Sampling distributions

1
Sampling distributions

• The sampling distribution of the mean
• The Central Limit Theorem
• The Normal Deviate Test (Z for samples)

2
Sampling distributions

• The distribution of a statistic (eg. mean,
  median, standard deviation) for the set of all
  possible samples from a population.
• For example, if we toss an unbiased coin
  repeatedly in sets of three tosses, scoring heads
  as 1 and tails as 0, the possible samples are as
  follows

3
An example

• Sample Mean
• HHH 1.00
• HHT .67
• HTH .67
• THH .67
• TTH .33
• THT .33
• HTT .33
• TTT .00

Sampling Distribution of the mean
Mean f 1.00 1 .67 3 .33 3
.00 1 8
p .125 .375 .375 .125 1.00
4
Characteristics of the sampling distribution

• It includes all of the possible values of a
  statistic for samples of a particular n
• It includes the frequency or probability of each
  value of a statistic for samples of a particular n

5
Another example

• Imaginary marbles
• Invisible vessels n 100
• Marking means Poker chips
• In the kitty The sampling distribution of the
  mean.

6
The null hypothesis population

• The entire set of scores as they are naturally,
  that is, if the treatment has not affected them.
• If the treatment has had no effect, then the null
  hypothesis is true thus, the name null
  hypothesis population.
• If a treatment has an effect, then the mean of
  the treated sample will not fit well in the null
  hypothesis population It will be weird.

7
The Central Limit Theorem

• If random samples of the same size are drawn from
  any population, then
• the mean of the sampling distribution of the mean
  approaches m , and
• the standard deviation of the sampling
  distribution, called the standard error of the
  mean, approaches s / n ...
• as n gets larger.

8
Generating a sampling distribution

• From a population of six people who are given
  grape Kool-Aid, persons 1, 2, and 3 have their
  IQs raised, and persons 4, 5, and 6 have their
  IQs go down.
• Sampling without replacement, form all of the
  possible unique samples of 2 people from the
  population of six. (Simplified example)
• In how may of the samples does the mean IQ
  increase?

9
The normal deviate test

• The normal deviate test is the Z test applied to
  sample means.
• To use it, you must know the population mean and
  standard deviation. You may know these as
• Population measurements
• TQM or CQI goals
• Design parameters
• Historical sample patterns

10
The normal deviate test...

• The only difference from the simple Z test is
  that the denominator is s / n , which is known
  as the standard error of the mean.
• To test our grape Kool-Aid gang, take a sample of
  100 Houghton students, and compute the mean IQ
  130. Compare that mean to a population mean of
  125, with a population standard deviation of 15.

11
The critical region

• You can simplify a set of decisions about sample
  means by establishing the critical region for
  sample means which fit a rejection criterion for
  Z.
• For a one-tailed test at the .05 level, the
  critical value of Z from table B-1 is 1.645
• For a two-tailed test at the .05 level, the
  critical value of Z is 1.96

12
Calculating the critical region

• Plug the appropriate critical value of Z (1.645
  or 1.96) into the equation for the normal deviate
  test, and solve for M.
• Remember that for a two-tailed test, the critical
  sample mean for each tail must be calculated by
  working above and below the population mean m.

13
Sample size and power

• Test the grape Kool-aid gang again, with sample
  sizes of 4, 9, 16, 25, 36, 49, 64, and 81.
• You will notice that as the sample size
  increases, the obtained Z-score for the same size
  difference between means also increases.
• If the same difference produces a larger Z-score,
  the test has more power.

14
When can we use the normal deviate Z-test?

• For a single sample mean
• When we know m and s
• When the sampling distribution of the mean is
  normally distributed, which we can usually assume
  when n is 30 or more
• Notable exception reaction time measures

15
Reporting standard error in APA format

• In text or in tables, report standard error with
  the abreviation SE.
• In graphs, indicate the size of the standard
  error with error bars, bracketed lines centered
  at the top of the bar of the graph for the mean,
  and extending one standard error above and below
  the mean.

16
Error bars in graphs
17
Normal deviate test in APA

• z 1.98, p lt .05
• z 1.95, p gt .05
• z 1.40, p .08