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8
Chapter
Sampling Distributions
and Estimation (Part 2)
Sample Size Determination for a
Mean
Sample Size Determination for a
Proportion
C.I. for the Difference of Two Means,
m1-m2
C.I. for the Difference of Two
Proportions, p1-p2
Confidence Interval for a Population
Variance, s2
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc.
Sample Size Determination for a
Mean
Sample Size to Estimate m
•
To estimate a population mean with a
precision of + E (allowable error), you
would need a sample of size
n = zs
E
8B-2
2
Sample Size Determination for a
Mean
How to Estimate s?
8B-3
•
Method 1: Take a Preliminary Sample
Take a small preliminary sample and use
the sample s in place of s in the sample
size formula.
•
Method 2: Assume Uniform Population
Estimate rough upper and lower limits a
and b and set s = [(b-a)/12]½.
Sample Size Determination for a
Mean
How to Estimate s?
•
•
8B-4
Method 3: Assume Normal Population
Estimate rough upper and lower limits a
and b and set s = (b-a)/4. This assumes
normality with most of the data with m + 2s
so the range is 4s.
Method 4: Poisson Arrivals
In the special case when m is a Poisson
arrival rate, then s = m
Sample Size Determination for a
Mean
Using LearningStats
•
8B-5
There is a
sample size
calculator in
LearningStats
for E = 1 and
E = .05.
Sample Size Determination for a
Mean
Using MegaStat
•
8B-6
There is a
sample size
calculator in
MegaStat. The
Preview button
lets you change
the setup and
see results
immediately.
Sample Size Determination for a
Mean
Caution 1: Units of Measure
•
When estimating a mean, the allowable error
E is expressed in the same units as X and s.
Caution 2: Using z
•
Using z in the sample size formula for a
mean is not conservative.
Caution 3: Larger n is Better
•
8B-7
The sample size formulas for a mean tend to
underestimate the required sample size.
These formulas are only minimum
guidelines.
Sample Size Determination
for a Proportion
•
To estimate a population proportion with a
precision of + E (allowable error), you
would need a sample of size
z
n=
E
•
8B-8
2
p(1-p)
Since p is a number between 0 and 1, the
allowable error E is also between 0 and 1.
Sample Size Determination
for a Proportion
How to Estimate p?
•
•
•
8B-9
Method 1: Take a Preliminary Sample
Take a small preliminary sample and use the
sample p in place of p in the sample size formula.
Method 2: Use a Prior Sample or Historical Data
How often are such samples available? p might
be different enough to make it a questionable
assumption.
Method 3: Assume that p = .50
This conservative method ensures the desired
precision. However, the sample may end up
being larger than necessary.
Sample Size Determination
for a Proportion
Using LearningStats
•
The sample size calculator in LearningStats
makes these calculations easy. Here are
some calculations for p = .5 and E = 0.02.
Figure 8.28
8B-10
Sample Size Determination
for a Proportion
Caution 1: Units of Measure
•
For a proportion, E is always between 0 and
1. For example, a 2% error is E = 0.02.
Caution 2: Finite Population
•
8B-11
For a finite population, to ensure that the
sample size never exceeds the population
size, use the following adjustment:
nN
n' =
n + (N-1)
`
Confidence Interval for the
Difference of Two Means m1 – m2
•
•
8B-12
If the confidence interval for the difference
of two means includes zero, we could
conclude that there is no significant
difference in means.
The procedure for constructing a confidence
interval for m1 – m2 depends on our
assumption about the unknown variances.
Confidence Interval for the
Difference of Two Means m1 – m2
Assuming equal variances:
2 + (n – 2)s 2
(n
–
1)s
1
1
2
2 1 +1
(x1 – x2) + t
n1 + n2 - 2
n1 n2
with n = (n1 – 1) + (n2 – 1) degrees of freedom
8B-13
Confidence Interval for the
Difference of Two Means m1 – m2
Assuming unequal variances:
2 s 2
s
(x1 – x2) + t 1+ 2
n1 n2
[s12/n1 + s22/n2]2 (Welch’s formula for
with n' = 2
(s1 /n1)2 + (s22/n2)2 degrees of freedom)
n1 – 1
n2 – 1
Or you can use a conservative quick rule for the
degrees of freedom: n* = min (n1 – 1, n2 – 1).
8B-14
Confidence Interval for the
Difference of Two Proportions p1 – p2
•
If both samples are large (i.e., np > 10 and
n(1-p) > 10, then a confidence interval for the
difference of two sample proportions is
given by
(p1 – p2) + z
8B-15
p1(1 - p1) + p2(1 - p2)
n1
n2
Confidence Interval for a
Population Variance s2
Chi-Square Distribution
•
•
•
If the population is normal, then the sample
variance s2 follows the chi-square distribution
(c2) with degrees of freedom n = n – 1.
Lower (c2L) and upper (c2U) tail percentiles for
the chi-square distribution can be found using
Appendix E.
Using the sample variance s2, the confidence
interval is
2
(n – 1)s2
(n
–
1)s
2 <
<
s
c 2U
c 2L
8B-16
Confidence Interval for a
Population Variance s2
8B-17
Confidence Interval for a
Population Variance s2
Confidence Interval for s
•
To obtain a confidence interval for the
standard deviation, just take the square root
of the interval bounds.
(n – 1)s2
(n – 1)s2
< s <
2
cU
c 2L
8B-18
Confidence Interval for a
Population Variance s2
Using MINITAB
•
MINITAB
gives
confidence
intervals for
the mean,
median, and
standard
deviation.
Figure 8.31
8B-19
Confidence Interval for a
Population Variance s2
Using LearningStats
•
Here is an example for n = 39. Because the
sample size is large, the distribution is
somewhat bell-shaped.
Figure 8.32
8B-20
Confidence Interval for a
Population Variance s2
Caution: Assumption of Normality
•
•
8B-21
The methods described for confidence
interval estimation of the variance and
standard deviation depend on the population
having a normal distribution.
If the population does not have a normal
distribution, then the confidence interval
should not be considered accurate.
Applied Statistics in
Business & Economics
End of Chapter 8B
8B-22
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc.