Download Chapter 6

Chapter
6
Confidence Intervals
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Chapter Outline
• 6.1 Confidence Intervals for the Mean (Large
Samples)
• 6.2 Confidence Intervals for the Mean (Small
Samples)
• 6.3 Confidence Intervals for Population Proportions
• 6.4 Confidence Intervals for Variance and Standard
Deviation
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Section 6.1
Confidence Intervals for the Mean
(Large Samples)
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Section 6.1 Objectives
• Find a point estimate and a margin of error
• Construct and interpret confidence intervals for the
population mean
• Determine the minimum sample size required when
estimating μ
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Point Estimate for Population μ
Point Estimate
• A single value estimate for a population parameter
• Most unbiased point estimate of the population mean
μ is the sample mean 𝑥
Estimate Population with Sample
Parameter…
Statistic
x
Mean: μ
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Interval Estimate
Interval estimate
• An interval, or range of values, used to estimate a
population parameter.
Point estimate
12.4
•
(
)
Interval estimate
How confident do we want to be that the interval estimate
contains the population mean μ?
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Level of Confidence
Level of confidence c
• The probability that the interval estimate contains the
population parameter.
c
-zc
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z=0
zc
z
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Level of Confidence
• If the level of confidence is 90%, this means that we
are 90% confident that the interval contains the
population mean μ.
c = 0.90
0.05
0.0500
zc
-zc = -1.645
z=0
zc =zc1.645
z
The corresponding z-scores are +1.645.
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Sampling Error
Sampling error
• The difference between the point estimate and the
actual population parameter value.
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Margin of Error
Margin of error
• The greatest possible distance between the point
estimate and the value of the parameter it is
estimating for a given level of confidence, c.
• Denoted by E.
E  zcσ x  zc
σ
n
When n  30, the sample
standard deviation, s, can
be used for .
• Sometimes called the maximum error of estimate or
error tolerance.
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Steps to Calculate a Confidence Interval
• Confidence interval =
[point estimate (mean)] ± [margin of error]
• 𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 =
Larson/Farber 4th ed
𝑥
𝑛
±
𝜎
𝑧𝑐
𝑛
11
Constructing Confidence Intervals for μ
Finding a Confidence Interval for a Population Mean
(n  30 or σ known with a normally distributed population)
In Words
1. Find the mean of the sample.
2. Specify , if known.
Otherwise, if n  30, find the
sample standard deviation s and
use it as an estimate for .
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In Symbols
x
x
n
(x  x )2
s
n 1
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Constructing Confidence Intervals for μ
In Words
3. Find the critical value zc that
corresponds to the given
level of confidence.
4. Find the margin of error E.
5. Find the left and right
endpoints and form the
confidence interval.
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In Symbols
Use the Standard
Normal Table.
E  zc

n
Interval:
xE  xE
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Example: Constructing a Confidence
Interval σ Known
A college admissions director wishes to estimate the
mean age of all students currently enrolled. In a random
sample of 20 students, the mean age is found to be 22.9
years. From past studies, the standard deviation is
known to be 1.5 years, and the population is normally
distributed. Construct a 90% confidence interval of the
population mean age.
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Solution: Constructing a Confidence
Interval σ Known
• First find the critical values
c = 0.90
zc
-zc = -1.645
z=0
zc =zc1.645
z
zc = 1.645
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Solution: Constructing a Confidence
Interval σ Known
• Margin of error:

E  zc
n
 1.645 
• Confidence interval:
Left Endpoint:
xE
 22.9  0.6
1.5
20
 0.6
Right Endpoint:
xE
 22.9  0.6
 23.5
 22.3
22.3 < μ < 23.5
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Solution: Constructing a Confidence
Interval σ Known
22.3 < μ < 23.5
Point estimate
22.3
(
x E
22.9
23.5
x
xE
•
)
With 90% confidence, you can say that the mean age
of all the students is between 22.3 and 23.5 years.
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Interpreting the Results
• μ is a fixed number. It is either in the confidence
interval or not.
• Incorrect: “There is a 90% probability that the actual
mean is in the interval (22.3, 23.5).”
• Correct: “If a large number of samples is collected
and a confidence interval is created for each sample,
approximately 90% of these intervals will contain μ.
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Interpreting the Results
The horizontal segments
represent 90% confidence
intervals for different
samples of the same size.
In the long run, 9 of every
10 such intervals will
contain μ.
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μ
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Section 6.2
Confidence Intervals for the Mean
(Small Samples)
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Section 6.2 Objectives
• Interpret the t-distribution and use a t-distribution
table
• Construct confidence intervals when n < 30, the
population is normally distributed, and σ is unknown
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The t-Distribution
• When the population standard deviation is unknown,
the sample size is less than 30, and the random
variable x is approximately normally distributed, it
follows a t-distribution.
x -
t
s
n
• Critical values of t are denoted by tc.
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The t-Distribution
The t-distribution is determined by a parameter called the
degrees of freedom. When you use a t-distribution to estimate
a population mean, the degrees of freedom are equal to one
less than the sample size.
 d.f. = n – 1
Degrees of freedom
As the degrees of freedom increase, the t-distribution
approaches the normal distribution. After 30 d.f., the tdistribution is very close to the standard normal z-distribution.
d.f. = 2
d.f. = 5
t
0
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Standard normal curve
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Example: Critical Values of t
Find the critical value tc for a 95% confidence when the
sample size is 15.
Solution: d.f. = n – 1 = 15 – 1 = 14
Table 5: t-Distribution
tc = 2.145
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Solution: Critical Values of t
95% of the area under the t-distribution curve with 14
degrees of freedom lies between t = +2.145.
c = 0.95
t
-tc = -2.145
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tc = 2.145
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Confidence Intervals for the Population
Mean
A c-confidence interval for the population mean μ
s
• x  E    x  E where E  tc
n
• The probability that the confidence interval contains μ
is c.
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Confidence Intervals and t-Distributions
In Words
1. Find the mean and s.d.
2. Identify the degrees of
freedom, the level of
confidence c, and the
critical value tc.
3. Find the margin of error E.
In Symbols
x
(x  x )2
x
s
n 1
n
d.f. = n – 1
s
E  tc
n
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Confidence Intervals and t-Distributions
In Words
4. Find the left and right
endpoints and form the
confidence interval.
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In Symbols
Left endpoint: x  E
Right endpoint: x  E
Interval:
xE  xE
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Example: Constructing a Confidence
Interval
You randomly select 16 coffee shops and measure the
temperature of the coffee sold at each. The sample mean
temperature is 162.0ºF with a sample standard deviation
of 10.0ºF. Find the 95% confidence interval for the
mean temperature. Assume the temperatures are
approximately normally distributed.
Solution:
Use the t-distribution (n < 30, σ is unknown,
temperatures are approximately distributed.)
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Solution: Constructing a Confidence
Interval
• n =16, x = 162.0 s = 10.0 c = 0.95
• df = n – 1 = 16 – 1 = 15
• Critical Value Table 5: t-Distribution
tc = 2.131
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Solution: Constructing a Confidence
Interval
• Margin of error:
s
10
E  tc
 2.131
 5.3
n
16
• Confidence interval:
Left Endpoint:
xE
 162  5.3
 156.7
Right Endpoint:
xE
 162  5.3
 167.3
156.7 < μ < 167.3
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Solution: Constructing a Confidence
Interval
• 156.7 < μ < 167.3
Point estimate
156.7
(
162.0
x E
•x
167.3
)
xE
With 95% confidence, you can say that the mean
temperature of coffee sold is between 156.7ºF and
167.3ºF.
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Normal or t-Distribution?
Is n  30?
Yes
No
Is the population normally,
or approximately normally,
distributed?
Use the normal distribution with
σ
E  zc
n
If  is unknown, use s instead.
No
Cannot use the normal
distribution or the t-distribution.
Yes
Use the normal distribution
with E  z σ
Yes
Is  known?
No
c
n
Use the t-distribution with
E  tc
s
n
and n – 1 degrees of freedom.
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Example: Normal or t-Distribution?
You randomly select 25 newly constructed houses. The
sample mean construction cost is $181,000 and the
population standard deviation is $28,000. Assuming
construction costs are normally distributed, should you
use the normal distribution, the t-distribution, or neither
to construct a 95% confidence interval for the
population mean construction cost?
Solution:
Use the normal distribution (the population is
normally distributed and the population standard
deviation is known)
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Homework
• Page 324-325: 27, 28, 31, 32
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Section 6.3
Confidence Intervals for Population
Proportions
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Section 6.3 Objectives
• Find a point estimate for the population proportion
• Construct a confidence interval for a population
proportion
• Determine the minimum sample size required when
estimating a population proportion
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Point Estimate for Population p
Population Proportion
• The probability of success in a single trial of a
binomial experiment.
• Denoted by p
Point Estimate for p
• The proportion of successes in a sample.
• Denoted by
x number of successes in sample
 pˆ  n 
number in sample
 read as “p hat”
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Confidence Intervals for p
A c-confidence interval for the population proportion p
•
pˆ  E  p  pˆ  E
where E  zc
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pq
ˆˆ
n
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Constructing Confidence Intervals for p
In Words
In Symbols
1. Identify the sample statistics n
and x.
2. Find the point estimate p̂.
3. Verify that the sampling
distribution of p̂ can be
approximated by the normal
distribution.
4. Find the critical value zc that
corresponds to the given level of
confidence c.
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pˆ 
x
n
npˆ  5, nqˆ  5
Use the Standard
Normal Table
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Constructing Confidence Intervals for p
In Words
5. Find the margin of error E.
6. Find the left and right
endpoints and form the
confidence interval.
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In Symbols
E  zc
pq
ˆˆ
n
Left endpoint: p̂  E
Right endpoint: p̂  E
Interval:
pˆ  E  p  pˆ  E
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Example: Confidence Interval for p
In a survey of 1000 U.S. adults, 662 said that it is
acceptable to check personal e-mail while at work.
Construct a 95% confidence interval for the population
proportion of adults in the U.S. adults who say that it is
acceptable to check personal e-mail while at work.
Solution: Recall
pˆ  0.662
qˆ  1  pˆ  1  0.662  0.338
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Solution: Confidence Interval for p
• Verify the sampling distribution of p̂ can be
approximated by the normal distribution
npˆ  1000  0.662  662  5
nqˆ  1000  0.3388  338  5
• Margin of error:
E  zc
pq
ˆ ˆ  1.96 (0.662)  (0.338)  0.029
n
1000
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Solution: Confidence Interval for p
• Confidence interval:
Left Endpoint:
Right Endpoint:
pˆ  E
 0.662  0.029
pˆ  E
 0.662  0.029
 0.633
 0.691
0.633 < p < 0.691
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Solution: Confidence Interval for p
• 0.633 < p < 0.691
Point estimate
p̂  E
p̂
p̂  E
With 95% confidence, you can say that the population
proportion of U.S. adults who say that it is acceptable
to check personal e-mail while at work is between
63.3% and 69.1%.
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Sample Size
• Given a c-confidence level and a margin of error E,
the minimum sample size n needed to estimate p is
2
 zc 
ˆ ˆ 
n  pq
E
• This formula assumes you have an estimate for p̂
and qˆ .
• If not, use pˆ  0.5 and qˆ  0.5.
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Example: Sample Size
You are running a political campaign and wish to
estimate, with 95% confidence, the proportion of
registered voters who will vote for your candidate. Your
estimate must be accurate within 3% of the true
population. Find the minimum sample size needed if
1. no preliminary estimate is available.
Solution:
Because you do not have a preliminary estimate
for p̂ use pˆ  0.5 and qˆ  0.5.
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Solution: Sample Size
• c = 0.95
zc = 1.96
2
E = 0.03
2
 zc 
 1.96 
ˆ ˆ    (0.5)(0.5) 
n  pq
  1067.11
 0.03 
E
Round up to the nearest whole number.
With no preliminary estimate, the minimum sample
size should be at least 1068 voters.
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Example: Sample Size
You are running a political campaign and wish to
estimate, with 95% confidence, the proportion of
registered voters who will vote for your candidate. Your
estimate must be accurate within 3% of the true
population. Find the minimum sample size needed if
2. a preliminary estimate gives pˆ  0.31 .
Solution:
Use the preliminary estimate pˆ  0.31
qˆ  1  pˆ  1  0.31  0.69
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Solution: Sample Size
• c = 0.95
zc = 1.96
2
E = 0.03
2
 zc 
 1.96 
ˆ ˆ    (0.31)(0.69) 
n  pq
  913.02
 0.03 
E
Round up to the nearest whole number.
With a preliminary estimate of pˆ  0.31, the
minimum sample size should be at least 914 voters.
Need a larger sample size if no preliminary estimate
is available.
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Section 6.3 Summary
• Found a point estimate for the population proportion
• Constructed a confidence interval for a population
proportion
• Determined the minimum sample size required when
estimating a population proportion
• Homework: Page 332-334: 6-18 every three
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