Chapter 4 Analysis of Variance

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Chapter 4 Analysis of Variance

• 4.3 Tests for Two Means

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Hypotheses About Mean Differences
Many research hypotheses compare two population
means

• Women live longer than men
• Republicans are more conservative than Democrats
• Happiness differs between old and young

Restate symbolically as population mean
differences
One-tailed
Two-tailed
H0 ?1 lt ?2 H1 ?1 gt ?2
H0 ?1 ?2 H1 ?1 ? ?2
Rearrange to show how the parameters differ from
zero
H0 ?1 - ?2 lt 0 H1 ?1 - ?2 gt 0
H0 ?1 - ?2 0 H1 ?1 - ?2 ? 0
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Apply the Central Limit Theorem
If large independent samples are drawn randomly
from two populations, then the sampling
distribution of their mean difference is also
normally distributed
In the sampling distribution of mean differences
Mean
Standard error
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Dual sampling distributions
?1 - ?2 8 6 2
_
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Combined sampling distribution
Standard error of the mean difference is wider
than the standard errors of the separate sampling
distributions
_
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Null and Research Hypotheses
Translate these English statements into symbolic
form null and research hypotheses. Determine
whether a one-tailed or two-tailed test is
required.
Older and younger people differ in church
attendance
?O ?Y ?O ? ?Y
H0 _________ H1 _________
Women express more intense religiosity than men
?W lt ?M ?W gt ?M
H0 _________ H1 _________
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Steps in Hypothesis Testing
12. State hypothesis pairs in English symbolic
forms rearrange to show numerical difference in
parameters
H0 ?1 ?2 H1 ?1 ? ?2
?
H0 ?1 - ?2 0 H1 ?1 - ?2 ? 0
3. Choose ?-level (Type I / false rejection error)
4. In the Z score table, find the critical
value(s) necessary to reject H0 at your chosen
?-level
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Steps (continued)
5. Estimate the standard error using sample
statistics
?
6. Calculate the t-test value, and compare it to
the critical value(s) then decide whether to
reject H0
Consistent with the null hypothesis, right-hand
side of the numerator equals zero (see rearranged
H0)
7. If you reject H0, what is probability of
making a Type I error (false rejection error)?
__?___
8. State your substantive conclusion.
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Test this research hypothesis with 2008 GSS data
Older and younger people differ in church
attendance
H0 ?O - ?Y 0 H1 ?O - ?Y ? 0
6.83
___________________________________________
Reject H0
Decision about null hypothesis
_____________________
p lt .001
Probability of Type I error _____________________
____
Older and younger differ in church attendance.
Conclusion ______________________________________
___
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Test this research hypothesis with 2008 GSS data
Women men differ in newspaper reading (times
per year)
H0 ?W ?M H1 ?W ? ?M
-0.53
___________________________________________
Do not reject H0
Decision about null hypothesis
_________________________
-----
Probability of Type I error _____________________
________
Men women dont differ in newspaper reading.
Conclusion ______________________________________
__
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Test this research hypothesis with 2008 GSS data
In the past 5 years, men had more sex partners
than women
H0 ?M lt ?W H1 ?M gt ?W
5.6
__________________________________________
Reject H0
Decision about null hypothesis
___________________
p lt .001
Probability of Type I error _____________________
__
Men had more sex partners than women.
Conclusion ___________________________________
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Difference in Two Proportions
For dichotomous dependent variables, form
hypotheses about population differences in two
proportions
Recall that a proportion is the relative
frequency of one outcome to the total number of
outcomes. The two proportions of a dichotomy sum
to unity
and p0 p1 1.00
(In BK, p. 124 p1 p and p0 q, thus p q
1.00)
Estimated the standard error for one populations
sampling distribution
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Hypotheses About Proportions

• Liberals favor legalizing pot more than
  conservatives

One-tailed
Rearranged
H0 ?L lt ?C H1 ?L gt ?C
H0 ?L - ?C lt 0 H1 ?L - ?C gt 0

• Blacks and whites differ in support for death
  penalty

Two-tailed
Rearranged
H0 ?B ?W H1 ?B ? ?W
H0 ?B - ?W 0 H1 ?B - ?W ? 0
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Test this research hypothesis with 2008 GSS data
Liberals visit art museums more than conservatives
H0 ?L - ?C lt 0 H1 ?L - ?C gt 0
9.30
_______________________________________________
Reject H0
Decision about null hypothesis
_______________________
p lt .001
Probability of Type I error _____________________
______
Liberals visit museums more than conservatives.
Conclusion ______________________________________
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Test this research hypothesis with 2008 GSS data
Protestants and Catholics differ on abortion for
any reason
H0 ?P - ?C 0 H1 ?P - ?C ? 0
1.67
______________________________________
Dont reject H0
Decision about null hypothesis
______________________
Probability of Type I error _____________________
_____
Conclusion ______________________________________
No religious differences on abortion for any
reason.
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Hypotheses About Paired Means

• Sometimes researchers want to compare the means
  of
• two matched samples, such as husbands and wives
• the same persons responses to one variable at
  two times (e.g., before and after some
  experience)
• two variables measured on identical scales for
  each person

• Who does more housework, husbands or wives?
• Do you feel about statistics today as you did
  last month?
• Which tastes better Coke or Pepsi?
• Do Americans like Japan or China more?

We cant apply the two-sample t-test. Although
the sample size is 2N cases (paired scores from N
cases), the members of each pair were not
selected independently. Instead, calculate t with
the difference in paired scores.
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Hypotheses about paired means ask whether the
difference is zero in the two populations ?D
?1 - ?2
H0 ?D 0 H1 ?D ? 0
In a sample, the difference for a pair of scores
is
Calculate the sample standard deviation of the
differences
Then estimate the standard error
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Test this research hypothesis with 2008 GSS data
Americans differ in their level of confidence in
business and confidence in Congress, each
measured on a 5-point scale.
H0 ?D 0 H1 ?D ? 0
Business Congress
Mean 2.91 2.57
Sample N 1,333 1,333
0.94 0.94
13.1
_____________________________________________
Reject H0
Decision about null hypothesis
______________________
p lt .001
Probability of Type I error _____________________
_____
Americans were more confident in business
Conclusion ______________________________________