Download ch07 03 6ed

Section 7.3
Hypothesis Testing for the Mean
( Unknown)
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
1
Section 7.3 Objectives
• Find critical values in a t-distribution
• Use the t-test to test a mean μ when σ is not known
• Use technology to find P-values and use them with a
t-test to test a mean μ
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
2
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.
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
3
Example: Finding Critical Values for t
Find the critical value t0 for a left-tailed test given
 = 0.05 and n = 21.
Solution:
• The degrees of freedom are
d.f. = n – 1 = 21 – 1 = 20.
• Use Table 5.
• Look at α = 0.05 in the “One
Tail, ” column.
• Because the test is left-tailed,
the critical value is negative.
.
0.05
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
-1.725 0
t
4
Example: Finding Critical Values for t
Find the critical values -t0 and t0 for a two-tailed test
given  = 0.10 and n = 26.
Solution:
• The degrees of freedom are
d.f. = n – 1 = 26 – 1 = 25.
• Look at α = 0.10 in the
“Two Tail, ” column.
• Because the test is twotailed, one critical value is
negative and one is positive.
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
5
t-Test for a Mean μ ( Unknown)
t-Test for a Mean
• A statistical test for a population mean.
• Random sample, σ not known
• The t-test can be used when the population is normally
distributed, or n  30.
• The test statistic is the sample mean x
• The standardized test statistic is t.
x 
t
s n
• The degrees of freedom are d.f. = n – 1.
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
6
Using P-values for a z-Test for Mean μ
( Unknown)
In Words
In Symbols
1. Verify that  is not known, the
sample is random, and either: the
population is normally distributed
or n  30.
.
2. State the claim mathematically
and verbally. Identify the null
and alternative hypotheses.
State H0 and Ha.
3. Specify the level of significance.
Identify .
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
7
Using P-values for a z-Test for Mean μ
( Unknown)
In Words
In Symbols
4. Identify the degrees of
freedom.
d.f. = n  1
5. Find the standardized test
statistic.
6. Determine the rejection
region(s).
Use Table 5 in
Appendix B.
7. Find the standardized test
statistic and sketch the
sampling distribution.
x 
t
s n
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
8
Using P-values for a z-Test for Mean μ
( Unknown)
In Words
In Symbols
8. Make a decision to reject or
fail to reject the null hypothesis.
If t is in the rejection
region, reject H0.
Otherwise, fail to reject
H0.
9. Interpret the decision in the
context of the original claim.
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
9
Example: Testing μ with a Small Sample
A used car dealer says that the mean price of a twoyear-old sedan is at least $20,500. You suspect this
claim is incorrect and find that a random sample of 14
similar vehicles has a mean price of $19,850 and a
standard deviation of $1084. Is there enough evidence
to reject the dealer’s claim at α = 0.05? Assume the
population is normally distributed.
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
10
Solution: Testing μ with a σ unknown
• Test Statistic:
•
•
•
•
•
.
H0: μ ≥ $20,500 (claim) x   19,850  20, 500
t

  2.244
s n
1084 14
Ha: μ < $20,500
α = 0.05
• Decision:
df = 14 – 1 = 13
Reject H0
Rejection Region:
At the 0.05 level of
significance, there is enough
evidence to reject the claim
that the mean price of a twoyear-old sedan is at least
$20,500.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
11
Example: Hypothesis Testing
An industrial company claims that the mean pH level of
the water in a nearby river is 6.8. You randomly select
39 water samples and measure the pH of each. The
sample mean and standard deviation are 6.7 and 0.35,
respectively. Is there enough evidence to reject the
company’s claim at α = 0.05? Assume the population is
normally distributed.
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
12
Solution: Testing μ with a σ unknown
•
•
•
•
•
H0: μ = 6.8 (claim)
Ha: μ ≠ 6.8
α = 0.05
df = 39 – 1 = 38
Rejection Region:
0.025
-2.024 0
0.025
2.024
t
• Test Statistic:
x - m 6.7 - 6.8
t=
=
» -1.784
s n 0.35 39
• Decision: Fail to reject H0
At the 0.05 level of
significance, there is not
enough evidence to reject
the claim that the mean pH
is 6.8.
-1.784
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
13
Example: Using P-values with t-Tests
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
14
Section 7.3 Summary
• Found critical values in a t-distribution
• Used the t-test to test a mean μ when  is not known
• Used technology to find P-values and used them with
a t-test to test a mean μ when  is not known
.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc.
15