Transcript Ch 6.2 Powerpoint

Chapter
6
Confidence Intervals
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Chapter Outline
•
•
•
•
.
6.1 Confidence Intervals for the Mean ( Known)
6.2 Confidence Intervals for the Mean ( Unknown)
6.3 Confidence Intervals for Population Proportions
6.4 Confidence Intervals for Variance and Standard
Deviation
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Section 6.2
Confidence Intervals for the Mean
( Unknown)
.
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Section 6.2 Objectives
• How to interpret the t-distribution and use a tdistribution table
• How to construct and interpret confidence intervals
for a population mean when  is not known
.
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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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Properties of the t-Distribution
1. The mean, median, and mode of the t-distribution
are equal to zero.
2. The t-distribution is bell shaped and symmetric
about the mean.
3. The total area under a t-curve is 1 or 100%.
4. The tails in the t-distribution are “thicker” than those
in the standard normal distribution.
5. The standard deviation of the t-distribution varies
with the sample size, but it is greater than 1.
.
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Properties of the t-Distribution
6. The t-distribution is a family of curves, each
determined by a parameter called the degrees of
freedom. The degrees of freedom are the number
of free choices left after a sample statistic such as
is calculated. 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
.
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Properties of the t-Distribution
7. As the degrees of freedom increase, the tdistribution approaches the normal distribution.
After 30 d.f., the t-distribution is very close to
the standard normal z-distribution.
d.f. = 2
d.f. = 5
Standard normal curve
.
t
0
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Example: Finding 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
.
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
In Symbols
1. Verify that  is not known,
the sample is random, and
the population is normally
distributed or n  30.
2. Identify the sample
statistics n, x , and s.
.
x
x
n
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(x  x )2
s
n 1
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Confidence Intervals and t-Distributions
In Words
In Symbols
3. Identify the degrees of
freedom, the level of
confidence c, and the
critical value tc.
d.f. = n – 1;
Use Table 5.
4. Find the margin of error E.
E  tc
5. Find the left and right
endpoints and form the
confidence interval.
.
s
n
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
(
x E
162.0
•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?
.
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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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Section 6.2 Summary
• Interpreted the t-distribution and used a t-distribution
table
• Constructed and interpreted confidence intervals for a
population mean when  is not known
.
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