Download 8.1 Testing the Difference Between Means (Large Independent

8.1 Testing the Difference
Between Means (Large
Independent Samples)
Statistics
Mrs. Spitz
Spring 2009
Objectives/Assignment
• An introduction to two-sample hypothesis
testing for the difference between two
population parameters.
• How to perform a two-sample z-test for the
difference between two means 1 and 2,
using large independent samples
• Assignment: pp. 373-378 #1-26
An overview of Two-Sample
Hypothesis Testing
• In Chapter 7, you studied methods for testing a
claim about the value of a population parameter.
In this chapter, you will learn how to test a claim
comparing parameters from two populations.
• For instance, suppose you are developing a
marketing plan for an Internet service provider
and want to determine whether there is a
difference in the amount of time male and
female college students spen online each day.
An overview of Two-Sample
Hypothesis Testing
• The only way you could conclude with
certainty that there is a difference is to
take a census of all college students,
calculate the mean daily times male
students and female students spend time
online, and find the difference. Of course
if it not practical to take such a census.
However, you can still determine with
some degree of certainty if such a
difference exists.
An overview of Two-Sample
Hypothesis Testing
• You can begin by assuming that there is
no difference in the mean times of the two
populations. That is 1 – 2 = 0. Then by
taking a random sample from each
population, and using the resulting twosample test statistic x1  x2 , you can
perform a two-sample hypothesis test.
Suppose you obtain the following results.
• The graph on the next slide shows the sampling
distribution of x1  x2 for many similar samples
taken from each population. From the graph,
you can see that it is quite unlikely to obtain
sample means that differ by 4 minutes when the
actual difference is 0. The difference of the
sample means is more than 2.5 standard errors
from the hypothesized difference of 0!
•So, you can conclude there is a significant difference in the
amount of time male college students and female college students
spend online each day.
•It is important to remember that when you perform a two-sample
hypothesis test, you are testing a claim concerning the difference
between the parameters in two populations, not the values of the
parameters themselves.
Null/Alternative Hypothesis
• To write a null and an alternative hypothesis for
a two-sample hypothesis test, translate the claim
made about the population parameters from a
verbal statement to a mathematical statement.
Then, write its complementary statement. For
instance, if the claim is about two population
parameters, 1 and 2, then some possible pairs
of null and alternative hypotheses are:
Study Tip:
• You can also write the null and alternative hypotheses as
follows.
Two-Sample z-test for the
Difference Between Means
• In the remainder of this section, you will learn
how to perform a z-test for the difference
between two population means 1 and 2. To
perform such a test, two conditions are
necessary.
1. The samples must be independent. Two
samples are independent if the sample
selected from one population is not related to
the sample selected from the second
population. (more in 8.3)
2. 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, then the
sampling distribution for x1  x2 (the
difference of the sample means) is a
normal distribution with mean and
standard deviation of
 x  x  1   2
1
2
And
Notice that the variance of the sampling
2
distribution

x1  x2
is the sum of the variances of the individual
sampling distributions for x1  x2
Two-Sample z-test for the
Difference Between Means
• Because the sampling distribution for x1  x2
is a normal distribution, you can use the z-test to
test the difference between two population
means 1 and 2. Notice that the standardized
test statistic takes the form of:
If the null hypothesis states that 1 = 2, then
the expression 1 – 2 is equal to 0 in the
preceding test.
Ex. 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 MasterCard
Gold. The results of a
random survey of 100
customers from each
group are shown. The two
samples are independent.
Do the results support the
executive’s claim? Use 
= 0.05..
Solution:
• You want to test the claim that there is a
difference in the mean household incomes
for Visa Gold and MasterCard Gold credit
card holders. So, the null and alternative
hypotheses are:
Ho: 1 = 2 and Ha: 1  2 (Claim)
Solution:
• Because the test is a twotailed test and the level of
significance is  = 0.05,
you look up the critical
values and find they are 1.96 and 1.96. The
rejection regions are z < 1.96 and z > 1.96.
Because both samples are
large, s1 and s2 are used to
calculate the standard
error.
Solution:
• Using the z-test, the
standardized test
statistic is:
z
( x1  x2 )  ( 1   2 )
 x x
1

2
(60,900  64,300)  (0)
z
 1.770
1921
Solution:
• The graph at the left
shows the location of the
rejection regions and the
standardized test statistic,
z. Because z Is not in the
rejection region, you
should fail to reject the
null hypothesis. At the
5% level, there is not
enough evidence to
conclude that there is a
significant difference in
the mean household
incomes of Visa Gold and
MasterCard Gold credit
card holders.
Ex. 2: Using Technology to
Perform a Two-Sample z-test
• The American Automobile
Association (AAA) claims that the
average daily cost for meals and
lodging when vacationing in Texas
is less than the same average
costs when visiting Washington
state. The table on the next slide
shows the results of a survey of
vacationers in each state. At  =
0.01, is there enough evidence to
support the claim?
Ho: 1 ≥ 2 and Ha: 1  2 (Claim)
Note: Texas is population 1 and
Washington is population 2
• The first two displays show
how to set up the hypothesis
test using a TI-83. The
remaining displays show the
possible results depending on
whether you select “Calculate”
or “Draw.”
1.
2.
3.
4.
5.
6.
7.
STAT
Arrow to TESTS
Choose 3: 2-SampZTest
Arrow to DATA and Enter
Enter as shown on the right
Arrow to < 2 and Arrow to
Calculate then Enter
To Draw 2-sample z
• The first two displays show
how to set up the hypothesis
test using a TI-83. The
remaining displays show the
possible results depending on
whether you select “Calculate”
or “Draw.”
1.
2.
3.
4.
5.
STAT
Arrow to TESTS
Choose 3: 2-SampZTest
Arrow down to
DRAW then Enter
Reject or Fail to Reject?
• Because the test is a left-tailed test and 
= 0.01, the rejection region is z < -2.33.
The standardized test statistic, z  -2.12, is
Not in the rejection region, so you should
fail to reject the null hypothesis. At the 1%
level, there is not evidence to support the
American Automobile Association’s claim.
One more on the calculator
• The American Automobile
Association claims that
the average daily meal
and lodging costs while
vacationing in Florida are
greater than the same
costs while vacationing in
Maryland. The table
shows the results of a
survey of vacationers in
each state. At  = 0.05,
is there enough evidence
to support the claim?
1. Use a calculator to find the
test statistic or P-value.
2. Determine whether the test
statistic is in the rejection
region or compare the P-value
to the level of significance, .
3. Make a decision.
Solution:
1. Z = 3.634, p = 0.000014
2. Rejection region is z > 1.645 or p-value <
0.05 =  (Look up value closest – or use
that table from Chapter 6 with the zvalues given)
3. Reject Ho because 3.634 is greater than
1.645 and because 0.00014 is less than
0.05.
Upcoming
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Last day for the quarter is March 11
Monday 2/23 – Notes 8.1
Tuesday 2/24 – work on 8.1 work
Wednesday 2/25– Notes 8.2
Thursday 2/26 – Work on 8.2
Friday 2/27 – Notes 8.3
Monday 3/2 – Work on 8.3
Tuesday 3/3 – Notes 8.4
Wednesday 3/4 – Work on 8.4
Thursday/Fri 3/5-6 – Review for Chapter 8 Exam
Monday 3/9 – Chapter 8 TEST – Everyone must be
here for the exam. I am tired of people flaking out.
There is no excuse for it.