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Statistics for Psychology

CHAPTER

SIXTH EDITION

Introduction to t Tests: Single Sample and Dependent Means

Estimated Population Variance (S2)

In order to compare a sample mean to a population with a known mean, but an unknown variance, the variance of the population must be estimated
Usually, the only information available about a population is a sample from the population
Therefore, the assumption that Populations 1 and 2 have the same variance is necessary

Representativeness of the Population

Variance of the sample should provide information about the population
If the sample variance is small – the population variance is probably small
If the sample variance is large – the population is probably large

Figure 7-1 The variation in a sample’s scores (shown in the lower distributions) is similar to the variation of scores in the population from which the sample is taken (shown in the upper distributions).

Estimating the Population Variance -1

A sample's variance cannot be used directly as an estimate of the population variance
It can be shown mathematically that a sample's variance will, on the average, be smaller than its population's variance

Estimating the Population Variance -2

Ordinarily, variance is figured as the sum of squared deviations from the mean divided by the number of participants in the sample: SD2 = SS/N, which gives a biased estimate of the population variance

Estimating the Population Variance -2

An unbiased estimate of the population variance (S2) is obtained by modifying the formula:

Estimating the Population Variance -3

Degrees of freedom
Number of scores that are “free to vary”
There are N-1 degrees of freedom because when figuring the deviations, each score is subtracted from the mean
Thus, if all the deviation scores but one are known, the last score can have only one value
Therefore,

Estimating the Population Variance -4

Therefore, the formula for S2 using degrees of freedom can be written

Important Distinction

When estimating population variance, divide the sum of squared deviations by the degrees of freedom (N-1)
When figuring the variance of the distribution of means, divide the estimated population variance by the full sample size (N)

Variance of the Distribution of Means

The variance of the distribution of means

The standard deviation of the distribution of means

Accuracy in Estimating S2

Accuracy is lost when estimating the population variance
Adjust for this loss by making the cutoff sample score for significance more extreme
An exact distribution takes this loss of accuracy into account

Figure 7-2 t distributions (dashed blue lines) compared to the normal curve (solid black line).

t Distributions -2

There is one t distribution for each number of degrees of freedom
The greater the number of degrees of freedom, the closer the t distribution is to the normal curve
When there is an infinite number of degrees of freedom, the t distribution is the same as the normal curve.

Table 7-2 Cutoff Scores for t Distributions with 1 Through 17 Degrees of Freedom (Highlighting Cutoff for Hours-Studied Example)

t Test for a Single Sample -1

Used to compare a sample mean to a known population mean, but the variance of the population is unknown

t Test for a Single Sample -2

Estimating the population variance from the sample scores
Biased estimate of the population variance
Unbiased estimate of the population variance (S2)

t Test for a Single Sample -3

Degrees of freedom
Number of scores that are
“free to vary”
Formula for S2 using degrees of freedom

t Test for a Single Sample -4

The variance of the distribution of means

The standard deviation of the distribution of means

t Test for a Single Sample -5

Locate the appropriate cutoff sample score for rejecting the null hypothesis in the t table
Locate the sample mean score on the comparison distribution by calculating a t score using

t Test for Dependent Means -1

Unknown population mean and variance
Two scores for each person
Repeated measures design
Same procedure as t test for single sample, except
Use difference scores
Assume that the population mean is 0

t Test for Dependent Means -2

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

Assumptions of the t Test

Normal population distribution
t tests are robust to moderate violations of this assumption

Effect Size for the t Test for Dependent Means

small d = .2
medium d = .5
large d = .8

Table 7-11 Approximate Power for Studies Using the t Test for Dependent Means for Testing Hypotheses at the .05 Significance Level

Table 7-12 Approximate Number of Research Participants Needed for 80% Power for the t Test for Dependent Means in Testing Hypotheses at the .05 Significance Level

Controversies and Limitations

Repeated measures designs
Have high power
Standard deviation of difference scores usually low
Weak research design without a control group

Table 7-13 Status Scale: Mean (and SE ) General Expectations for Female and Male Targets

End of Chapter 7

Statistics for Psychology

CHAPTER

SIXTH EDITION

The t Test for Independent Means

t Test for Independent Means

Comparing two samples like an experimental group and a control group
Scores from the groups are independent because they are obtained from different participants

Figure 8-1 Diagram of the logic of a distribution of differences between means.

The Distribution of Differences between Means -2

Mean
If null hypothesis true, the two populations have equal means
If null hypothesis true, the two distributions of means have equal means
If null hypothesis true, the mean of the distribution of differences between equals 0

The Distribution of Differences Between Means -3

Estimating the population variance
Assume the populations have the same variance
Pooled estimate of the population variance

The Distribution of Differences Between Means -4

Figuring the variance of each of the two distributions of means

The Distribution of Differences Between Means -5

Figuring the variance of the distribution of differences between means

Figuring the standard deviation of the distribution of differences between means

The Distribution of Differences Between Means -6

Shape
t distribution

t score for the difference between the two actual means

Assumptions

Each of the population distributions follows a normal curve
The two populations have the same variance

Effect Size for the t Test for Independent Means -1

small d = .2
medium d = .5
large d = .8

Effect Size for the t Test for Independent Means -2

Estimated effect size after a completed study

Table 8-5 Approximate Power for Studies Using the t Test for Independent Means Testing Hypotheses at the .05 Significance Level

Table 8-6 Approximate Number of Participants Needed in Each Group (Assuming Equal Sample Sizes) for 80% Power for the t Test for Independent Means, Testing Hypotheses at the .05 Significance Level

Controversies and Limitations

The problem of too many t tests
Multiple t tests in the same study
Possibility any one of them turns out significant at .05 level by chance is greater than .05
How do researchers adjust for this problem?

Power for the t Test for Independent Means

Power when sample sizes are not equal
Harmonic mean

Table 8-8 Mean Differences Between Pet Owners and Nonowners in Study 1 on Well-Being, Personality, and Attachment Style Measures in Study 1

Table 8-8 (continued) Mean Differences Between Pet Owners and Nonowners in Study 1 on Well-Being, Personality, and Attachment Style Measures in Study 1

End of Chapter 8