statistics for Business and Economics, 6/e

Instructor: Nahid Farnaz (Nhn)
North South University
Statistics for
Business and Economics
6th Edition
Sampling and
Sampling Distributions
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-1
Populations and Samples
 A Population is the set of all items or individuals
of interest

Examples:
All likely voters in the next election
All parts produced today
All sales receipts for November
 A Sample is a subset of the population

Examples:
1000 voters selected at random for interview
A few parts selected for destructive testing
Random receipts selected for audit
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-2
Why Sample?
 Less time consuming than a census
 Less costly to administer than a census
 It is possible to obtain statistical results of a
sufficiently high precision based on samples.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-3
Simple Random Samples
 Every object in the population has an equal chance of
being selected
 Objects are selected independently
 Samples can be obtained from a table of random
numbers or computer random number generators
 A simple random sample is the ideal against which
other sample methods are compared
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-4
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-5
Chapter Outline
Sampling
Distributions
Sampling
Distribution of
Sample
Mean
Sampling
Distribution of
Sample
Proportion
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Sampling
Distribution of
Sample
Variance
Chap 7-6
Developing a
Sampling Distribution
 Assume there is a population …
 Population size N=4
A
B
C
D
 Random variable, X,
is age of individuals
 Values of X:
18, 20, 22, 24 (years)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-7
Developing a
Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
X

μ
P(x)
i
N
18  20  22  24

 21
4
σ
 (X  μ)
i
N
.25
0
2
 2.236
18
20
22
24
A
B
C
D
x
Uniform Distribution
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-8
Developing a
Sampling Distribution
(continued)
Now consider all possible samples of size n = 2
1st
Obs
2nd Observation
18
20
22
24
18 18,18 18,20 18,22 18,24
16 Sample
Means
20 20,18 20,20 20,22 20,24
1st 2nd Observation
Obs 18 20 22 24
22 22,18 22,20 22,22 22,24
18 18 19 20 21
24 24,18 24,20 24,22 24,24
20 19 20 21 22
16 possible samples
(sampling with
replacement)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
22 20 21 22 23
24 21 22 23 24
Chap 7-9
Developing a
Sampling Distribution
(continued)
Sampling Distribution of All Sample Means
Sample Means
Distribution
16 Sample Means
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
_
P(X)
.3
.2
.1
0
18 19
20 21 22 23
24
_
X
(no longer uniform)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-10
Developing a
Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution:
X

E( X) 
18  19  21   24

 21  μ
N
16
i
σX 

2
(
X

μ)
i

N
(18 - 21)2  (19 - 21)2    (24 - 21)2
 1.58
16
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-11
Expected Value of Sample Mean
 Let X1, X2, . . . Xn represent a random sample from a
population
 The sample mean value of these observations is
defined as
1 n
X   Xi
n i1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-12
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:
σ
σX 
n
 Note that the standard error of the mean decreases as
the sample size increases
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-13
If the Population is Normal
 If a population is normal with mean μ and
standard deviation σ, the sampling distribution
is also normally distributed with
X of
μX  μ
and
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
σ
σX 
n
Chap 7-14
Z-value for Sampling Distribution
of the Mean
 Z-value for the sampling distribution of X :
( X  μ) ( X  μ)
Z

σ
σX
n
where:
X = sample mean
μ = population mean
σ = population standard deviation
n = sample size
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-15
Sampling Distribution Properties
μx  μ

Normal Population
Distribution
μ
(i.e.
x is unbiased )
x
Normal Sampling
Distribution
(has the same mean)
μx
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
x
Chap 7-16
Sampling Distribution Properties
(continued)
 For sampling with replacement:
As n increases,
Larger
sample size
σ x decreases
Smaller
sample size
μ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
x
Chap 7-17
Example 7.2:
“Executive Salary Distributions”
 Suppose that the annual percentage salary
increases for the chief executive officers of all
midsize corporations are normally distributed
with mean 12.2% and standard deviation 3.6%.
 A random sample of nine observations is
obtained from this population and the sample
mean computed.
 What is the probability that the sample mean
will be less than 10%?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-18
Example 7.3
“Spark Plug Life”
 A spark plug manufacturer claims that the lives
of its plugs are normally distributed with mean
36,000 miles and standard deviation 4000
miles.
 A random sample of 16 plugs had an average
life of 34,500 miles.
 If the manufacturer’s claim is correct, what is
the probability of finding a sample mean of
34,500 or less?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-19
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
and
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
σ
σx 
n
Chap 7-20
Central Limit Theorem
As the
sample
size gets
large
enough…
n↑
the sampling
distribution
becomes
almost normal
regardless of
shape of
population
x
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-21
If the Population is not Normal
(continued)
Population Distribution
Sampling distribution
properties:
Central Tendency
μx  μ
σ
σx 
n
Variation
μ
x
Sampling Distribution
(becomes normal as n increases)
Larger
sample
size
Smaller
sample size
μx
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
x
Chap 7-22
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?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-23
Example
(continued)
Solution:
 Even if the population is not normally
distributed, the central limit theorem can be
used (n > 25)
 … so the sampling distribution of
approximately normal
x
is
 … with mean μx = 8
σ
3
 …and standard deviation σ x  n  36  0.5
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-24
Example
(continued)
Solution (continued):


μ
μ
 7.8 - 8
8.2 - 8 
X
P(7.8  μ X  8.2)  P



3
σ
3


36
n
36 

 P(-0.5  Z  0.5)  0.3830
Population
Distribution
???
?
??
?
?
?
?
?
μ8
Sampling
Distribution
Standard Normal
Distribution
Sample
.1915
+.1915
Standardize
?
X
7.8
μX  8
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
8.2
x
-0.5
μz  0
0.5
Z
Chap 7-25
Acceptance Intervals
 Goal: determine a range within which sample means are
likely to occur, given a population mean and variance
 By the Central Limit Theorem, we know that the distribution of X
is approximately normal if n is large enough, with mean μ and
standard deviation σ X
 Let zα/2 be the z-value that leaves area α/2 in the upper tail of the
normal distribution (i.e., the interval - zα/2 to zα/2 encloses
probability 1 – α)
 Then
μ  z/2σ X
is the interval that includes X with probability 1 – α
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-26
Sampling Distributions of
Sample Proportions
Sampling
Distributions
Sampling
Distribution of
Sample
Mean
Sampling
Distribution of
Sample
Proportion
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Sampling
Distribution of
Sample
Variance
Chap 7-27
Population Proportions, P
P = the proportion of the population having
some characteristic
 Sample proportion (Pˆ ) provides an estimate
of P:
X
number of items in the sample having the characteristic of interest
Pˆ 

n
sample size
 0 ≤ Pˆ ≤ 1
 Pˆ has a binomial distribution, but can be approximated
by a normal distribution when nP(1 – P) > 9
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-28
^
Sampling Distribution of P
 Normal approximation:
Sampling Distribution
P( Pˆ )
.3
.2
.1
0
0
Properties:
ˆ) p
E( P
and
.2
.4
.6
8
ˆ
1 P
 X  P(1  P)
σ  Var  
n
n
2
ˆ
P
(where P = population proportion)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-29
Z-Value for Proportions
Standardize Pˆ to a Z value with the formula:
Pˆ  P
Z

σ Pˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Pˆ  P
P(1  P)
n
Chap 7-30
Example 7.8
Business Course Selection (Prob. Of Sample Proportion)
 It has been estimated that 43% of business
graduates believe that a course in business
ethics is important.
 Find the probability that more than one-half of a
random sample of 80 business graduates have
this belief.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-31
Example (for Practice)
 If the true proportion of voters who support
Proposition A is P = .4, what is the probability
that a sample of size 200 yields a sample
proportion between .40 and .45?
 i.e.: if P = .4 and n = 200, what is
ˆ ≤ .45) ?
P(.40 ≤ P
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-32
Example
(continued)

Find σ ˆ :
P
Convert to
standard
normal:
if P = .4 and n = 200, what is
ˆ ≤ .45) ?
P(.40 ≤ P
P(1  P)
.4(1 .4)
σPˆ 

 .03464
n
200
.45  .40 
 .40  .40
ˆ
P(.40  P  .45)  P
Z

.03464 
 .03464
 P(0  Z  1.44)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-33
Example
(continued)

if p = .4 and n = 200, what is
ˆ ≤ .45) ?
P(.40 ≤ P
Use standard normal table:
P(0 ≤ Z ≤ 1.44) = .4251
Standardized
Normal Distribution
Sampling Distribution
.4251
Standardize
.40
.45
Pˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
0
1.44
Z
Chap 7-34
Sampling Distributions of
Sample Proportions
Sampling
Distributions
Sampling
Distribution of
Sample
Mean
Sampling
Distribution of
Sample
Proportion
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Sampling
Distribution of
Sample
Variance
Chap 7-35
Sample Variance
 Let x1, x2, . . . , xn be a random sample from a
population. The sample variance is
n
1
2
s2 
(x

x
)

i
n  1 i1
 the square root of the sample variance is called
the sample standard deviation
 the sample variance is different for different
random samples from the same population
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-36
Sampling Distribution of
Sample Variances

The sampling distribution of s2 has mean σ2
E(s 2 )  σ 2

If the population distribution is normal, then
4
2σ
Var(s2 ) 
n 1

If the population distribution is normal then
(n - 1)s2
σ2
has a 2 distribution with n – 1 degrees of freedom
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-37
The Chi-square Distribution
 The chi-square distribution is a family of distributions,
depending on degrees of freedom:
 d.f. = n – 1
0 4 8 12 16 20 24 28
d.f. = 1
2
0 4 8 12 16 20 24 28
d.f. = 5
2
0 4 8 12 16 20 24 28
2
d.f. = 15
 Text Table 7 contains chi-square probabilities
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-38
Degrees of Freedom (df)
Idea: Number of observations that are free to vary
after sample mean has been calculated
Example: Suppose the mean of 3 numbers is 8.0
Let X1 = 7
Let X2 = 8
What is X3?
If the mean of these three
values is 8.0,
then X3 must be 9
(i.e., X3 is not free to vary)
Here, n = 3, so degrees of freedom = n – 1 = 3 – 1 = 2
(2 values can be any numbers, but the third is not free to vary
for a given mean)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-39
Chi-square Example
 A commercial freezer must hold a selected
temperature with little variation. Specifications call
for a standard deviation of no more than 4 degrees
(a variance of 16 degrees2).
 A sample of 14 freezers is to be
tested
 What is the upper limit (K) for the
sample variance such that the
probability of exceeding this limit,
given that the population standard
deviation is 4, is less than 0.05?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-40
Finding the Chi-square Value
2
(n
1)s
χ2 
σ2
Is chi-square distributed with (n – 1) = 13
degrees of freedom
 Use the the chi-square distribution with area 0.05
in the upper tail:
213 = 22.36 (α = .05 and 14 – 1 = 13 d.f.)
probability
α = .05
2
213 = 22.36
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-41
Chi-square Example
(continued)
213 = 22.36
So:
(α = .05 and 14 – 1 = 13 d.f.)
 (n  1)s2
2 

P(s  K)  P
 χ13   0.05
 16

2
(n  1)K
 22.36
16
or
so
K
(where n = 14)
(22.36)(16)
 27.52
(14  1)
If s2 from the sample of size n = 14 is greater than 27.52, there is
strong evidence to suggest the population variance exceeds 16.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-42
Example 7.9
George Sampson is responsible for quality assurance at
Integrated Electronics. He has asked you to establish a
quality monitoring process for the manufacturer of control
device A. The variability of the electrical resistance,
measured in ohms, is critical for this device.
Manufacturing standards specify a standard deviation of
3.6 and normal distribution. The monitoring process
requires that a random sample of n=6 observations be
obtained from the population of devices and the sample
variance be computed.
Determine an upper limit for the sample variance such that
the probability of exceeding this limit, given a population
S.D. of 3.6, is less than 0.05.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-43