The OneSample Ttest

1
The One-Sample T-test

• Testing hypotheses without knowing the population
  standard deviation

2
The Basics

• Z-test formula
• (Sample mean null mean)/(pop.stand.dev./sqr.
  Root N)
• What if we dont know pop.stand.dev.?
• We have to use our sample standard deviation to
  estimate the pop.stand.dev.

3
Estimating

• Pop standard deviation sqr.root(SS/n)
• Estimate of pop.stand.dev sqr.root(SS/n-1)
• N-1 degrees of freedom

4
Degrees of Freedom

• The number of independent pieces of information a
  sample of observations can provide for purpose of
  statistical inference.
• If the mean of a sample is 10 and has 3 numbers
  in the sample, how many are independent (free to
  vary)?
• First number is 110, second number is 85, what
  is the third number?
• 110-85530/103
• If we know the mean, only n-1 numbers can vary
  freely.

5
Degrees of Freedom, cont.

• When estimating standard deviations, the numbers
  are bound by the fact that the sum of the
  deviations from the mean must 0
• Thus, only n-1 numbers are free to vary.
• Degrees of freedom n-1

6
How do we use degrees of freedom?

• Estimating standard deviation
• Finding critical values for our hypothesis test.
• Finding the t-values that cut off an appropriate
  5 of the distribution.
• The critical values define the critical region
• If our OBSERVED t falls in the critical region,
  the probability associated with OBSERVED t must
  be lt .05
• Practice table on page 357-359
• Critical values for two-tailed .05, n 20
• One-tailed .01, n 10
• One-tailed .05, n 15

7
Students t-distribution

• Similar to z (normal) and identical when n is
  large.
• When n is small, t is heavier in the tails,
  narrower in the shoulders.
• Is slightly different for each degree of freedom

8
Doing a t-test

• State null and alternative
• Is alternative directional or not?
• Decide on alpha, identify critical values
• t (sample mean null mean)/estimate of
  pop.stand.dev/sqr.root(n)
• Cant compute the exact probability of observed
  t, so, we ask the question Does t fall in
  critical region(s)?

9
Confidence intervals

• The same as previously, except
• Use standard error derived from estimate of
  pop.stand.dev.
• Use critical value identified in t-distribution
  using degrees of freedom.
• Formula sample mean - tcrit(stand.error)