Student

1
Students t test

• The case of the single sample

2
Comparing t with Z

• The formulas are nearly identical
• The difference is that t uses the sample standard
  deviation (s) where Z uses the population
  standard deviation (s)
• t is useful when you do not know the population
  standard deviation (s)
• You still need to know the population mean.

3
Comparing formulas

• If Z X - m , then t X - m
• s s
• __
• If Z X - m , then t X - m
• s/ N s/ N

4
When do we know m but not s ?

• When a population is incompletely described
• When the sample history includes only means
• When goals are stated without acceptable
  variation
• When design parameters do not specify tolerances

5
The t distribution

• A family of distributions
• Shaped like the normal curve, but flatter
• Approaches the normal curve as N increases
• Selected by the degrees of freedom
• The t table of critical values

6
Computing t from raw scores

• Recall that with raw scores, the easiest way to
  compute the sample standard deviation is via the
  sum of squares.
• Since SS SX2 - (SX)2 / N,
• and s ( SS / (N - 1),
• and sx s / (N),
• then t (X - m)
• SS/(N(N-1))

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7
Confidence intervals

• Confidence intervals may simplify decisions about
  the null hypothesis
• For Z, the 95 confidence limits are found by
  taking the population mean m and adding and
  subtracting from it the critical value from the
  table multiplied by the standard error of the
  mean m /- 1.96 (sx)

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8
Confidence intervals for t

• Recall that a confidence interval is the range of
  values that we believe to be part of the null
  hypothesis population.
• In other words, a confidence interval contains
  the values that would lead us to retain the null
  hypothesis.
• The confidence limits are the border values of a
  confidence interval.

9
Finding confidence intervals for t

• To find the confidence interval for Z, we first
  found the critical (table) value of Z and
  multiplied it by sX
• Then we added and subtracted that from the
  population mean m.
• Follow the same procedure for t, but use the
  critical value of t.
• Control df, a, and tailedness (control dfat).

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10
Factors affecting power

• Power is the likelihood of rejecting a false null
  hypothesis, and thus supporting a true alternate
  hypothesis.
• Power is affected by a and by sample size.
• Lower a levels decrease power.
• Larger samples increase power.
• Power is also affected by the standard deviation
  (variability) of the scores.
• Finally, the greater the effect size, the greater
  the power.

11
t-test for Pearson r

• Treating r as a sample mean, you can calculate a
  corresponding t score, and test it for
  significance.
• However, the dirty work has already been done, so
  you can use a computer program or table A-5
  (Pearson) or A-6 (Spearman).
• If robt gt rcrit, then reject H0