Introduction to the t statistic

Steps to calculate the denominator for the t-test

1. Calculate variance or SD

s2= SS/n-1

2. Calculate the standard error

n

sOR

n

ssM

2

T-test Procedure

Set the hypotheses (null and alternative) Set the decision criteria (by selecting the alpha

level, directionality, and critical values). Compute the test-statistic (t) Compare obtained t-statistic to critical value(s) Make a statistical decision Concluding statement

Conducting a t-test

It’s not a normal distribution, and you don’t use the unit normal table

Using the t-distribution (William Sealy Gosset, 1908) instead

T-distribution

Bell-shaped and symmetrical

A ‘family’ of distributions - depends on the degrees of freedom

The mean of the t-distribution is zero

T-distribution table

When the df is NOT listed in the table provided in Appendix B, you should look up the critical t for both of the surrounding df values listed and use the LARGER value for t

Measuring Effect Size with the t Statistic (cont.)

Estimated Cohen’s d

s

M

Measuring Effect Size with the t Statistic (cont.)

percentage of variance accounted for = r2 (aka 2)

dft

tr

2

22

Assumptions for a One-sample t-Test

1. Independent observations

2. The sample data come from a normal population distribution

Example 1

The legal highway limit near populated areas in many parts of the US is 55 mph. To determine whether people exceed the 55mph, suppose a state monitored the speeds of 25 cars at selected highway locations. Can we conclude that the average speed is higher than the 55 mph limit?

Example 1 - Data

55 60 60 55 57 60 55 58 6354 65 56 61 58 55 57 59 5359 65 56 61 55 54 59

Find the following:MeanSSs2

s