3.
Basic Principle of the t Test: Estimating the Population Variance from the Sample Scores
You can estimate the variance of the population of individuals from the scores of people in your sample.
The variance of the scores from your sample will be slightly smaller than the variance of scores from the population.
Using the variance of the sample to estimate the variance of the population produces a biased estimate.
Unbiased Estimate
estimate of the population variance based on sample scores, which has been corrected so that it is equally likely to overestimate or underestimate the true population variance
The bias is corrected by dividing the sum of squared deviation by the sample size minus 1
Eight participants are tested after being given an experimental procedure. Their scores are 14, 8, 6, 5, 13, 10, 10, 6. The population (of people not given this procedure) is normally distributed with a mean of 6. Using the .05 level, does the experimental procedure make a difference? (a) Use the five steps of hypothesis testing and (b) sketch the distinctions involved.
8.
Step 1: Restate the question about the research hypothesis and a null hypothesis about the populations.
Research Question – “Does the experimental procedure make a difference for the participants?”
9.
Step 1: Restate the question about the research hypothesis and a null hypothesis about the populations.
Research Question – “Does the experimental procedure make a difference for the participants?”
Population 1: Participants given the experimental procedure.
Population 2: The general population
H a : “Participants given the experimental procedure will score differently than the general population”
H o : “Participants given the experimental procedure will not score differently than the general population”
10.
Step 2: Determine the characteristics of the comparison distribution
population mean
This is the same as the known population mean.
population variance
Figure the estimated population variance.
(df = N-1)
(N = # scores in sample)
Figure the variance of the distribution of means.
(N = # scores in sample)
standard deviation of the distribution of means
Figure the standard deviation of the distribution of means.
shape of the comparison distribution
t distribution with N – 1 degrees of freedom
11.
Step 2: Determine the characteristics of the comparison distribution (cont.)
19.
Population of Difference Scores with a Mean of 0
Null hypothesis in a repeated measured design
On average, there is no difference between the two groups of scores.
When working with difference scores, you compare the population of difference scores from which your sample of difference scores comes (Population 1) to a population of difference scores (Population 2) with a mean of 0.
A researcher tests 10 individuals before and after an experimental procedure.
Test the hypothesis that there is an increase in scores, using the .05 significance level. (a) Use the five steps of hypothesis testing and (b) sketch the distribution involved.
22.
Step 1: Restate the question about the research hypothesis and a null hypothesis about the populations.
Population 1: Participants given the experimental procedure.
Population 2: Those who show no change from before or after the procedure
H a : “Participants given the experimental procedure will have an increase in scores following the procedure”
H o : “Participants given the experimental procedure will not have an increase in scores following the procedure”
23.
Step 2: Determine the characteristics of the comparison distribution
Make each person’s two scores into a difference score.
Do all of the remaining steps using these difference scores.
Figure the mean of the difference scores.
Assume the mean of the distribution of means of difference scores = 0.
Find the standard deviation of the distribution of means of difference scores.
Figure the estimated population variance of difference scores.
df = N-1
Figure the variance of the distribution of means of difference scores.
Figure the standard deviation of the distribution of means of difference scores.
The shape is a t distribution with N – 1 degrees of freedom.
24.
Step 2: Determine the characteristics of the comparison distribution (cont.)
Make each person’s two scores into a difference score.
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