Statistics and Research methods

1
Statistics and Research methods

• Wiskunde voor HMI
• Bijeenkomst 4

2
The Distribution of Means

• Comparison distributions considered so far were
  distributions of individual scores
• Mean of a group of scores
• Comparison distribution is distribution of means
• Used in hypothesis tests with means of samples

3
The Distribution of Means

• Distribution of means
• Distribution of the means of each of a very large
  number of samples of the same size (with each
  sample randomly taken from the same population of
  individuals)

4
The Distribution of Means

• Characteristics
• Its mean is the same as
• the mean of the population
• of individuals
• Its variance is the variance
• of the population divided by
• the number of individuals in
• each of the samples

5
The Distribution of Means

• Characteristics
• Its standard deviation is the square root of its
  variance
• Shape it is approximately normal if either
• Each sample is of 30 or more individuals or
• The distribution of the population of individuals
  is normal

6
Review of the Different Kinds of Distributions

• Distribution of a population of individuals
• Distribution of a particular sample
• Distribution of means
• See also Table 7-1

7
Z-test

• Hypothesis testing with a distribution of means
• Confidence intervals (betrouwbaarheidsintervallen)
  
• Example Opgave 1 bijeenkomst 4

8
t Test for a Single Sample

• Population mean known, variance not known
• Compare mean of sample scores with population
  mean
• Hypothesis testing steps with population variance
  estimated from sample scores
• Shape of the comparison distribution the t
  distribution
• Degrees of freedom df N-1

9
t Test for a Single Sample (continued)

• The t distribution
• Varies in shape according to the degrees of
  freedom

10
t Test for Dependent Means

• Unknown population mean and variance
• Two scores for each person
• Repeated measures (within-subjects) design
• Use difference scores
• Assume that the population mean is 0
• Assumption Normal population distribution
• t tests are robust to moderate violations of this
  assumption

11
t Test for Dependent Means

• Difference scores
• For each person, subtract one score from the
  other
• Carry out hypothesis testing with the difference
  scores
• Population of difference scores with a mean of 0
• Population 2 has a mean of 0