Download Confidence Interval

Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
6-1
Statistics for Business and
Economics
Chapter 6
Inferences Based on a Single Sample
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6-2
Content
1. Identifying and Estimating the Target
Parameter
2. Confidence Interval for a Population Mean:
Normal (z) Statistic
3. Confidence Interval for a Population Mean:
Student’s t-Statistic
4. Large-Sample Confidence Interval for a
Population Proportion
5. Determining the Sample Size
6. Finite Population Correction for Simple
Random Sampling
7. Confidence Interval for a Population Variance
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6-3
Learning Objectives
1. Estimate a population parameter (means,
proportion, or variance) based on a large
sample selected from the population
2. Use the sampling distribution of a statistic to
form a confidence interval for the population
parameter
3. Show how to select the proper sample size
for estimating a population parameter
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6-4
Thinking Challenge
Suppose you’re
interested in the
average amount of
money that students
in this class (the
population) have on
them. How would
you find out?
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6-5
Statistical Methods
Statistical
Methods
Descriptive
Statistics
Inferential
Statistics
Estimation
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Hypothesis
Testing
6-6
6.1
Identifying and Estimating
the Target Parameter
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6-7
Estimation Methods
Estimation
Point
Estimation
Interval
Estimation
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6-8
Target Parameter
The unknown population parameter (e.g., mean or
proportion) that we are interested in estimating is
called the target parameter.
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6-9
Target Parameter
Determining the Target Parameter
Parameter Key Words of Phrase Type of Data
µ
Mean; average
Quantitative
p
Proportion; percentage
fraction; rate
Qualitative
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6-10
Point Estimator
A point estimator of a population parameter is a
rule or formula that tells us how to use the sample
data to calculate a single number that can be used
as an estimate of the target parameter.
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6-11
Point Estimation
1. Provides a single value
• Based on observations from one sample
2. Gives no information about how close the
value is to the unknown population
parameter
3. Example: Sample mean x = 3 is the
point estimate of the unknown
population mean
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Interval Estimator
An interval estimator (or confidence interval) is
a formula that tells us how to use the sample data
to calculate an interval that estimates the target
parameter.
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6-13
Interval Estimation
1. Provides a range of values
•
Based on observations from one sample
2. Gives information about closeness to unknown
population parameter
•
Stated in terms of probability
– Knowing exact closeness requires knowing
unknown population parameter
3. Example: Unknown population mean lies between
50 and 70 with 95% confidence
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6-14
6.2
Confidence Interval for a
Population Mean:
Normal (z) Statistic
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6-15
Estimation Process
Population


Mean, , is
unknown

Random Sample
Mean

 x = 50
I am 95%
confident that 
is between 40 &
60.



Sample





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Key Elements of
Interval Estimation
Sample statistic
Confidence interval
(point estimate)
Confidence limit
(lower)
Confidence limit
(upper)
A confidence interval provides a range of
plausible values for the population parameter.
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6-17
Confidence Interval
According to the Central Limit Theorem, the
sampling distribution of the sample mean is
approximately normal for large samples. Let us
calculate the interval estimator:
1.96
x  1.96 x  x 
n
That is, we form an interval from 1.96 standard
deviations below the sample mean to 1.96 standard
deviations above the mean. Prior to drawing the
sample, what are the chances that this interval will
enclose µ, the population mean?
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Confidence Interval
If sample measurements yield a value of x that falls
between the two lines on either side of µ, then the
interval x  1.96 x will contain µ.
The area under the
normal curve between
these two boundaries
is exactly .95. Thus,
the probability that a
randomly selected
interval will contain µ
is equal to .95.
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Confidence Coefficient
The confidence coefficient is the probability that
a randomly selected confidence interval encloses
the population parameter - that is, the relative
frequency with which similarly constructed
intervals enclose the population parameter when
the estimator is used repeatedly a very large
number of times. The confidence level is the
confidence coefficient expressed as a percentage.
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95% Confidence Level
If our confidence level is 95%, then in the long run,
95% of our confidence intervals will contain µ and 5%
will not.
For a confidence coefficient of 95%, the area in the
two tails is .05. To choose a different confidence
coefficient we increase or decrease the area (call it )
assigned to the tails. If we place /2 in each tail
and z/2 is the z-value, the
confidence interval with
coefficient (1 – ) is
 
x  z 2  x .
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Conditions Required for a Valid
Large-Sample
Confidence Interval for µ
1. A random sample is selected from the target
population.
2. The sample size n is large (i.e., n ≥ 30). Due to
the Central Limit Theorem, this condition
guarantees that the sampling distribution of x is
approximately normal. Also, for large n, s will be
a good estimator of .
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6-22
Large-Sample (1 – )% Confidence
Interval for µ
  
 x  x  z 2 
 n 
 
x  z 2
where z/2 is the z-value with an area /2 to its right
and in the standard normal distribution. The
parameter  is the standard deviation of the
sampled population, and n is the sample size.
Note: When  is unknown and n is large (n ≥ 30),
the confidence interval is approximately equal to
 s 
x  z 2 
 n 
where s is the sample standard deviation.
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Thinking Challenge
You’re a Q/C inspector for
Gallo. The  for 2-liter bottles
is .05 liters. A random
sample of 100 bottles showed
x = 1.99 liters. What is the
90% confidence interval
estimate of the true mean
amount in 2-liter bottles?
22 liter
liter
© 1984-1994 T/Maker Co.
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Confidence Interval
Solution*
x  z /2 
1.99  1.645

n
.05
   x  z /2 

n
   1.99  1.645
100
.05
100
1.982    1.998
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6-25
6.3
Confidence Interval for a
Population Mean:
Student’s t-Statistic
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6-26
Small Sample  Unknown
Instead of using the standard normal statistic
xµ
z
x
use the t–statistic
xµ

 n
xµ
t
s n
in which the sample standard deviation, s, replaces
the population standard deviation, .
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Student’s t-Statistic
The t-statistic has a sampling distribution very
much like that of the z-statistic: mound-shaped,
symmetric, with mean 0.
The primary
difference between
the sampling
distributions of t and
z is that the t-statistic
is more variable than
the z-statistic.
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Degrees of Freedom
The actual amount of variability in the sampling
distribution of t depends on the sample size n. A
convenient way of expressing this dependence is
to say that the t-statistic has (n – 1) degrees of
freedom (df).
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Student’s t Distribution
Standard
Normal
Bell-Shaped
t (df = 13)
Symmetric
t (df = 5)
‘Fatter’ Tails
0
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z
t
6-30
t - Table
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6-31
t-value
If we want the t-value with an area of .025 to its
right and 4 df, we look in the table under the
column t.025 for the entry in the row corresponding
to 4 df. This entry is t.025 = 2.776. The
corresponding standard normal z-score is z.025 =
1.96.
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Small-Sample
Confidence Interval for µ
 s 
x  t 2 
 n 
where ta/2 is based on (n – 1) degrees of freedom.
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Conditions Required for a
Valid Small-Sample
Confidence Interval for µ
1. A random sample is selected from the target
population.
2. The population has a relative frequency
distribution that is approximately normal.
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Estimation Example
Mean ( Unknown)
A random sample of n = 25 has x = 50 and s = 8.
Set up a 95% confidence interval estimate for .
s
s
x  t /2 
   x  t /2 
n
n
8
8
50  2.064 
   50  2.064 
25
25
46.70    53.30
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Thinking Challenge
You’re a time study analyst
in manufacturing. You’ve
recorded the following task
times (min.):
3.6, 4.2, 4.0, 3.5, 3.8, 3.1.
What is the 90% confidence
interval estimate of the
population mean task time?
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Confidence Interval Solution*
• x = 3.7
• s = 3.8987
• n = 6, df = n – 1 = 6 – 1 = 5
• t.05 = 2.015
3.7  2.015
.38987
   3.7  2.015
6
.492    6.908
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.38987
6
6-37
6.4
Large-Sample Confidence
Interval for a Population
Proportion
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Sampling Distribution of p̂
1. The mean of the sampling distribution of p̂ is p;
that is, p̂ is an unbiased estimator of p.
2. The standard deviation of the sampling
distribution of p̂ is pq n ; that is,  p̂ 
where q = 1–p.
pq n
3. For large samples, the sampling distribution of p̂
is approximately normal. A sample size is
considered large if both np̂  15 and nq̂  15.
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Large-Sample Confidence
Interval for p̂
pq
p̂q̂
p̂  z 2 p̂  p̂  z 2 
 p̂  z 2 
n
n
x
where p̂ 
n
and q̂  1  p̂.
Note: When n is large, p̂ can approximate the
value of p in the formula for  p̂ .
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Conditions Required for a
Valid Large-Sample
Confidence Interval for p
1. A random sample is selected from the target
population.
2. The sample size n is large. (This condition will be
satisfied if both np̂  15 and nq̂  15 . Note that np̂
and nq̂ are simply the number of successes and
number of failures, respectively, in the sample.).
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Estimation Example
Proportion
A random sample of 400 graduates showed
32 went to graduate school. Set up a 95%
confidence interval estimate for p.
ˆˆ
ˆˆ
pq
pq
pˆ  Z /2 
 p  pˆ  Z /2 
n
n
.08  1.96 
.08 .92 
400
pˆ 
 p  .08  1.96 
32
 0.08
400
.08 .92 
400
.053  p  .107
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Thinking Challenge
You’re a production
manager for a newspaper.
You want to find the %
defective. Of 200
newspapers, 35 had
defects. What is the 90%
confidence interval estimate
of the population
proportion defective?
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Confidence Interval
Solution*
pˆ  z /2 
pˆ  qˆ
 p  pˆ  z /2 
n
pˆ  qˆ
n
.175(.825)
.175(.825)
.175  1.645 
 p  .175  1.645 
200
200
.1308  p  .2192
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Adjusted (1 – )100%
Confidence Interval for a
Population Proportion, p
p  z 2
p 1  p 
n4
x2
p
where
n  4 is the adjusted sample proportion
of observations with the characteristic of interest, x
is the number of successes in the sample, and n is
the sample size.
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6.5
Determining the Sample Size
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Sampling Error
In general, we express the reliability associated
with a confidence interval for the population mean
µ by specifying the sampling error within which
we want to estimate µ with 100(1 –)% confidence.
The sampling error (denoted SE), then, is equal to
the half-width of the confidence interval.
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Sample Size Determination
for 100(1 – ) %
Confidence Interval for µ
In order to estimate µ with a sampling error (SE)
and with 100(1 – )% confidence, the required
sample size is found as follows:
  
z 2 
 SE

 n
The solution for n is given by the equation
z   

n
2
2
 2
SE 
2
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Sample Size Example
What sample size is needed to be 90%
confident the mean is within  5? A pilot
study suggested that the standard deviation
is 45.
(z 2 ) 
2
n
(SE) 2
2
1.645 45


 219.2  220
5
2
2
2
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Sample Size Determination
for 100(1 – ) %
Confidence Interval for p
In order to estimate p with a sampling error SE and
with 100(1 – )% confidence, the required sample
size is found by solving the following equation for
n:
pq
z 2
 SE
n
The solution for n can be written as follows:
z   pq 

n
2
 2
SE 
2
Note: Always round n
up to the nearest
integer value.
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Sample Size Example
What sample size is needed to estimate p
within .03 with 90% confidence?
width .03
SE 

 .015
2
2
(Z 2 )  pq 
2
n
(SE) 2
1.645  .5 .5 


 3006.69  3007
2
.015 
2
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Thinking Challenge
You work in Human
Resources at Merrill Lynch.
You plan to survey
employees to find their
average medical expenses.
You want to be 95% confident
that the sample mean is
within ± $50.
A pilot study showed that 
was about $400. What
sample size do you use?
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Sample Size Solution*
n
(z 2 )2  2
(SE)2
1.96  400 


50
2
2
2
 245.86  246
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Key Ideas
Population Parameters, Estimators, and
Standard Errors
Parameter
Estimator
 
̂ 
Standard
Error of
Estimator
Estimated
Std Error
̂ 
 
̂
̂
Mean, µ
x
Proportion, p
p̂

n
pq n
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s
n
p̂q̂ n
6-54
Key Ideas
Population Parameters, Estimators, and
Standard Errors
Confidence Interval: An interval that encloses
an unknown population parameter with a certain
level of confidence (1 – )
Confidence Coefficient: The probability (1 – )
that a randomly selected confidence interval
encloses the true value of the population
parameter.
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Key Ideas
Key Words for Identifying the Target
Parameter
µ – Mean, Average
p – Proportion, Fraction, Percentage, Rate,
Probability
2 - Variance
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Key Ideas
Commonly Used z-Values for a LargeSample Confidence Interval
90% CI:
(1 – ) = .10
z.05 = 1.645
95% CI:
(1 – ) = .05
z.025 = 1.96
98% CI:
(1 – ) = .02
z.005 = 2.326
99% CI:
(1 – ) = .01
z.005 = 2.575
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Key Ideas
Determining the Sample Size n
   ME 
Estimating µ: n   z 2  
2
2
2
Estimating p: n   z 2   pq   ME 
2
2
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Key Ideas
Finite Population Correction Factor
Required when n/N > .05
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Key Ideas
Confidence Interval for Population
Variance
Uses chi-square (2) distribution

Need to know
and df.
2
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Key Ideas
Illustrating the Notion of “95%
Confidence”
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Key Ideas
Illustrating the Notion of “95%
Confidence”
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