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Chapter Contents
Chapter 8
Confidence Intervals
Confidence Interval for a Mean (μ) with Known σ
Confidence Interval for a Mean (μ) with Unknown σ
Confidence Interval for a Proportion (π)
Estimating from Finite Populations
Sample Size Determination for a Mean
Sample Size Determination for a Proportion
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Chapter 8
Confidence Intervals
Chapter Learning Objectives (LO’s)
Construct a 90, 95, or 99 percent confidence interval for μ.
Know when to use Student’s t instead of z to estimate μ.
Construct a 90, 95, or 99 percent confidence interval for π.
Construct confidence intervals for finite populations.
Calculate sample size to estimate a mean or proportion.
Construct a confidence interval for a variance (optional).
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Chapter 8
Confidence Interval for a Mean () with
known ()
What is a Confidence Interval?
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What is a Confidence Interval?
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Chapter 8
Confidence Interval for a Mean () with known ()
The confidence interval for  with known  is:
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A higher confidence level leads to a wider confidence interval.
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Greater confidence
implies loss of precision
(i.e. greater margin of
error).
95% confidence is
most often used.
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Chapter 8
Choosing a Confidence Level
Confidence Intervals for Example 8.2
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Chapter 8
Interpretation
A confidence interval either does or does not contain .
The confidence level quantifies the risk.
Out of 100 confidence intervals, approximately 95% may contain ,
while approximately 5% might not contain  when constructing 95%
confidence intervals.
When Can We Assume Normality?
If  is known and the population is normal, then we can safely use the
formula to compute the confidence interval.
• If  is known and we do not know whether the population is normal, a common
rule of thumb is that n  30 is sufficient to use the formula as long as the
distribution
Is approximately symmetric with no outliers.
• Larger n may be needed to assume normality if you are sampling from a strongly
skewed population or one with outliers.
•
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Chapter 8
Confidence Interval for a Mean () with Unknown ()
Student’s t Distribution
•
Use the Student’s t distribution instead of the normal distribution
when the population is normal but the standard deviation  is
unknown and the sample size is small.
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Chapter 8
Student’s t Distribution
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Chapter 8
Student’s t Distribution
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t distributions are symmetric and shaped like the standard normal
distribution.
The t distribution is dependent on the size of the sample.
Comparison of Normal and Student’s t
Figure 8.11
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Chapter 8
Degrees of Freedom
Degrees of Freedom (d.f.) is a parameter based on the sample
size that is used to determine the value of the t statistic.
Degrees of freedom tell how many observations are used to
calculate , less the number of intermediate estimates used in
the calculation. The d.f for the t distribution in this case, is given
by d.f. = n -1.
As n increases, the t distribution approaches the shape of the
normal distribution.
For a given confidence level, t is always larger than z, so a
confidence interval based on t is always wider than if z were used.
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Chapter 8
Comparison of z and t
For very small samples, t-values differ substantially from the
normal.
As degrees of freedom increase, the t-values approach the
normal z-values.
For example, for n = 31, the degrees of freedom, d.f. = 31 – 1 =
30.
So for a 90 percent confidence interval, we would use
t = 1.697, which is only slightly larger than z = 1.645.
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Chapter 8
Example GMAT Scores Again
Figure 8.13
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Construct a 90% confidence interval for the mean GMAT score of
all MBA applicants.
x = 510
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Chapter 8
Example GMAT Scores Again
s = 73.77
Since  is unknown, use the Student’s t for the confidence interval
with d.f. = 20 – 1 = 19.
First find t/2 = t.05 = 1.729 from T-table.
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Chapter 8
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For a 90% confidence
interval, use Appendix
D to find t0.05 = 1.729
with d.f. = 19.
Note: One can use Excel,
Minitab, etc. to
obtain these values
as well as to
construct confidence
Intervals.
We are 90 percent confident
that the true mean GMAT
score might be within the
interval [481.48, 538.52]
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Chapter 8
Confidence Interval Width
Confidence interval width reflects
- the sample size,
- the confidence level and
- the standard deviation.
To obtain a narrower interval and more precision
- increase the sample size or
- lower the confidence level (e.g., from 90% to 80% confidence).
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Chapter 8
Using T-Table
Beyond d.f. = 50, Appendix D shows d.f. in steps of 5 or 10.
If the table does not give the exact degrees of freedom, use the
t-value for the next lower degrees of freedom.
This is a conservative procedure since it causes the interval to be
slightly wider.
A conservative statistician may use the t distribution for
confidence intervals when σ is unknown because
using z would underestimate the margin of error.
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Chapter 8
Confidence Interval for a Proportion ()
A proportion is a mean of data whose only values are 0 or 1.
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Chapter 8
Applying the CLT
The distribution of a sample proportion p = x/n is symmetric if  = .50
and regardless of , approaches symmetry as n increases.
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Chapter 8
When is it Safe to Assume Normality of p?
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Rule of Thumb: The sample proportion p = x/n may be assumed to
be normal if both n 10 and n(1- ) 10.
Sample size to assume
normality:
Table 8.9
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Chapter 8
Confidence Interval for 
Since  is unknown, the confidence interval for p = x/n
(assuming a large sample) is
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Chapter 8
Example Auditing
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Chapter 8
N = population size; n = sample size
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Chapter 8
Sample Size determination for a Mean
Sample Size to Estimate 
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To estimate a population mean with a precision of + E (allowable
error), you would need a sample of size. Now,
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Chapter 8
How to Estimate ?
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Method 1: Take a Preliminary Sample
Take a small preliminary sample and use the sample s in place of
 in the sample size formula.
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Method 2: Assume Uniform Population
Estimate rough upper and lower limits a and b and set
 = [(b-a)/12]½.
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Method 3: Assume Normal Population
Estimate rough upper and lower limits a and b and set  = (b-a)/4.
This assumes normality with most of the data with  ± 2 so the
range is 4.
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Method 4: Poisson Arrivals
In the special case when  is a Poisson arrival rate, then  =  .
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To estimate a population proportion with a precision of ± E
(allowable error), you would need a sample of size
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Since  is a number between 0 and 1, the allowable error E is
also between 0 and 1.
Chapter 8
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Chapter 8
How to Estimate ?
Method 1: Assume that  = .50
This conservative method ensures the desired precision. However,
the sample may end up being larger than necessary.
Method 2: Take a Preliminary Sample
Take a small preliminary sample and use the sample p in place of 
in the sample size formula.
Method 3: Use a Prior Sample or Historical Data
How often are such samples available? Unfortunately,  might be
different enough to make it a questionable assumption.
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