T-test

“A t-test is used to determine whether a set or sets of scores are from the same population.” - Coakes & Steed (1999), p.61

History
The t-test is years old. The t statistic was introduced by William Sealy Gosset for cheaply monitoring the quality of beer brews. Gosset was a statistician for the Guinness brewery in Dublin, Ireland. He was hired due to Guiness' innovative policy of recruiting the best graduates from Oxford and Cambridge to apply biochemistry and statistics to Guinness' industrial processes. Gosset published the t test in Biometrika in 1908, but was forced to use a pen name by his employer who regarded the fact that they were using statistics as a trade secret. Hence, the name Student's t-test. Read more here: t-test history.

Types
There are three types of t-test:

One-sample t-test

 * 1) Used to compare a sample mean with a known population mean or some other meaningful, fixed value

Independent samples t-test

 * 1) Used to compare two means from independent groups

Paired samples t-test

 * 1) Used to compare two means that are repeated measures for the same participants - scores might be repeated across different measures or across time.
 * 2) Used also to compare paired samples, as in a two treatment randomized block design.

Assumptions
Dependent variables must be:
 * 1) Measured at interval or ratio level level of measurement - i.e., needs to be continuous.
 * 2) Normally distributed in all groups of the independent variable.
 * 3) * Robust to violations of this assumption if sample sizes are large and approximately equal (> 15 cases per group)
 * 4) Have approximately equal variance across all groups of the IV (homogeneity of variance e.g., tested by Levene's test).
 * 5) * If not the p-values for significance tests are inaccurate.
 * 6) * If the variances are different SPSS has post-hoc tests to adjust for this.
 * 7) Cases represent random samples from the populations and the scores of the test variable are independent of each other.
 * 8) * Inaccurate p-values if the independence assumption is violated.

Graphing

 * 1) Bar chart (1 IV) and clustered bar charts (> 1 IV)
 * 2) Box and whisker plot
 * 3) Error-bar chart
 * 4) Stem and leaf plot

Checklist

 * 1) Purpose? (e.g., hypothesis?)
 * 2) Variables / design
 * 3) Descriptive statistics (4 moments)
 * 4) Assumptions
 * 5) Figure / graph? (optional)
 * 6) One-tailed or two-tailed?
 * 7) Test statistics, including effect size and direction of effects

Example 1
An independent t-test was used to determine whether there was a difference in mean grip strength between males and females. This revealed a significant difference (t (88) = 2.04, p = .04), with males having significantly higher mean strength scores than females (males, M = 2.93, SD = 1.23; females, M = 2.71, SD = '1.41).

Note: Usually means, standard deviations and effect sizes are reported in a Table.

Non-parametric

 * Mann-Whitney U
 * Wilcoxon signed-rank test