1. Overview of Lecture Parametric Analysis is used for
Single Sample Experiments
Between Group Experiments
Within Subject Experiments
Homogeneity of Variance is important for
Two Level Tests
The statistic we use for all the above is the t-test

2. Single Sample Experiments
These kind of inference are usually done under three conditions:
The sample is relatively small in size
The population is normally distributed (or assumed to be normal distributed)
The population standard deviation, s , is unknown.
The parameter we can examine are:
The population mean
The population variance

3. Single Sample Experiments
When the population standard deviation is unknown, and
The sample size is small, and
The population is assumed to be normally distributed
Then the sampling distribution of the statistic
is not normal so we do not have a distribution against which we can compare.

4. Student’s t
W.S. Gosset, writing as “Student”, developed the sampling distribution based on the statistic
The sampling distribution of this statistic is known as Student's t distribution
The t statistic has a family of sampling distributions that vary with respect to the degrees of freedom associated with the sample estimates used.
t is associated with a sampling distribution that has N-1 degrees of freedom

5. One Sample t-test
We can test if a sample mean is significantly different from a population mean using:
The null hypothesis states that the difference between the sample mean and the population mean is zero
The alternative hypothesis states that the difference between the sample mean and the population mean is greater than zero

6. One Sample t-test: Example
We have measured the social desirability scores of a group of people.
The population mean is 3
We are testing whether the sample mean is different to the population mean using:

7. One Sample t-test: Example
First we have to calculate:
The estimate of the population standard error of the mean:

8. One Sample t-test: Example
So,
t=2.049 is the single sample t associated with this data.
We find a critical value for the t statistic by looking it up in tables with the appropriate degrees of freedom
The critical value of the t statistic is equal to for a two-tailed test at p<0.05.
If tobserved  tcritical then reject the null hypothesis
Since the observed value (2.049) doesn't exceed the critical value (2.262), we fail to reject the null hypothesis

9. One Sample t-test: Example
When reporting a result we include four items
The comparison made
The statistical test used
The degrees of freedom
The significance level
For this example:
This sample has a mean social desirability score of 3.75 which is not significantly different from the population mean of 3 (t7=2.049, p<0.05)

10. Parametric Analysis of Two Condition Experiments
When comparing the central tendency of two or more sets of scores we make a particular assumption about the variance:
The variances of the two groups of scores are equal.
This is known as the homogeneity of variance assumption

11. Homogeneity of Variance
In the between groups design we use the test
A critical value of F is looked up using N-1 degrees of freedom for both the numerator and denominator of the F ratio
If the critical value in the tables exceeds the observed value then we reject the null hypothesis of no difference between the variances

12. Homogeneity of Variance
With within subject designs alternative equation is use.
Where

13. Between Group t-test
Both z scores and t tests have relied upon standardization using the following
We can use this again to derive the distribution for differences between the means of two independent samples

14. Between Groups t-test
If the null hypothesis is true then the difference between the population means will be equal to zero since
In other words, under the null hypothesis
is an accurate representation of the difference between two means
If this statistic is significantly greater than zero then we can reject the null hypothesis

15. Between Groups t-test
The standard deviation of the sampling distribution of the mean differences is estimated using the statistic
This formula simplifies to
when the sample sizes are the same

16. Between Groups t-test
A Doctor is interested in looking at the effectiveness of a drug that reduces pulse rates after exercise
One group of subjects are given a placebo
Another group of subjects are given the drug
The pulse rates are measured after 3 minutes resting time

17. Between Groups t-test
For this data
and
Therefore

18. Between Groups t-test
Looking up t with 17 ( n1+n2-2 ) degrees of freedom the critical value is at p<0.05
The observed value is 1.91
Since the observed value is not greater than the critical value we cannot reject the null hypothesis.
For this data, there is no significant difference between the mean pulse rates of the placebo and drug groups.

19. Between Groups t-test When reporting a result we include four items
The comparison made
The statistical test used
The degrees of freedom
The significance level
For this example:
This the mean pulse rate of the drug group (91.11) is not significantly different to the mean pulse rate of the placebo group (107)with t17=1.91, p>0.05)

20. Repeated Measures t test
The repeated measures t test is also known as
the within subjects t test
the matched subjects t test
the related samples t test
It is used when:
there are two sets of scores provided by one group of subjects (a within subjects or repeated measures design)
there are two set of scores provided by two groups of matched subjects

21. Repeated Measures t test
The rationale of the matched t-test is only moderately different from the independent groups t-test.
It begins by assuming that the relevant null hypothesis is that there is no difference between the scores that each individual subject in an experiment provides.
If we adapt the single sample version of student's t to look at difference scores
becomes

22. Repeated Measures t test
The null hypothesis for this test is that the mean of the difference scores will be equal to zero
The alternative hypothesis is that the mean of the difference scores will be greater than zero
Under the null hypothesis the population mean of the difference scores is equal to zero therefore appropriate statistic for this test is:
with N-1 degrees of freedom

23. Repeated Measures t test
In this case the standard error of the mean difference scores is given by
where
is the estimate of the population standard deviation of the difference scores

24. Repeated Measures t test
In this experiment subjects have be asked to judge the speed of two drivers.
They have been told that one of the drivers is irresponsible and the other responsible

25. Repeated Measures t test
For this data
and So

26. Repeated Measures t test
Looking up t with 7 ( N-1 ) degrees of freedom the critical value is at p<0.05
The observed value is 5.196
Since the observed value is greater than the critical value we can reject the null hypothesis.
For this data, there is a significant difference between the estimated speeds of the responsible and irresponsible drivers

27. Repeated Measures t tests
When reporting a result we include four items
The comparison made
The statistical test used
The degrees of freedom
The significance level
For this example:
The mean rating of speed of the irresponsible driver (40.375) is significantly different to the mean rating of the speed of the responsible driver (32.625) with t7=5.196, p<0.05)