Chapter 9 Hypothesis Testing Using the Two Sample t-test

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Chapter 9Hypothesis Testing Using the Two Sample
t-test
Related Samples
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Two-sample t-testIndependent vs. Related Samples

• If we have two samples and we want to know
  whether the samples come from a single or two
  different populations, we already learned that we
  would perform a t-test.
• If the samples are independent, that is the
  people in each sample are totally different
  people, we would perform the independent-samples
  t-test, the one we already learned.
• However, if the people that are in the two
  different sample are the same people, then a
  special form of the t-test is used.

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Related SamplesExamples

• If I wanted to know whether a new drug works to
  increase attention I could take a measure of
  reaction time without the drug, then a second
  measure of reaction time with the drug testing
  the exact same people.
• However, potential problems could arise. If the
  reaction times get better with the drug, it could
  be because the drug works, or because the
  subjects have practice with the reaction time
  task.
• Well discuss some of the other problems and
  solutions to this later.

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Related SamplesExamples

• Related samples can also occur with matched
  samples.
• Matched samples means that the researcher matched
  the subjects that are put into the two groups.
• In this example, perhaps the researcher decided
  that age was an important determinant of reaction
  time.
• If the researcher created two groups, one that
  received the drug and one that did not and
  matched the samples on age, then we would have a
  repeated measures design.
• Matching the samples means that if a 21 year-old
  was placed into the drug treatment, a 21 year-old
  would also be placed into the no drug control.

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Related SamplesAdvantages/Uses

• Repeated samples tests are performed in some
  situations where an independent measures test is
  impractical.
• If the population from which the subjects are
  drawn is rare or the researcher has access to
  only a limited sampling pool, then a repeated
  measures test is more economical in the sense
  that the researcher has to find fewer rare
  subjects.

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Related SamplesAdvantages/Uses

• Some research questions are simply better asked
  with a repeated measures design, such as studies
  looking at response change over time, or
  developmental studies.

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Related SamplesAdvantages/Uses

• A repeated measures design would also be used if
  greater statistical control is desired.
• Variability due to individual differences is a
  major source of bad variability for hypothesis
  testing in that this variance goes straight into
  the standard error.
• If the same subjects are measured repeatedly,
  then a large amount of this variance due to
  individual differences is reduced or eliminated
  because the same subjects are contributing two
  scores.
• Repeated measures tests are more powerful than
  independent measures designs.

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Related SamplesDisadvantages/Problems

• Carry-over Effects Occurs when the subjects
  responses in the second treatment they are
  exposed to are in some way affected by their
  exposure to the first condition.
• Progressive Error Occurs when a subjects
  performance changes over time, such as with
  practice.

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Related SamplesDisadvantages/Problems

• Both of these problems can be dealt with using
  counterbalancing the order of the presentation of
  the treatments. That is, half of the subjects are
  exposed to treatment 1 followed by treatment 2,
  the other half of the subjects are exposed to
  treatment 2 followed by treatment 1.
• Counterbalancing creates its own statistical and
  control issues which you will be happy to know
  that you will learn about in great depth in your
  research methods course.

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Related Samples t-testAssumptions

• The observations within each treatment condition
  must be independent.
• The population of difference scores must be
  normal.

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Related Samples t-test

• In order to save time in the next class period
  for us to review, I will do the next homework
  assignment now and use this as our opportunity to
  learn how to do the related samples t-test.
• What we need to understand is that the
  related-samples t-test is performed by conducting
  a one-sample t-test on the difference scores.

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Hypothesis Testing for Two Sample Means The
t-test

• 9-4 9-5

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9-4

• A researcher investigate whether children exhibit
  a higher number of aggressive acts after watching
  a violent television show. The number of
  aggressive acts for the same 10 participants
  before and after watching the show are as
  follows
• (a) Subtracting before-scores from after-scores,
  what are H0 and Ha? (b) Compute tobt. (c) With a
  .05, what is tcrit? (d) What should the
  researcher conclude about this relationship? (e)
  Compute the appropriate confidence interval. (f)
  If you want to understand childrens aggression,
  how important is it to consider whether they
  watch violent television shows?

After Before
5 4
6 6
4 3
4 2
7 4
3 1
2 0
1 0
4 5
3 2
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Difference Scores
After Before D
5 4 5-41
6 6 6-60
4 3 4-31
4 2 4-22
7 4 7-43
3 1 3-12
2 0 2-02
1 0 1-01
4 5 4-5-1
3 2 3-21
Difference scores can be calculated by
subtracting before-after or after-before. The
same answer will be obtained (opposite sign
though). I personally choose the order which
creates the fewest negative numbers. When we
interpret the results we need to be careful to
remember the order we used.
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9-4a
After Before D
5 4 5-41
6 6 6-60
4 3 4-31
4 2 4-22
7 4 7-43
3 1 3-12
2 0 2-02
1 0 1-01
4 5 4-5-1
3 2 3-21

• (a) Subtracting before-scores from after-scores,
  what are H0 and Ha?
• H0
• Ha

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9-4b
After Before D
5 4 5-41
6 6 6-60
4 3 4-31
4 2 4-22
7 4 7-43
3 1 3-12
2 0 2-02
1 0 1-01
4 5 4-5-1
3 2 3-21

• (b) Compute tobt.
• SD 10123221-1112
• SD2 12 02 12 22 32 22 22 12 -12
  12 26
• N 10

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9-4b

• (b) Compute tobt.

After Before D
5 4 5-41
6 6 6-60
4 3 4-31
4 2 4-22
7 4 7-43
3 1 3-12
2 0 2-02
1 0 1-01
4 5 4-5-1
3 2 3-21
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9-4c

• (c) With a .05, what is tcrit?
• nD 10
• df n 1 9
• Researcher predicts higher aggressive acts after
  watching violence, therefore, this is a
  one-tailed test.
• tcrit(9)a.05 1.833

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9-4d

• (d) What should the researcher conclude about
  this relationship?
• Since the tobt is in the tail created by tcrit,
  we reject H0 and conclude the results are
  significant. In the population, children exhibit
  more aggressive acts after watching the show
  (with m about 3.9) than they do before the show
  (with m about 2.7).

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9-4e

• (e) Compute the appropriate confidence interval.

I am 95 confident that the true difference
between the population means after and before
watching violent television is between .39 and
2.01.
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9-4f

• (f) If you want to understand childrens
  aggression, how important is it to consider
  whether they watch violent television shows?
• Violence on television is an important variable
  to consider here. 55 of the variance in violence
  is predictable knowing television watching.

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9-5

• You investigate whether the older or younger male
  in pairs of brothers tend to be more extroverted.
  You obtain the following extroversion scores
• (a) What are H0 and Ha? (b) Compute tobt. (c)
  With a .05, what is tcrit? (d) What should you
  conclude about this relationship? (e) Is this a
  scientifically informative relationship?

Younger Older
10 18
11 17
18 19
12 16
15 15
13 19
19 13
15 20
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9-5a

• (a) What are H0 and Ha?
• H0
• Ha

Younger Older
10 18
11 17
18 19
12 16
15 15
13 19
19 13
15 20
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9-5b

• (b) Compute tobt.
• SD 8 6 1 4 0 6 (-6) 5 24
• SD2 82 62 12 42 02 62 (-6)2 52
  214
• N 8

Younger Older D
10 18 8
11 17 6
18 19 1
12 16 4
15 15 0
13 19 6
19 13 -6
15 20 5
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9-5c

• (c) With a .05, what is tcrit?
• nD 8
• df n 1 7
• The researcher did not predict whether older or
  younger siblings would be more extraverted,
  therefore, this is a two-tailed test.
• tcrit(7)a.05 2.365

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9-5d

• (d) What should you conclude about this
  relationship?
• Since the tobt does not fall in the regions of
  rejection created by tcrit, we conclude the
  results are not significant, we have no evidence
  of a relationship. In the population, neither
  older nor younger brothers are significantly more
  extraverted than the other.

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9-5e

• (e) Is this a scientifically informative
  relationship?
• We cannot determine effect size with a
  nonsignificant result because we do not know if a
  relationship exists or not.