Section 7.3
Hypothesis Testing for the Mean
(Small Samples)
Larson/Farber 4th ed.
Section 7.3 Objectives
• Find critical values in a t-distribution
• Use the t-test to test a mean μ
Larson/Farber 4th ed.
Finding Critical Values in a t-Distribution
1. Identify the level of significance  .
2. Identify the degrees of freedom d.f. = n – 1.
3. Find the critical value(s) using Table 5 in Appendix B in
the row with n – 1 degrees of freedom. If the hypothesis
test is
a. left-tailed, use “One Tail,  ” column with a negative
sign,
b. right-tailed, use “One Tail,  ” column with a
positive sign,
c. two-tailed, use “Two Tails,  ” column with a
negative and a positive sign.
Larson/Farber 4th ed.
Example: Finding Critical Values for t
Find the critical value t0 for a left-tailed test given
 = 0.05 and n = 15.
Solution:
• The degrees of freedom are
d.f. = n – 1 = 15 – 1 = 14.
• Look at α = 0.05 in the “One
Tail,  ” column.
• Because the test is lefttailed, the critical value is
negative.
Larson/Farber 4th ed.
0.05
-1.761 0
t
Example: Finding Critical Values for t
Find the critical values t0 and -t0 for a two-tailed test
given  = 0.10 and n = 23.
Solution:
• The degrees of freedom are
d.f. = n – 1 = 23 – 1 = 22.
• Look at α = 0.10 in the “Two
0.05
Tail,  ” column.
• Because the test is two-tailed,
-1.717 0
one critical value is negative
and one is positive.
Larson/Farber 4th ed.
0.05
1.717
t
t-Test for a Mean μ (n < 30,  Unknown)
t-Test for a Mean
• A statistical test for a population mean.
• The t-test can be used when the population is normal
or nearly normal,  is unknown, and n < 30.
• The test statistic is the sample mean
• The standardized test statistic is t.
• The degrees of freedom are d.f. = n – 1.
Larson/Farber 4th ed.
Using the t-Test for a Mean μ
(Small Sample)
In Words
1. State the claim mathematically
and verbally. Identify the null and
alternative hypotheses.
In Symbols
State H0 and Ha.
2. Specify the level of significance.
Identify  .
3. Identify the degrees of freedom
and sketch the sampling
distribution.
d.f. = n – 1.
4. Determine any critical value(s).
Use Table 5 in
Appendix B.
Larson/Farber 4th ed.
Using the t-Test for a Mean μ
(Small Sample)
In Words
In Symbols
5. Determine any rejection
region(s).
6. Find the standardized test
statistic.
7. Make a decision to reject or
fail to reject the null
hypothesis.
8. Interpret the decision in the
context of the original claim.
Larson/Farber 4th ed.
If t is in the rejection
region, reject H0.
Otherwise, fail to
reject H0.
Example: Testing μ with a Small Sample
7.3 #26
A computer company believes the mean repair cost for
a damaged computer is more than $95. To test this
claim, you determine the repair costs for 7 randomly
selected computers and find that the mean repair cost is
$100 per computer with a standard deviation of $42.50.
At α = 0.01, do you have enough information to support
the repair’s claim?
Larson/Farber 4th ed.
Solution: Testing μ with a Small Sample
•
•
•
•
•
H0: μ ≤ $95
Ha: μ > $95 (claim)
α = 0.01
df = 7 – 1 = 6
Rejection Region:
0.01
0 3.143
0.311
Larson/Farber 4th ed.
z
• Test Statistic:
• Decision: Fail to reject H0
At the 0.01 level of
significance, there is not
enough evidence to support
the computer repair’s claim
that the mean repair cost for
damaged computers is more
than $95.
Example: Testing μ with a Small Sample
7.3, #28
An employment information service claims the mean
annual pay for full-time female workers over age 25
and without a high school diploma is $19,100. The
annual pay for a random sample of 12 full time female
workers without a high school diploma is gathered. At α
= 0.05, test the claim that the mean salary is $19,100
Larson/Farber 4th ed.
Solution: Testing μ with a Small Sample
•
•
•
•
•
H0: μ = $19,100
Ha: μ ≠ $19,100
α = 0.05
df = 12 – 1 = 11
Rejection Region:
0.025
• Test Statistic:
• Decision: Fail to reject H0
0.025
-2.101 0
-0.529
Larson/Farber 4th ed.
2.101
t
At the 0.05 level of
significance, there is not
enough evidence to reject
the claim that the mean
salary is $19,100 for fulltime female workers, over
25, without a high school
Section 7.3 Summary
• Found critical values in a t-distribution
• Used the t-test to test a mean μ
• HW: 7, 17, 19, 23 - 27 EO, 31, 33
Larson/Farber 4th ed.