Section 8.1
Testing the Difference Between
Means (Large Independent Samples)
Section 8.1 Objectives
• Determine whether two samples are independent or
dependent
• Perform a two-sample z-test for the difference
between two means μ1 and μ2 using large independent
samples
Two Sample Hypothesis Test
•
•
Compares two parameters from two populations.
Sampling methods:
 Independent Samples
• The sample selected from one population is not
related to the sample selected from the second
population.
 Dependent Samples (paired or matched samples)
• Each member of one sample corresponds to a
member of the other sample.
Independent and Dependent Samples
Independent Samples
Dependent Samples
Sample 1
Sample 1
Sample 2
Sample 2
Example: Independent and Dependent
Samples
Classify the pair of samples as independent or
dependent.
• Sample 1: Weights of 65 college students before their
freshman year begins.
• Sample 2: Weights of the same 65 college students
after their freshmen year.
Solution:
Dependent Samples (The samples can be paired with
respect to each student)
Example: Independent and Dependent
Samples
Classify the pair of samples as independent or dependent.
• Sample 1: Scores for 38 adults males on a psychological screening test for attention-deficit hyperactivity
disorder.
• Sample 2: Scores for 50 adult females on a psychological screening test for attention-deficit hyperactivity
disorder.
Solution:
Independent Samples (Not possible to form a pairing
between the members of the samples; the sample sizes
are different, and the data represent scores for different
individuals.)
Two Sample Hypothesis Test with
Independent Samples
1. Null hypothesis H0
 A statistical hypothesis that usually states there is
no difference between the parameters of two
populations.
 Always contains the symbol ≤, =, or ≥.
2. Alternative hypothesis Ha
 A statistical hypothesis that is true when H0 is
false.
 Always contains the symbol >, ≠, or <.
Two Sample Hypothesis Test with
Independent Samples
H0: μ1 = μ2
Ha: μ1 ≠ μ2
H0: μ1 ≤ μ2
Ha: μ1 > μ2
H0: μ1 ≥ μ2
Ha: μ1 < μ2
Regardless of which hypotheses you use, you always
assume there is no difference between the population
means, or μ1 = μ2.
Note that if μ1 = μ2, that is the same as μ1 - μ2 = 0
Two Sample z-Test for the Difference
Between Means
Three conditions are necessary to perform a z-test for
the difference between two population means μ1 and μ2.
1. The samples must be randomly selected.
2. The samples must be independent.
3. Each sample size must be at least 30, or, if not, each
population must have a normal distribution with a
known standard deviation.
Two Sample z-Test for the Difference
Between Means
If these requirements are met, the sampling distribution
for x1  x2 (the difference of the sample means) is a
normal distribution with
Mean:  x  x   x   x  1   2
1
2
1
2
 12  22
Standard error:  x  x   x2   x2  n  n
1
2
1
2
1
2
Sampling distribution
for x1  x2 :
σ x  x
1
1  2
2
σ x x
1
2
x1  x2
Two Sample z-Test for the Difference
Between Means
• Test statistic is x1  x2.
• The standardized test statistic is
x1  x2  1  2
 12  22

z
where  x  x 
 .
 x x
n1 n2
1
1
2
2
µ1 - µ2 = 0
• When the samples are large, you can use s1 and s2 in place
of σ1 and σ2. If the samples are not large, you can still use
a two-sample z-test, provided the populations are
normally distributed and the population standard
deviations are known.
Using a Two-Sample z-Test for the
Difference Between Means (Large
Independent Samples)
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 α.
3. Determine the critical value(s).
Use Table 4 in
Appendix B.
4. Determine the rejection region(s).
Using a Two-Sample z-Test for the
Difference Between Means (Large
Independent Samples)
In Words
5. Find the standardized test
statistic and sketch the
sampling distribution.
6. Make a decision to reject or
fail to reject the null
hypothesis.
In Symbols
z
x1  x2  1  2
 x x
1
2
If z is in the rejection
region, reject H0.
Otherwise, fail to
reject H0.
7. Interpret the decision in the
context of the original claim.
µ1 - µ2 = 0
Example: Two-Sample z-Test for the
Difference Between Means
A credit card watchdog group claims that there is a
difference in the mean credit card debts of households
in New York and Texas. The results of a random survey
of 250 households from each state are shown below.
The two samples are independent. Do the results
support the group’s claim? Use α = 0.05.
New York
Texas
x1  $4446.25
x2  $4567.24
σ1 = $1045.70
n1 = 250
σ2 = $1361.95
n2 = 250
Solution: Two-Sample z-Test for the
Difference Between Means
•
•
•
•
•
H0: μ 1 = μ 2
Ha: μ1 ≠ μ2 (claim)
α  0.05
n1= 250 , n2 = 250
Rejection Region:
z = –1.11
• Test Statistic:
(4446.25  4567.24)  0
z
 1.11
108.5983
• Decision: Fail to Reject H0.
At the 5% level of significance,
there is not enough evidence to
support the group’s claim that
there is a difference in the mean
credit card debts of households
in New York and Texas.
Example: Two-Sample z-Test for the
Difference Between Means
An energy company wants to choose between two regions to
install energy producing wind turbines. A researcher claims the
wind speed in Region C and Region D are equal. The average
wind speeds for 75 days are observed in Region C and for 80
days in Region D. Do the results support the claim? Use α = 0.03.
Region C
𝑥 1 = 14.0 mph
Region D
𝑥 2 = 15.1 mph
σ1 = 2.9 mph
n1 = 75
σ2 = 3.3 mph
n2 = 80
Solution: Two-Sample z-Test for the
Difference Between Means
•
•
•
•
•
H0: μ1 = μ2 (claim)
Ha: μ1 ≠ μ2
α  0.03
n1= 75 , n2 = 80
Rejection Region:
• Test Statistic:
Solution: Two-Sample z-Test for the
Difference Between Means
•
•
•
•
H0: μ1 = μ2 (claim)
Ha: μ1 ≠ μ2
α  0.03
n1= 75 , n2 = 80
𝒛=
𝒛=
• Test Statistic:
𝟏𝟒 − 𝟏𝟓. 𝟏
𝟐. 𝟗𝟐 𝟑. 𝟑𝟐
+
𝟖𝟎
𝟕𝟓
−𝟏.𝟏
𝟖.𝟒𝟏𝟎 𝟏𝟎.𝟖𝟗𝟎
+ 𝟖𝟎
𝟕𝟓
=
−1.1
.112+.136
=
−1.1
.248
=
−1.1
.498
= −2.21
Solution: Two-Sample z-Test for the
Difference Between Means
•
•
•
•
•
H0: μ1 = μ2 (claim)
Ha : μ 1 ≠ μ 2
α  0.03
n1= 75 , n2 = 80
Rejection Region:
α/2 = .015
So, z0 = -2.17, +2.17
(from table)
Decision: Reject H0
Interpret?
• Test Statistic: z= -2.21
Section 8.1 Summary
• Determined whether two samples are independent or
dependent
• Performed a two-sample z-test for the difference
between two means μ1 and μ2 using large independent
samples