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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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Properties of the t-Distribution
The t-distribution is bell shaped and symmetric about the
mean.
2. 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 x 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.
1.

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d.f. = n – 1
Degrees of freedom
Properties of the t-Distribution
3. The total area under a t-curve is 1 or 100%.
4. The mean, median, and mode of the t-distribution are
equal to zero.
5. 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 zdistribution.
The tails in the tdistribution are “thicker”
than those in the standard
normal distribution.
d.f. = 2
d.f. = 5
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Standard normal curve
t
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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
Confidence Intervals for the Population
Mean
A c-confidence interval for the population mean μ

x E   x E
where E  tc
s
n
 The probability that the confidence interval contains μ is c.
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Confidence Intervals and t-Distributions
In Words
1. Identify the sample
statistics n, x , and s.
2. Identify the degrees of
freedom, the level of
confidence c, and the
critical value tc.
3. Find the margin of error E.
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In Symbols
x
(x  x )2
x
s
n 1
n
d.f. = n – 1
E  tc
s
n
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
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:
E  tc
s  2.131 10  5.3
n
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 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
156.7
(
x E
Point estimate
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?
Is n  30?
Yes
No
Is the population normally,
or approximately normally,
distributed?
No
Cannot use the normal
distribution or the t-distribution.
Yes
Use the normal distribution
with E  z σ
Yes
Is  known?
No
c
Use the t-distribution with
E  tc
s
n
and n – 1 degrees of freedom.
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Use the normal distribution with
σ
E  zc
n
If  is unknown, use s instead.
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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 tdistribution, or neither to construct a 95% confidence interval for
the population mean construction cost?
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Solution:
Use the normal distribution (the population is
normally distributed and the population standard deviation
is known)
Section 6.2 Summary
 Interpreted the t-distribution and used a t-distribution table
 Constructed confidence intervals when n < 30, the
population is normally distributed, and σ is unknown
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