Correlated t-Tests

Mg-10 and Mg-

Perez, Sugay, Torres, Villafuerte, Alcos, Manaid

What is a t-Test?The t-Test is used when you want to

compare the means of two groups.

Two distributions would be shown: the control group and the treated group.

The test answers whether the two groups are statistically different.

t-Test for Two Correlated Samples

In this type of t-Test, the two groups are tested before and after the independent variable is introduced to the treated group.

This is done to statistically remove the chance of coincidence.

t-Test for Two Correlated Samples

For example, an experiment is done to determine whether a certain shoe would make people run faster than ordinary shoes.

Statistics are shown as follows:

Runner Speed w/ordinary shoes

Speed w/ special shoes

A 12.1 mi/h 12.5 mi/h

B 14.4 mi/h 15.0 mi/h

C 11.6 mi/h 11.4 mi/h

D 11.2 mi/h 11.6 mi/h

E 12.4 mi/h 11.8 mi/h

F 11.6 mi/h 12.0 mi/h

G 15.0 mi/h 14.9 mi/h

t-Test for Two Correlated Samples

We then follow the procedures in testing the statistics:

State the null hypothesis to be tested:

“ Ho = uA = uB ”This states that the special shoe provides no significant

change in the speeds of the people tested

Runner Speed w/ordinary

shoes

Speed w/ special shoes

A 12.1 mi/h 12.5 mi/h

B 14.4 mi/h 15.0 mi/h

C 11.6 mi/h 11.4 mi/h

D 11.2 mi/h 11.6 mi/h

E 12.4 mi/h 11.8 mi/h

F 11.6 mi/h 12.0 mi/h

G 15.0 mi/h 14.9 mi/h

t-Test for Two Correlated Samples

State the alternative hypothesis:

The special shoe provides an increase in speed for the runners

“ HA : uA = uB ”

Runner Speed w/ordinary

shoes

Speed w/ special shoes

A 12.1 mi/h 12.5 mi/h

B 14.4 mi/h 15.0 mi/h

C 11.6 mi/h 11.4 mi/h

D 11.2 mi/h 11.6 mi/h

E 12.4 mi/h 11.8 mi/h

F 11.6 mi/h 12.0 mi/h

G 15.0 mi/h 14.9 mi/h

t-Test for Two Correlated Samples

State the level of significance, α = 0.05

Set-up the following table:

Runner Speed w/ordinary

shoes

Speed w/ special shoes

Speed before – Speed after

D D^2

A 12.1 mi/h 12.5 mi/h -0.4 0.16

B 14.4 mi/h 15.0 mi/h -0.6 0.36

C 11.6 mi/h 11.4 mi/h 0.2 0.04

D 11.2 mi/h 11.6 mi/h -0.4 0.16

E 12.4 mi/h 11.8 mi/h -0.4 0.16

F 11.6 mi/h 12.0 mi/h -0.4 0.16

G 15.0 mi/h 14.9 mi/h 0.1 0.01

∑D = -1.9∑D2 = 1.05

t-Test for Two Correlated Samples

Compute for t using the following steps:

Calculate sum of squares of the difference score:

∑d2 = ∑d2 - = 1.05 – (-1.9)2 = 0.534 (∑D)2

N 7

-Calculate the standard error of the mean differencesqrt(∑d2 / N(N-1)) = sqrt(0.534/7(6)) = 0.11

-Calculate D = ∑D = (-1.9) = -0.27

N 7

t-Test for Two Correlated Samples

Calculate t = D = -0.27 = -2.45

Find the critical or tabular value of t, df = 6, α = 0.05

t critical = t0.05 = +/-

Formulate your conclusion:

Since the calculated t is larger than the critical t, we shall accept HA, which states that the special shoe is effective in increasing the speed of runners wearing them.

SD 0.11