Download Ch7-Sec7.3

Section 7.3
Hypothesis Testing for the Mean (Small Samples)
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Section 7.3 Objectives
 Find critical values in a t-distribution
 Use the t-test to test a mean μ
 Use technology to find P-values and use them with a t-test to
test a mean μ
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Finding Critical Values in a tDistribution
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.
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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.
• Look at α = 0.05 in the
“One Tail, ” column.
• Because the test is lefttailed, the critical value is
negative.
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0.05
-1.725 0
t
Example: Finding Critical Values for t
Find the critical values t0 and -t0 for a two-tailed test given  =
0.05 and n = 26.
Solution:
• The degrees of freedom are
d.f. = n – 1 = 26 – 1 = 25.
• Look at α = 0.05 in the
“Two Tail, ” column.
• Because the test is twotailed, one critical value is
negative and one is positive.
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0.025
-2.060 0
0.025
2.060
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 x
 The standardized test statistic is t.
x 
t
s n
 The degrees of freedom are d.f. = n – 1.
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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.
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Using the t-Test for a Mean μ
(Small Sample)
In Words
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.
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In Symbols
x 
t
s n
If t is in the rejection
region, reject H0.
Otherwise, fail to
reject H0.
Example: Testing μ with a Small
Sample
A used car dealer says that the mean price of a 2005 Honda Pilot
LX is at least $23,900.You suspect this claim is incorrect and find
that a random sample of 14 similar vehicles has a mean price of
$23,000 and a standard deviation of $1113. Is there enough
evidence to reject the dealer’s claim at α = 0.05? Assume the
population is normally distributed. (Adapted from Kelley Blue Book)
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Solution: Testing μ with a Small Sample
•
•
•
•
•
H0: μ ≥ $23,900
Ha: μ < $23,900
α = 0.05
df = 14 – 1 = 13
Rejection Region:
• Test Statistic:
t
-3.026
10
s
n

23, 000  23,900
1113 14
• Decision: Reject H0
0.05
-1.771 0
x 
t
 3.026
At the 0.05 level of significance,
there is enough evidence to
reject the claim that the mean
price of a 2005 Honda Pilot LX
is at least $23,900
Example: Testing μ with a Small
Sample
An industrial company claims that the mean pH level of the water
in a nearby river is 6.8.You randomly select 19 water samples and
measure the pH of each. The sample mean and standard deviation
are 6.7 and 0.24, respectively. Is there enough evidence to reject
the company’s claim at α = 0.05? Assume the population is
normally distributed.
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Solution: Testing μ with a Small Sample
•
•
•
•
•
H0: μ = 6.8
Ha: μ ≠ 6.8
α = 0.05
df = 19 – 1 = 18
Rejection Region:
0.025
-2.101 0
-1.816
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0.025
2.101
t
• Test Statistic:
t
x 
s
n

6.7  6.8
0.24 19
 1.816
• 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.
Example: Using P-values with t-Tests
The American Automobile Association claims that the mean daily
meal cost for a family of four traveling on vacation in Florida is
$118. A random sample of 11 such families has a mean daily meal
cost of $128 with a standard deviation of $20. Is there enough
evidence to reject the claim at α = 0.10? Assume the population is
normally distributed. (Adapted from American Automobile Association)
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Solution: Using P-values with t-Tests
• H0: μ = $118
• Ha: μ ≠ $118
TI-83/84set up:
Calculate:
Draw:
• Decision: 0.1664 > 0.10
Fail to reject H0. At the 0.10 level of significance, there
is not enough evidence to reject the claim that the
mean daily meal cost for a family of four traveling on
vacation in Florida is $118.
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Section 7.3 Summary
 Found critical values in a t-distribution
 Used the t-test to test a mean μ
 Used technology to find P-values and used them with a t-test
to test a mean μ
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