Title: Hypothesis Testing:
1Chapter 9
- Hypothesis Testing
- Two Sample Test for Means and Proportions
2Introduction
- The two sample test is similar to the one sample
test except that we are now testing for
differences between two populations rather than a
sample and a population. There are three types of
two sample tests - Hypothesis Testing with Sample Means (Large
Samples) - Hypothesis Testing with Sample Means (Small
Samples) - Hypothesis Testing with Sample Proportions (Large
Samples)
3The Question to be Answered
- Is the difference between sample statistics
large enough to conclude that the populations
represented by the samples are significantly
different
4Null Hypothesis
- The H0 is that the populations are the same.
-
- H0 µ1 µ2
- 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.
5Null Hypothesis (cont.)
- 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.
6Alternate Hypothesis
- The alternate hypothesis is the research
hypothesis. - If the null hypothesis is rejected then we will
have found evidence to support the research
hypothesis. - H1 µ1 µ2
7Formula for Hypothesis Testing with Sample Means
(Large Samples)
8Explanation of formula
- The numerator is the
difference in sample means. - The denominator is the pooled
estimate of the standard error for both samples. - The pooled estimate is calculated by using the
sample information in the following formula
9The Five Step Model
- Make assumptions and meet test requirements.
- State the H0 and H1.
- Select the Sampling Distribution and Determine
the Critical Region. - Calculate the test statistic.
- Make a Decision and Interpret Results.
10Example Hypothesis Testing in the Two Sample Case
- Text P. 244 Problem 9.5 b (Email messages)
- 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
11Problem Information
- E-Mail Messages
- Sample 1 (M.Class) Sample 2 (W.Class)
- 8.7
5.7 - S1 0.3 S2 1.1
- N1 89 N2 55
12Step 1 Make Assumptions and Meet Test Requirements
- We have
- Independent Random Samples
- Level of Measurement is Interval Ratio
- Sampling Distribution is normal in shape because
we have a large sample - N1 N2 100 (in this case N1 N2 144)
13Step 2 State the Null Hypothesis
- H0 µ1 µ2
- The Null asserts there is no significant
difference between the populations. - H1 µ1 µ2
- The research hypothesis contradicts the H0 and
asserts there is a significant difference between
the populations.
14Step 3 Select the Sampling Distribution and
Establish the Critical Region
- Sampling Distribution Z distribution
- Alpha (a) 0.05
- Z (critical) 1.96
15Using the formula
- Compute the pooled estimate (S.E.)
- Solve for Z
16Step 5 Make a Decision
- The obtained test statistic (Z 19.74) falls in
the Critical Region so reject the null
hypothesis. - The difference between the sample means is so
large that we can conclude (at a 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
(Z19.74 a.05)
17Two-tailed Hypothesis Test
- When a .05 then .025 of the area is
distributed on either side of the curve in area
(C ) - The .95 in the middle section represents no
significant difference between the two
populations. - The cut-off between the middle section and /-
.025 is represented by a Z-value of /- 1.96.
18Factors 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.
19Significance 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.
20Significance 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 critical region (region of rejection) and be
declared significant.
21Significance Vs Importance
- Significance and importance are different things.
- A sample outcome could be
- significant and important
- significant but unimportant
- not significant but important
- not significant and unimportant
22Using SPSS to do Independent Samples Test for
Difference in Two Means
- SPSS uses a t-test rather than a z-test for both
large and small samples. - Follow guidelines in text at the end of the
chapter. - In interpreting your printout look at the
Levenes test (shown in the first two columns F
and sig.) first. - If the p-value (sig) is greater than alpha.05
focus on interpreting the top row of the t-test
for Equality of Means. If it is less than .05
use the bottom row of the t-test. - If the significance level (Sig. 2-tailed) is less
than a.05 then the difference between the
sample means is significant. Report t df and
your a-level in your interpretation.
23Formula for Hypothesis Testing with Sample
Proportions (Large Samples)
- Formula for proportions
- See nextfor how to calculate the standard
deviation of the sampling distribution and the
pooled estimate of the population proportion. - Note that you need to calculate both these
values in order to solve the denominator of the
above equation!
24Calculating Pu (the Pooled Estimate of the
Population Proportion) and the Standard Deviation
of the Sampling Distribution
- To calculate Pu (the pooled estimate see p.
231) - Standard Deviation of the S.D. (see p. 231)
25Example
- Using the same guidelines as for the large sample
test for means (above) and the 5-step method
work with a partner and try 9.11 to test for a
difference in proportions. - The answer to this question can be found at the
back of your text.
26Formula (t-test) for Hypothesis Testing with
Sample Means (Small Samples N1 N2 lt 100)
FormulaS.D Note Use t-table with df
N1 N2 - 2
27Example
- Using the same format as for the large sample
test (above) and the 5-step method work with a
partner and try 9.7a - The answer to this question can be found at the
back of your text.