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Confidence Intervals for the
Mean (Small Samples)
Larson/Farber 4th ed
1
 Interpret
the t-distribution and use a tdistribution table
 Construct confidence intervals when n <
30, the population is normally
distributed, and σ is unknown
Larson/Farber 4th ed
2
 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 tdistribution.
x -
t
s
n
 Critical values of t are denoted by tc.
Larson/Farber 4th ed
3
The t-distribution is bell shaped and
symmetric about the mean.
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
x
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.
2.
•
d.f. = n – 1
Degrees of freedom
Larson/Farber 4th ed
4
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
Standard normal curve
t
0
Larson/Farber 4th ed
5
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
Larson/Farber 4th ed
6
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
Larson/Farber 4th ed
7
A c-confidence interval for the
s
x  E    x mean
 E where
population
μ E  tc
n

 The
probability that the confidence
interval contains μ is c.
Larson/Farber 4th ed
8
In Words
In Symbols
x
(x  x )2
x
s
n 1
n
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.
Larson/Farber 4th ed
d.f. = n – 1
E  tc
s
n
9
In Words
In Symbols
4. Find the left and right
endpoints and form the
confidence interval.
Larson/Farber 4th ed
Left endpoint: x  E
Right endpoint: x  E
Interval:
xE  xE
10
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
Solution:
temperature. Assume the temperatures are
Use
the
t-distribution
(n
<
30,
σ
is
unknown,
approximately normally distributed.
temperatures are approximately distributed.)
Larson/Farber 4th ed
11



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
Larson/Farber 4th ed
12

Margin of error:

Confidence interval:
s
E  tc

n
10
2.131
 5.3
16
Left Endpoint:
xE
 162  5.3
 156.7
Right
xE
Endpoint:
 162  5.3
 167.3
156.7 < μ < 167.3
Larson/Farber 4th ed
13
 156.7
< μ < 167.3
156.7
(
x E
Point
162.0
estimate
•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.
Larson/Farber 4th ed
14
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.
Larson/Farber 4th ed
15
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)
Larson/Farber 4th ed
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 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
Larson/Farber 4th ed
17