Hypothesis Test: Comparing Two Groups

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Hypothesis TestComparing Two Groups
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Review One-sample Test

• 1. Assumption
• 2. State H0 and H1
• 3. Choose a a-level
• typically .05, sometimes .10 or .01
• 4. Look up value of test statistic corresponding
  to the a-level (called the critical value)
• 5. Calculate the relevant test statistic
• 6. Compare test statistic to critical value
• If test statistic is larger, we reject H0, and
  accept H1
• If it is smaller, we fail to reject H0, and
  cannot accept H1

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Small Sample
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Todays TopicHypothesis Test (two-sample)

• Elements of two-sample test
• Transforming two-sample into one-sample
• Extension of C.L.T.
• Comparing means of two groups
• Comparing means of two paired groups
• Comparing proportions of two groups
• Hypothesis test errors
• How to conduct hypothesis test in SPSS

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Two-sample Test

• Hypothesis tests on the difference between two
  different groups
• Examples
• Gender difference in wage/academic performance
• Racial difference in homeownership/ wage/
  education
• Is drug A better than drug B in reducing the
  likelihood of heart disease?
• Students who did well in high school perform well
  in college (time series)

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Two-sample Test Difference in Means

• Question Is the difference due to the particular
  sample? Can we infer that the population means
  are different?
• Example test scores for 20 boys, 20 girls
• Y-barboys 72.75, s 8.80
• Y-bargirls 78.20, s 9.5

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Example Test Scores
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Example Hypothesis

• Issue
• How likely is it to have sample means with a
  difference of -5.45, if population means are the
  same (difference0)?
• Hypothesis

OR
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Strategy for Mean Difference

• The C.L.T. defines how sample means (Y-bars)
  cluster around the true mean
• The center and width of the sampling distribution
• This tells us the range of values where Y-bars
  fall
• For any two means, the difference will also fall
  in a certain range
• Group 1 means range from 6.0 to 8.0
• Group 2 means range from 1.0 to 2.0
• Difference in means will range from 4.0 to 7.0

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Strategy for Mean Difference

• Visually If each population has a sampling
  distribution, the difference does too

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The Central Limit Theorem

• As n grows large, the sampling distribution of
  the sample mean (Y-bar) approaches normality

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A Corollary of the C.L.T

• For any two random samples (of size n1, n2), with
  m1, m2 and s1, s2 for two sampling distributions,
  the sampling distribution for the difference of
  two means is normal, with a mean and S.D.
• The mean is the difference of the means of two
  sampling distributions
• The variance is equal to the sum of variances of
  two sampling distributions

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Types of Two-sample Test

• Samples
• Independent random samples
• Dependent samples (time-series, husband-wife
  pair)
• Data type
• Quantitative comparing two means
• Qualitative comparing two proportions
• Sample size
• Large sample, use z test, calculate z-score
• Small sample, use t test, calculate t-score
• Hypothesis one-tail vs. two-tail test
• Different combinations

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Two-sample Test Steps

• 1. Assumption
• 2. State H0 and H1
• 3. Calculate the relevant test statistic
• 4. Find out corresponding P-value
• 5. Choose a a-level
• typically .05, sometimes .10 or .01
• 6. Compare P-value with a-level
• If P is smaller, significant, we reject H0
• If P is larger, not significant, we fail to
  reject H0

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Two-sample Test Difference in Means

• Two-tailed test
• H0 m1m2
• H1 m1?m2
• E.g. boys mean test score is different from
  girls.
• One-tailed test
• H0 m1ltm2
• H1 m1gtm2
• E.g. boys mean test score is higher than girls
• H0 m1gtm2
• H1 m1ltm2
• E.g. boys mean test score is lower than girls

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Two-sample Test Difference in Means

• Two independent samples
• n1, Y-bar1, s1 n2, Y-bar2,s2 assume large n1,
  n2 (gt20)
• Hypotheses
• H0 ?1?2 (or ?1-?20)
• H1 ?1??2 (or ?1-?2?0)
• Z-test, calculate z-score

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Z-Values for Mean Differences

• Visually,
• Q In which case can we reject H0?
• When Z is large, it is highly unlikely that the
  difference in populations means is zero

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Example Elements

• Test score differences for boys and girls
• n130, Y-barboys 72.75, sboys 8.80
• n230, Y-bargirls 78.20, sgirs 9.55
• Assume sample sizes are large, use z-test
• Choose a.05, two-tailed test
• Critical Z 1.96
• Hypotheses

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Example Z-value

• Calculate Z-value
• P20.0107 0.0214lt 0.05
• (OR Observed Z 2.30, critical Z 1.96)
• Q can we reject H0?
• YES! We are 95 confident that H0 can be
  rejected, and we can accept that the two groups
  are different.

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Small Sample Mean Difference

• But we often have small samples medical test,
  spatial units
• Use T-test
• The computation of df value is very complex
• Assume the same variability for the two groups
  (homoscedasticity) ? a simpler df expression
• The standard error can be derived from the
  variances of both groups (i.e. pooled)

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Example T-test

• Example setup
• Boys n115, Y-bar 72.75, s1 8.80
• Girls n215, Y-bar 78.20, s2 9.55
• Hypothesis
• a-level 0.1
• Calculate t-score

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Example Two-tailed Test

• Calculate the standard error of mean difference
• The critical value for a0.1, two-tailed
  t(28)1.701
• Observed t-value 1.63 lt Critical Value
• Can we reject the null hypothesis (H0)?
• No! We fail to reject H0, and we cannot accept
  H1. We are not sure boys are different from
  girls.

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Example One-tailed Test

• One-tailed t test Girls mean test score is
  higher than boys
• Critical value for a0.1, one-tailed t-test,
  df28
• One-tailed test t(28)1.313
• Observed t -1.63
• observed t gt Critical Value
• We can reject the null hypothesis! We accept H1
  that girls perform better than boys
• Moral of the story
• If you have strong directional suspicions ahead
  of time, use a one-tailed test. It increases
  your chance of rejecting H0.

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Dependent Two-sample Test

• Each observation in sample 1 has a matching
  observation in sample 2 matched-pairs
• Same sample, repeated measurement (time series)
• Two samples, but paired observations
  (husband-wife father-son)
• Same sample sizes n1 n2
• Difference between the means of the two samples
  equals the mean of the difference in scores
• If DiV1i-V2i, then ?D?1-?2

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Dependent Two-sample Test

• So, we can base analyses about ?1-?2 on inference
  about ?D , thus, reduces a two-sample problem to
  a one-sample problem
• Previous hypothesis H0 ?1?2 H1 ?1??2
• Now H0 ?D0 H1 ?D
  ?0

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Dependent Two-sample Test

• Calculate test statistic (t-score, z-score)

When n1gt30, z-score can substitute for the t-score
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Example

• Critical value for two-tailed t-test at 0.05
  level, with df2?
• Observed t-score 6.9 gt critical value 4.303
• Reject H0 two therapies are different.
• Benefits of using dependent samples
• Many sources of potential bias are controlled
• Same sample same gender, personality
• Couples same lifestyle

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Review Two-sample Tests of Means

• Hypothesis is based on theory
• Often conduct a two-tailed test. After accepting
  the H1 that two groups are different, develop
  one-tailed H1 and conduct one-tailed test
• Same calculation for one-tailed and two-tailed
  tests difference is at the stage of finding out
  the critical value in Z or t-table based on
  predetermined significance level

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Review Two-sample Tests of Means

• Two-tailed test
• H0 m1m2 H1 m1?m2
• E.g. boys mean score is different from girls.
• calculated t gt critical value, reject H0
• One-tailed test
• H0 m1ltm2 H1 m1gtm2
• E.g. boys mean test score is higher than girls
• calculated t gt0 and t gt C.V., reject H0
• H0 m1gtm2 H1 m1ltm2
• E.g. boys mean test score is lower than girls
• Calculated t lt 0 and t gt C.V., reject H0

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Two-sample Test Difference in Proportions

• Qualitative data
• Two samples n1, ?-hat1 n2, ?-hat2
• Similar steps as two-sample test for means
• Hypotheses
• H0 ?1?2 H1 ?1? ?2 (two-tail)
• H0 ?1gt?2 H1 ?1lt ?2 OR
• H0 ?1lt?2 , H1 ?1gt?2 (one-tail)
• Choose the a-level get the critical value
• Calculate test statistic (z-score or t-score)
• Compare test statistic with the critical value
• Conclusion

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Two-sample Test Difference in Proportions

• Sampling distribution of the difference in
  proportion
• Z-score
• Although we can use the sum of standard errors of
  two sampling distributions, its preferable to
  use the following pooled estimate
• p-hat is the proportion of the total sample (two
  samples combined)

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Example

• Do you agree or disagree that women should take
  care of running their homes and leave running the
  country up to men?
• n1345, 122 agree (0.35), 223 disagree n21900,
  268 agree (.14), 1632 disagree
• ?-hat10.35, ?-hat20.14
• ?-hat(122268)/(3451900)0.174

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Example

• H0 ?1?2 H1 ?1? ?2
• Choose the a-level0.05 critical value1.96
• Calculate the z-score z-9.6
• Significant at 0.05 level reject H0, and accept
  H1, which means two samples are from two
  different populations.

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Example

• H0 ?1lt?2 H1 ?1 gt ?2
• Choose the a-level0.05 critical value1.65
• Calculate the z-score z-9.6 (same calculation
  as two-tailed test)
• Significant at 0.05 level reject H0, and accept
  H1, which means the second population is less
  likely to agree with the statement.

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Hypothesis Test Errors

• Due to the probabilistic nature of tests, there
  will be errors.
• Sometimes the null hypothesis is true, but we
  will reject it
• Our alpha-level determines the probability of
  this
• Sometimes we do not reject the null hypothesis,
  even though it is false

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Hypothesis Test Errors

• When we falsely reject H0 ?Type I error
• Alpha level of the test
• When we falsely fail to reject H0, ?Type II error
• Inversely related to alpha level

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• Cannot minimize both types of error.
• In general, we are most concerned about Type I
  error try to be conservative (smaller alpha).
• Use a large sample to reduce both types of error

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Summary

• Elements of two-sample test
• Corollary of C.L.T
• General steps
• Independent two-sample test of means
• Dependent two-sample test of means
• Two-sample test of proportions
• The issue of errors
• How to do these tests in SPSS

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One-sample Test of Mean in SPSS
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Independent Two-sample Test
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• Levenes Test for Equality of Variance (F-test)
• H0 ?1 ?2
• H1 ?1? ?2
• 0.795 gt 0.05, fail to reject H0, equality of
  variance assumed, use the first row
• If Sig. lt 0.05, reject H0, equality of variance
  not assumed, use the second row

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Paired Two-sample Test in SPSS
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