Hypothesis Testing:

1
Chapter 9

• Hypothesis Testing
• Two Sample Test for Means and Proportions

2
Introduction

• 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)

3
The Question to be Answered

• Is the difference between sample statistics
  large enough to conclude that the populations
  represented by the samples are significantly
  different

4
Null 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.

5
Null 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.

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Alternate 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

7
Formula for Hypothesis Testing with Sample Means
(Large Samples)
8
Explanation 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

9
The 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.

10
Example 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

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Problem 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

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Step 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)

13
Step 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.

14
Step 3 Select the Sampling Distribution and
Establish the Critical Region

• Sampling Distribution Z distribution
• Alpha (a) 0.05
• Z (critical) 1.96

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Using the formula

• Compute the pooled estimate (S.E.)
• Solve for Z

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Step 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)

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Two-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.

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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.

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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.

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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 critical region (region of rejection) and be
  declared significant.

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Significance 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

22
Using 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.

23
Formula 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!

24
Calculating 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)

25
Example

• 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.

26
Formula (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
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Example

• 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.