Transcript Powerpoint 5b

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)
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, so that 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/wk (s = 3.1, N=125) and
working class families average
5.7(s=2.9, N=104) 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
 each N is 100 or more so the Central Limit
Theorem applies
Step 2 State the Null
Hypothesis
 H0: μ1 = μ2
 The Null asserts there is no significant
difference between the populations.
Step 2 State the Alternative or
Research Hypothesis
 H1: μ1  μ2
 The research hypothesis contradicts the
H0 and asserts there is a significant
difference between the populations.
 This will be a two-tailed test. Why?
Step 3 Select the S. D. and
Establish the C. R.
 Sampling Distribution = Z distribution
because N1 + N2 = 100 or more
 Alpha (α) = 0.05
 Z (critical) = ± 1.96
 We could also use a t-distribution
 Degrees of freedom = N1+N22=125+104=229 t (critical) = 1.96
 But then we use formula 9.5 and 9.6
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 = 7.5) 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
 We test for significance whenever we
generalize from a sample to a
population
 Significance is not the same thing as
importance.
 Differences that are otherwise trivial or
uninteresting may be significant.
Significance Vs Importance
 A sample outcome could be:
 significant and important
 significant but unimportant
 not significant but important
 not significant and unimportant
Test of hypothesis between two
sample proportions
Appropriate only with large samples (N1
+N2 = 100 or more)
The logic is basically the same, but the
formulae get more and more complicated
See formulas 9.7, 9.8, 9.9,. 9.10 on p. 226.
Again, the basic idea to establish the
standard error (the standard deviation of
the sampling distribution) and then do the
five-step testing procedure.