# The Statistical t-Test

The statistical t-test is used to compare two conditions, specifically the means of two conditions. A t-test can be applied to both a between participants and within participants design. This test can only be done on normally distributed data, and as such is a parametric test. The purpose of the t-test is to decide whether or not the difference between the means of the two conditions is statistically significant. If the difference is statistically significant we are able to except our experimental hypothesis and also give some directionality to our hypothesis. If the difference between the means of the two conditions is not statistically significant we must reject our experimental hypothesis and accept the null hypothesis. The t-score is technically more than just the difference between the means. Just like normal data distribution, the t-score also has a 95% confidence interval, which means for the difference between the means to be statistically significant, the alpha level needs to be less than 0.05. The alpha level was decided on 0.05 to try and reduce the amount of type I and type II errors. Type I errors is when we reject the null hypothesis but should not have, and type II errors is when we reject the experimental hypothesis but we should not have. If the sample size is large and the null hypothesis is true, the distribution of the t-scores is also normal. The smaller the sample size becomes, the more tail-heavy the distribution becomes.

The way this is interpreted is if two groups come from the same population,  then 95% of the time, the t-score (reflecting the difference in the means) will be within the 95% area under the graph of the data.

## Degrees of Freedom

Degrees of freedom for within-participants design is the same as the number of participants.

Degrees of freedom for between-participants design is the (number of participants in group 1 -1) + (number of Ps in group 2 -1)

SPSS will do the math for you!

## Basically…

When you start with your mean scores, assume that the null hypothesis is true and that there is no significant difference between the means.

Then set your significance level at p<0.05 or the alpha level, which is the same thing. SPSS should do this automatically.

Then using SPSS calculate the t-score.

If the t-score is within the 95% interval: accept the null hypothesis and reject the experimental hypothesis.

If the t-score is outside the 95% interval: reject the null hypothesis and accept the experimental hypothesis. You have now established that there is a significant difference between the two means.

This is an example of the type of output that will be given by SPSS. From this output you can answer the following questions:

Question 1: Is the experimental design within or between participants?

Answer: The experimental design is within. You can tell this from the heading where it says paired difference.

Question 2: What is the t-score?

Question 3: What are the degrees of freedom?

Answer: The degrees of freedom (df) is 77.

Question 4: Is it two-tailed or one-tailed test?

Answer: It is two tailed as shown in the last box. Sig. (2-tailed).

Question 5: Is the result significant at an alpha level of 0.05? Why?

Answer: The result is significant at the alpha level because p<0.001, which obviously is less than 0.05.

## Reporting the Results

This is an example of how you would report the following data for the results section of a lab report:

The mean and standard deviation of participants’ reaction time under conditions 1 and 2 are given in Table (not in this post). The data were analysed using a two-tailed within-participants t-test and an alpha level of 0.05.There is a statistically significant difference between the ideal IQ and the estimated IQ, with the estimated IQ significantly lower than IQ for an ideal job, t(77) = -9.60, p <0.001.