Hypothesis Tests Applied to Means: Two Related Samples

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Hypothesis Tests Applied to Means Two Related
Samples

• Chapter 13

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Testing Means

• In chapter 12, we tested if the mean of the one
  sample we drew was different from a population
  mean
• In this chapter, we will test for differences
  between 2 sample means
• Specifically, we will test the sample means from
  samples that are related in some way

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2 Related Samples

• Related samples (repeated measures, matched
  samples)
• An experimental design in which the same
  individual is observed under more than 1
  treatment (i.e. we collect data from an
  individual more than once)
• Ex we could record an individuals anxiety
  levels both before and after an exam. We would
  expect the 2 scores to be related. A highly
  anxious individual will be so both before and
  after the exam, but we will also expect the
  scores to be different from time 1 and 2
• When we collect data from an individual more than
  once, we are usually interested in changes in
  that variable

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2 Related Samples

• This is also the situation when the data from one
  individual is related to the data from another
  individual
• Ex We would expect the marital satisfaction of
  husbands and wives to be related
• If we know something about one member of a pair
  of scores, we also know something about the other
  member of the pair (even if it is something small)

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Example 1 Same Person

• We measure anxiety on a 10 point scale where 1 is
  not at all and 10 is very. We do this both
  BEFORE and AFTER an exam. Here are the scores
• Before After
• Person 1 9 7
• Person 2 5 2
• Person 3 3 1
• Person 4 7 5
• Person 5 8
  7

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Example 2 Pairs of People

• We measure marital satisfaction on a 10 point
  scale where 1 is not at all and 10 is very.
  We do this with both HUSBANDS and WIVES. Here
  are the scores
• HUSBANDS WIVES
• Pair 1 5 3
• Pair 2 6 7
• Pair 3 1 2
• Pair 4 2 4
• Pair 5 8 10

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Using Example 1

• Person 1 2 3 4 5
• Before 9 1 3 7 8 x15.6
• After 7 2 1 5 7 x24.4
• Diff. 2 -1 2 2 1
• Difference scores are the set of scores
  representing the difference between the
  individuals performance on two occasions
• D x1 - x2

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Difference Scores

• Once we have the D score for each person, we can
  get the mean D score (D)
• We sum up the D scores and divide by n
• We can also compute the variance and SD of the D
  scores
• For the variance (sD2)
• ?(D-D)2 / n
• For the SD (sD)
• Sqrt (sD2 )

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Hypothesis testing

• When testing hypotheses using related samples
• The null hypothesis will be that there is no
  difference between the scores or the population
  of difference scores (?D) is 0
• H0 ?D ?Before - ?After 0
• The alternative hypothesis will be that there are
  differences between the 2 sets of scores or that
  on set is larger/smaller than the other
• H1 ?Before ? ?After or ?D ? 0

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The t statistic

• t (D - 0) / sD
• Where sD sD / sqrt(n)
• This is the standard error of the difference
• n the number of difference scores
• This is the same computation as the one sample
  case, we are just substituting D for X
• We use the t-tables in the same way as for the
  one sample tests
• The interpretation is that either there is or is
  not a difference between the 2 sets of scores

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Example

• In a study on attraction, people were asked to
  rate pictures of people taken before and after
  they had braces. Ratings of attractiveness are
  given in the table. The researcher wants to know
  if people differ in attractiveness after they
  have had braces.
• Picture Pre-Rating
  Post-rating
• 1 113
  115
• 2 105
  117
• 3 120
  125
• 4 119
  117
• 5 104
  107
• 6 100
  105
• 7 111
  110

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Advantages of Repeated Measures

• Avoid problem of individual differences (i.e.
  variability from subject to subject) thus, giving
  us more power by keeping sample variance small
• Control for extraneous variables ex ask 5 people
  anxiety before exam (10, 10, 10, 10, 10) and
  another 5 people after (1, 1, 1, 1, 1) how do we
  know that first 5 arent always anxious and
  second always calm?
• Requires fewer people because we have multiple
  observations per individual

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Disadvantages of Repeated Measures

• Order effect
• The effect of performance attributable to the
  order in which treatments were administered
• Ex. I have you run a mile, I measure your heart
  rate, I have you study for 1 minute and then I
  measure your heart rate. If I did this in reverse
  order it would have a major impact on the data
  collected
• Carry-over effect
• The effect of previous trials (conditions) on a
  subjects performance on subsequent trials
• Ex I have you take an exam, 1 week later I give
  you exam again. It is likely that seeing the
  questions before will influence your performance