Case II: Testing Difference Between Means

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Case II Testing Difference Between Means

• Case II Do observations taken on two groups of
  subjects differ from one another? Do the two sets
  represent samples from identical or different
  populations?
• Case IIa Known population standard deviation
• Case IIb Population standard deviation not
  known

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Case IIa Z test of Difference between Means

• Do observations take on two groups of subjects
  differ from one another? Do the two sets
  represent samples from identical or different
  populations?
• known population standard deviations

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Teacher Expectancy Study

• Sample 60 elementary children randomly selected
  and randomly assigned to experimental (bloomers)
  or control group.
• Mean IQ of Bloomers 110.23
• Mean IQ of control group109.57
• Known population s.d.s (sigmas) experimental
  group 10 and control group 10
• Is their a significant difference between
  post-test IQ between Bloomers and Controls?

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Case IIa Design Requirements

• One independent variable with two levels
• Dependent variable is continuous (interval or
  ratio)
• A subject appears in one group only
• Population s.d. is known

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Case IIa Assumptions

• Scores in two groups are independent of one
  another
• Scores in the respective populations are normally
  distributed
• Variances of scores in two populations are equal
  (homogeneity of variance)

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Teacher Expectancy Null and Alternative
Hypotheses

• Assume the null hypothesis is true until proven
  guilty
• State the Null Hypothesis
• H0 ?Bloomers ? Control
• H0 ? Bloomers- ? Control 0
• State the Alternative (Research) Hypothesis
• H1 ? Bloomers gt ? Control (directional,
  one-tailed)
• H1 ? Bloomers- ? Control gt 0 (directional,
  one-tailed)
• Or could have been
• H1 ? Bloomers not ? Control
  (non-directional, two-tailed)
• H1 ? Bloomers- ? Control not 0
  (non-directional, two-tailed)

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Case II Same or Different Population
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Sampling Distribution of Differences between Means

68
95
99
-1se
-2se
1se
2se
-3se
3se
0
Null Hypothesis
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Sampling Distribution of Difference between Means
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Teacher Expectancy Testing Hypothesis (cont)

• Calculate the standard error of the difference of
  means 2.57
• Calculate Zobserved by subtracting the sample
  difference of means from the known
  (hypothesized) population difference of means and
  dividing by the standard error of the difference
  of means
• ((110.23-109.57)- (0)) /2.57
• .66/2.57.26 standard error points
• Evaluate against criteria (set level of
  significance)
• Significance level .05 one-tailed 1.65
  (Zcritical)
• Significance level .01 one-tailed 2.33
  (Zcritical)

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Defining the Critical Area - Teacher Expectancy
Unlikely at .05
Unlikely at .01
1.65se
2.33se
0 population mean difference
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Teacher Expectancy (cont)

• Decision Rule
• If Zobserved falls outside of the chosen
  Zcritical then reject the null hypothesis (H0)
• If observed Z falls inside of the chosen
  Zcritical then do not reject the null hypothesis
  (H0)
• Decision
• Having picked a significance levelof .05 our
  Zobserved of .26 falls inside of the chosen
  Zcritical of 1.65 so we would not reject the null
  hypothesis
• Conclusion

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Making Decision -Teacher Expectancy
Z Distribution Normal Curve
One tailed test
H0 ?E - ?c 0
.26
?diff 0
2se
-2se
1se
-1se
Critical Values
1..65se
2.33se
.01
.05
p
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Case IIb t-Test of Differences between
Independent Means

• Do the two samples belong to an identical
  population or to a different population?
• Population s.d. not known

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Case IIb Design Requirements for Use of t Test

• One independent variable with two levels
• Dependent variable is continuous (interval or
  ratio)
• A subject appears in one group only
• Population s.d.s not known

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Case IIb Assumptions

• Scores in two groups are independent of one
  another
• Scores in the respective populations are normally
  distributed
• Variances of scores in two populations are equal
  (homogeneity of variance)

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Example T-test Two Sample Independent Means
Teacher Expectancy
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T-test for Two Sample (Independent Means) Designs

• Set Null Hypothesis ?e - ? c 0
• Set Alternative Hypothesis ? e - ? c gt 0
  (one-tailed)
• Examine Assumptions of t-test
• Independence and random sampling
• Normality within each population
• Homogeneity of variance (equal variance in two
  pops)
• Decide Significance Level alpha .01
• Compute standard error for difference between
  means 6.44

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T-test for Two Sample (cont)

• Locate tcritical with N-2 df tcritical (.01/1,8)
  2.896
• Compute tobserved
• ((116.4 -105.2)- 0)/6.44
• 11.2/6.44 1.74
• Decision Rule
• Reject H0 if tobserved gt tcritical
• Do Not Reject H0 if tobserved lt tcritical 1.74
  lt 2.896
• Decision???
• Conclude although difference was in expected
  direction, there is insufficient evidence to lead
  us to believe that the observed difference in our
  sample is due to anything other than chance

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Teacher Expectancy E.g.
t Distribution Sample Sizes 5 each df8 Alpha
.01
One tailed test
H0 ?E - ?c 0
1.74
0
2se
-2se
1se
-1se
Critical Values
1.860e
2..89se
.01
.05
P
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Teacher Expectancy 99 CI

• (Xe-Xc) - tcritical(.01/2,8) (sxe-xc)
• lt ue-uc lt
• (Xe-Xc) tcritical(.01/2,8) (sxe-xc)
• 11.2 - 2.306 (6.44) lt ue-uc lt 11.2 2.306
  (6.44)
• 11.2 - 14.85 lt ue-uc lt 11.2 14.85
• -3.65 lt ue-uc lt 26.05
• 95 CI tells us that over all random samples of
  difference between means, the true value of ?e-
  ?c lies within this interval of IQ points with a
  probability of ..95

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Assessing Practical Significance

• Significant findings may not be practically
  significant
• Three ways of assessing practical significance or
  strength of association
• Strength of Association
• Estimated Effect Size
• Statistical Power

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Practical Significance Strength of Association

• After making decision to reject H0
• The degree to which the sample data are found to
  be incompatible with the H0
• Proportion of variance in dependent variable
  accounted for, explained by, independent variable
• Ranges from 0-1.00

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Practical Significance Effect Size (d)

• .20 - weak effect
• .33 - meaningful
• .50 - medium effect
• .80 - strong effect