Statistics 8.1 Testing the Difference Between Means (Large Independent Samples)
LEQ: How do you do a hypothesis test on the difference between two means?
1. An Overview of Two-Sample Hypothesis Testing:
a. Definition 1: For a two-sample hypothesis test:
i. the ____________________________________ is a statistical
hypothesis that usually states there is no difference between the
parameters of two populations. The null hypothesis always contains
ii. The ___________________________________ is a statistical
hypothesis that is true when is false. The alternative hypothesis
contains the symbol
2. Two-Sample z-Test for the Difference Between Means:
a. Definition 2: A _________________________ can be used to test the
difference between two population means and when a large sample
(at least 30) is randomly selected from each population and the samples
are independent. The ________________________ is , and the
When the samples are large, you can use and in place of and .
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.
b. Guidelines: Using a Two-Sample z-Test for the Difference Between
Means (Large Independent Samples):
i. State the claim mathematically. Identify the null and alternative
ii. Specify the level of significance.
iii. Sketch the sampling distribution.
iv. Determine the critical values.
v. Determine the rejection regions.
vi. Find the standardized test statistic.
vii. Make a decision to reject or fail to reject the null hypothesis.
viii. Interpret the decision in the context of the original claim.
c. Example 1: A two-sample z-test for the difference between means:
An advertising executive claims that there is a difference in the mean
household income for credit card holders of Visa Gold and of Gold
MasterCard. The results of a random survey of 100 customers from each
group are shown below. The two samples are independent. Do the results
support the executive’s claim? Use α = 0.05.
Visa Gold Gold MasterCard
d. Example 2: A two-sample z-test for the difference between means:
A survey indicates that the mean per capita credit card charge for
residents of New Hampshire and New York is $3900 and $3500 per year,
respectively. The survey included a randomly selected sample of size 50
from each state, and sample standard deviations are $900 (NH) and $500
(NY). The two samples are independent. At α = 0.01, is there enough
evidence to conclude that there is a difference in the mean credit card
e. Example 3: Using technology to perform a two-sample z-test:
The American Automobile Association claims that the average daily cost
for meals and lodging for vacationing in Texas is less than the same
average costs for vacationing in Washington State. The table below shows
the results of a random survey of vacationers in each state. The two
samples are independent. At α = 0.01, is there enough evidence to
support the claim?
f. Example 4: Using technology to perform a two-sample z-test:
A sociologist claims that children ages 3 – 12 spent more time watching tv
in 1981 than children ages 3 – 12 do today. A study was conducted in
1981 to find the time that children ages 3 – 12 watched tv on weekdays.
The results (in hours per weekday) are shown below.
2.0 2.5 2.1 2.3 2.1 1.6 2.6 2.1 2.1 2.4 2.1 2.1 1.5
1.7 2.1 2.3 2.5 3.3 2.2 2.9 1.5 1.9 2.4 2.2 1.2 3.0
1.0 2.1 1.9 2.2
Recently, a similar study was conducted. The results are shown below.
1.9 1.8 0.9 1.6 2.0 1.7 1.1 1.1 1.6 2.0 1.4 1.5 1.7
1.6 1.6 1.7 1.2 2.0 2.2 1.6 1.5 2.0 1.6 1.8 1.7 1.3
1.1 1.4 1.2 2.0
At α = 0.025, can you support the sociologist’s claim?
3. HW: p. 409 (10 – 20 evens)