Download Example: Finding Critical Values for t

Have HW out to be checked
Hypothesis Testing for the Mean
(Small Samples)
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
Larson/Farber 4th ed.
2
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 (𝑡0 ) 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,  ”( 𝑡0 will be negative)
b. right-tailed, use “One Tail,  ” ”( 𝑡0 will be positive)
c. two-tailed, use “Two Tails,  ” ”( 𝑡0 will be negative
and positive)
Larson/Farber 4th ed.
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.
• Look at α = 0.05 in the
“One Tail, ” column.
• Because the test is lefttailed, the critical value is
negative.
Larson/Farber 4th ed.
4
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.
Larson/Farber 4th ed.
5
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.
• t=
−µ
𝑠
𝑛
• The degrees of freedom are d.f. = n – 1.
Larson/Farber 4th ed.
6
Using the t-Test for a Mean μ
(Small Sample)
In Words
In Symbols
1. State the claim mathematically
and verbally. Identify the null
and alternative hypotheses.
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.
7
Using the t-Test for a Mean μ
(Small Sample)
In Words
In Symbols
5. Determine any rejection
region(s).
x 
t
s n
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.
8
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)
Larson/Farber 4th ed.
10
Solution: Testing μ with a Small
Sample
•
•
•
•
•
• Test Statistic:
H0: μ ≥ $23,900
Ha: μ < $23,900
α = 0.05
df = 14 – 1 = 13
Rejection Region:
t
n
23, 000  23,900
1113 14
 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
t
-3.026
Larson/Farber 4th ed.
s

• Decision: Reject H0
0.05
-1.771 0
x 
negative
11
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.
Larson/Farber 4th ed.
12
Solution: Testing μ with a Small
Sample
•
•
•
•
•
• Test Statistic:
H0: μ = 6.8
Ha: μ ≠ 6.8
α = 0.05
df = 19 – 1 = 18
Rejection Region:
0.025
-2.101 0
t
n
6.7  6.8
0.24 19
 1.816
At the 0.05 level of
significance, there is not
enough evidence to reject
the claim that the mean pH
is 6.8.
t
-1.816
Larson/Farber 4th ed.
s

• Decision: Fail to reject H0
0.025
2.101
x 
13
Using calculator
•
•
•
•
Stat
Tests
t-test
Stats if putting in µ , σ, etc
Data if putting in a sample list
Still must figure out 𝐻𝑜 , 𝐻𝑎 , etc.
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 $132. A random
sample of 11 such families has a mean daily meal
cost of $141 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)
Larson/Farber 4th ed.
15
Solution: Using P-values with t-Tests
• H0: μ = $132
• Ha: μ ≠ $132
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 $132.
Larson/Farber 4th ed.
16
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
Larson/Farber 4th ed.
17
assignment
• Page 393 3-35, 37
• More we get done less you have to do