Download Chapter 9 Hypothesis Testing II

Chapter 9
Hypothesis Testing II
Chapter Outline
 Introduction
 Hypothesis Testing with Sample
Means (Large Samples)
 Hypothesis Testing with Sample
Means (Small Samples)
 Hypothesis Testing with Sample
Proportions (Large Samples)
Chapter Outline
 The Limitations of Hypothesis Testing:
Significance Versus Importance
 Interpreting Statistics: Are There
Significant Differences in Income
Between Men and Women?
In This Presentation
 The basic logic of the two sample
case.
 Test of significance for sample means
(large samples).
 The difference between “statistical
significance” and “importance”.
Basic Logic
 We begin with a difference between
sample statistics (means or
proportions).
 The question we test:
 “Is the difference between statistics
large enough to conclude that the
populations represented by the
samples are different?”
Basic Logic
 The H0 is that the populations are the
same.
 There is no difference between the parameters
of the two populations
 If the difference between the sample
statistics is large enough, or, if a difference
of this size is unlikely, assuming that the
H0 is true, we will reject the H0 and
conclude there is a difference between the
populations.
Basic Logic
 The H0 is a statement of “no difference”
 The 0.05 level will continue to be our
indicator of a significant difference
 We change the sample statistics to a Z
score, place the Z score on the sampling
distribution and use Appendix A to
determine the probability of getting a
difference that large if the H0 is true.
The Five Step Model
1. Make assumptions and meet test
requirements.
2. State the H0.
3. Select the Sampling Distribution and
Determine the Critical Region.
4. Calculate the test statistic.
5. Make a Decision and Interpret
Results.
Example: Hypothesis Testing in the
Two Sample Case
 Problem 9.7b.
 Middle class families average 8.7 email
messages and working class families
average 5.7 messages.
 The middle class families seem to use
email more but is the difference
significant?
Step 1 Make Assumptions and
Meet Test Requirements
 Model:
 Independent Random Samples
 The samples must be independent of each
other.
 LOM is Interval Ratio
 Number of email messages has a true 0 and
equal intervals so the mean is an appropriate
statistic.
 Sampling Distribution is normal in shape
 N = 144 cases so the Central Limit Theorem
applies and we can assume a normal shape.
Step 2 State the Null
Hypothesis
 H0: μ1 = μ2
 The Null asserts there is no significant
difference between the populations.
Step 2 State the Null
Hypothesis
 H1: μ1  μ2
 The research hypothesis contradicts the
H0 and asserts there is a significant
difference between the populations.
Step 3 Select the S. D. and
Establish the C. R.
 Sampling Distribution = Z distribution
 Alpha (α) = 0.05
 Z (critical) = ± 1.96
Step 4 Compute the Test
Statistic
 Use Formula 9.4 to compute the
pooled estimate of the standard error.
 Use Formula 9.2 to compute the
obtained Z score.
Step 5 Make a Decision
 The obtained test statistic (Z = 20.00) falls
in the Critical Region so reject the null
hypothesis.
 The difference between the sample means
is so large that we can conclude (at α =
0.05) that a difference exists between the
populations represented by the samples.
 The difference between the email usage of
middle class and working class families is
significant.
Factors in Making a Decision
 The size of the difference (e.g.,
means of 8.7 and 5.7 for problem
9.7b)
 The value of alpha (the higher the
alpha, the more likely we are to reject
the H0
Factors in Making a Decision
 The use of one- vs. two-tailed tests
(we are more likely to reject with a
one-tailed test)
 The size of the sample (N). The
larger the sample the more likely we
are to reject the H0.
Significance Vs. Importance
 As long as we work with random
samples, we must conduct a test of
significance.
 Significance is not the same thing as
importance.
 Differences that are otherwise trivial or
uninteresting may be significant.
Significance Vs. Importance
 When working with large samples,
even small differences may be
significant.
 The value of the test statistic (step 4) is
an inverse function of N.
 The larger the N, the greater the value of
the test statistic, the more likely it will
fall in the C.R. and be declared
significant.
Significance Vs Importance
 Significance and importance are different
things.
 In general, when working with random
samples, significance is a necessary but not
sufficient condition for importance.
Significance Vs Importance
 A sample outcome could be:
 significant and important
 significant but unimportant
 not significant but important
 not significant and unimportant