Transcript April 23

BHS 204-01
Methods in Behavioral
Sciences I
April 23, 2003
Chapter 5 (Ray)
Experimental Decision-Making
Inferential Statistics
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We use information about groups (samples) to
make inferences about populations.
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Population – all possible cases.
Sample – a subset of cases drawn as randomly as
possible from the larger population.
Probability tells us whether the sample is
representative of the larger population.
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If the experiment were repeated would we get the
same result?
Examples With Coins
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Observed coin tosses are a sample of all
possible coin tosses.
Probability rules:
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P(A & B) = P(A) x P(B)
P(A or B) = P(A) + P(B)
Figure it out from the rules – or figure it out
by listing all possible events and seeing how
many produce the expected outcome.
Probabilities can be Observed
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Probabilities depend on the frequencies of
events possible within a population:
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Probability of selecting a male subject from a
group is % of males within that group.
Probability of observing H on a coin toss is the
frequency of H among the coins sides (H & T).
Probabilities can be combined:
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Probability of drawing someone male or under 20
is 20/40 + 15/40 – (overlap) .375 = .825
Normal Distribution
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The normal distribution consists of the
frequencies likely to be observed with
repeated sampling from a population.
It is useful because we can use it to know
what percentage of cases are likely to fall
within different portions of the curve.
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Central Limit Theorem – most of variability will
fall within 2 standard deviations of the mean.
Figure 5.3. (p. 116)
Probability distribution for combinations resulting from tossing
10 coins.
Figure 5.5. (p. 117)
Normal distribution showing standard deviations.
Standard Error of the Mean
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With a normal distribution, the means of most
sampled groups are likely to fall within a
small range of the observed sample mean.
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Standard error of the mean – the standard
deviation of all means possible to be observed
with repeated sampling.
Divide standard deviation by square root of the
number of scores (cases).
Standard error defines a confidence interval.
Figure 5.7. (p. 119)
Hypothetical distribution of means derived from giving a standardized test to
either a large or a small number of students a large number of times.
Hypothesis Testing
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We wish to know where the observed mean of
a sample falls within the distribution of means
from all possible samples.
With two scores, we want to know whether
the difference between them is due to
sampling variation or the manipulation.
T-test – a widely used statistic for testing
differences between means of two groups.
Population vs Sample Statistics
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Population statistics:
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Mean
Variance
Standard deviation
m
s2
s
Sample statistics:
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Mean
Variance
Standard deviation
M or X
S2
S or SD
Adjusting for the Population
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When inferential statistics are used, an
adjustment is made to allow the sample mean
to more closely approximate the population
mean.
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Population variance is calculated by dividing by
N-1, not N.
Other statistics show this same adjustment – t-test
uses N-1 not N in denominator.
Degrees of Freedom
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Degrees of freedom (df) – how many numbers
can vary and still produce the observed result.
Population statistics include the degrees of
freedom.
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Calculated differently depending upon the
experimental design – based on the number of
groups.
T-Test df = (Ngroup1 -1) + (Ngroup2 -1)
Reporting T-Test Results
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Include a sentence that gives the direction of
the result, the means, and the t-test results:
Example:
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The experimental group showed significantly
greater weight gain (M = 55) compared to the
control group (M = 21), t(12) = 3.97, p=.0019,
two-tailed.
Give the exact probability of the t value.
Underline all statistics.
When to Use a T-Test
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When two independent groups are compared.
When sample sizes are small (N< 30).
When the actual population distribution is
unknown (not known to be normal).
When the variances within the two groups are
unequal.
When sample sizes are unequal.
Using Error Bars in Graphs
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Error bars show the standard error of the
mean for the observed results.
To visually assess statistical significance, see
whether:
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The mean (center point of error bar) for one
group falls outside the error bars for the other
group.
Also compare how large the error bars are for the
two groups.
Figure 5.8. (p. 124)
Graphic illustration of cereal experiment.