PSY 307

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PSY 307 Statistics for the Behavioral Sciences

• Chapter 13 Single Sample t-Test
• Chapter 15 -- Dependent Sample t-Test

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Midterm 2 Results
Score Grade N
45-62 A 7
40-44 B 3
34-39 C 4
29-33 D 4
0-28 F 3
The top score on the exam and for the curve was
50 2 people had it.
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Students t-Test

• William Sealy Gossett published under the name
  Student but was a chemist and executive at
  Guiness Brewery until 1935.

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What is the t Distribution?

• The t distribution is the shape of the sampling
  distribution when n lt 30.
• The shape changes slightly depending on the
  number of subjects in the sample.
• The degrees of freedom (df) tell you which t
  distribution should be used to test your
  hypothesis
• df n - 1

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Comparison to Normal Distribution

• Both are symmetrical, unimodal, and bell-shaped.
• When df are infinite, the t distribution is the
  normal distribution.
• When df are greater than 30, the t distribution
  closely approximates it.
• When df are less than 30, higher frequencies
  occur in the tails for t.

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The Shape Varies with the df (k)
Smaller df produce larger tails
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Comparison of t Distribution and Normal
Distribution for df4
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Finding Critical Values of t

• Use the t-table NOT the z-table.
• Calculate the degrees of freedom.
• Select the significance level (e.g., .05, .01).
• Look in the column corresponding to the df and
  the significance level.
• If t is greater than the critical value, then the
  result is significant (reject the null
  hypothesis).

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Link to t-Tables
http//www.statsoft.com/textbook/sttable.html
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Calculating t

• The formula for t is the same as that for z
  except the standard deviation is estimated not
  known.
• Sample standard deviation (s) is calculated using
  (n 1) in the denominator, not n.

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Confidence Intervals for t

• Use the same formula as for z but
• Substitute the t value (from the t-table) in
  place of z.
• Substitute the estimated standard error of the
  mean in place of the calculated standard error of
  the mean.
• Mean (tconf)(sx)
• Get tconf from the t-table by selecting the df
  and confidence level

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Assumptions

• Use t whenever the standard deviation is unknown.
• The t test assumes the underlying population is
  normal.
• The t test will produce valid results with
  non-normal underlying populations when sample
  size gt 10.

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Deciding between t and z

• Use z when the population is normal and s is
  known (e.g., given in the problem).
• Use t when the population is normal but s is
  unknown (use s in place of s).
• If the population is not normal, consider the
  sample size.
• Use either t or z if n gt 30 (see above).
• If n lt 30, not enough is known.

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What are Degrees of Freedom?

• Degrees of freedom (df) are the number of values
  free to vary given some mathematical restriction.
• Example if a set of numbers must add up to a
  specific toal, df are the number of values that
  can vary and still produce that total.
• In calculating s (std dev), one df is used up
  calculating the mean.

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Example

• What number must X be to make the total 20?
• 5 100
• 10 200
• 7 300
• X X
• 20 20

Free to vary
Limited by the constraint that the sum of all the
numbers must be 20
So there are 3 degrees of freedom in this example.
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A More Accurate Estimate of s

• When calculating s for inferential statistics
  (but not descriptive), an adjustment is made.
• One degree of freedom is used up calculating the
  mean in the numerator.
• One degree of freedom must also be subtracted in
  the denominator to accurately describe
  variability.

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Within Subjects Designs

• Two t-tests, depending on design
• t-test for independent groups is for Between
  Subjects designs.
• t-test for paired samples is for Within Subjects
  designs.
• Dependent samples are also called
• Paired samples
• Repeated measures
• Matched samples

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Examples of Paired Samples

• Within subject designs
• Pre-test/post-test
• Matched-pairs

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Independent samples separate groups
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Dependent Samples

• Each observation in one sample is paired
  one-to-one with a single observation in the other
  sample.
• Difference score (D) the difference between
  each pair of scores in the two paired samples.
• Hypotheses
• H0 mD 0 mD 0
• H1 mD ? 0 mD gt 0

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Repeated Measures

• A special kind of matching where the same subject
  is measured more than once.
• This kind of matching reduces variability due to
  individual differences.

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Calculating t for Matched Samples

• Except that D is used in place of X, the formula
  for calculating the t statistic is the same.
• The standard error of the sampling distribution
  of D is used in the formula for t.

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Degrees of Freedom

• Subtracting values for two groups gives a single
  difference score.
• The differences, not the original values, are
  used in the t calculation, so degrees of freedom
  n-1.
• Because observations are paired, the number of
  subjects in each group is the same.

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Confidence Interval for mD

• Substitute mean of D for mean of X.
• Use the tconf value that corresponds to the
  degrees of freedom (n-1) and the desired a level
  (e.g., 95 .05 two tailed).
• Use the standard deviation for the difference
  scores, sD.
• Mean D (tconf)(sD)

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When to Match Samples

• Matching reduces degrees of freedom the df are
  for the pair, not for individual subjects.
• Matching may reduce generality of the conclusion
  by restricting results to the matching criterion.
• Matching is appropriate only when an uncontrolled
  variable has a big impact on results.

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Deciding Which t-Test to Use

• How many samples are there?
• Just one group -- treat as a population.
• One sample plus a population is not two samples.
• If there are two samples, are the observations
  paired?
• Do the same subjects appear in both conditions
  (same people tested twice)?
• Are pairs of subjects matched (twins)?

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Population Correlation Coefficient

• Two correlated variables are similar to a matched
  sample because in both cases, observations are
  paired.
• A population correlation coefficient (r) would
  represent the mean of rs for all possible pairs
  of samples.
• Hypotheses
• H0 r 0
• H1 r ? 0

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t-Test for Rho (r)

• Similar to a ttest for a single group.
• Tests whether the value of r is significantly
  different than what might occur by chance.
• Do the two variables vary together by accident or
  due to an underlying relationship?

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Formula for t
Standard error of prediction
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Calculating t for Correlated Variables

• Except that r is used in place of X, the formula
  for calculating the t statistic is the same.
• The standard error of prediction is used in the
  denominator to calculate the standard deviation.
• Compare against the critical value for t with df
  n 2 (n pairs).

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Importance of Sample Size

• Lower values of r become significant with greater
  sample sizes
• As n increases, the critical value of t
  decreases, so it is easier to obtain a
  significant result.
• Cohens rule of thumb
• .10 weak relationship
• .30 moderate relationship
• .50 strong relationship