The following data are from a repeated-measures study examining the effect of treatment by measuring a group of n=6 participants before and after they receive a treatment.

Question

Participants Before Treatment: A7, B2, C4, D5, E5, F3

Participants After Treatment: A8, B9, C6, D7, E6, F8

a. Calculate the difference scores and Md .

b. Compute SS, sample variance, and estimated standard error.

c. find t-critical and t value to reject or fail to reject the null hypothesis

Answer 1

This is a dependent sample t-test. To add a little levity to our calculations, we will imagine that this data is from six students who participated in a study to see whether drinking coffee during a statistics class had a change in their alertness. The stats professor measured each student's alertness, gave them a Venti Americano with 2 extra shots (that's like 6 shots of espresso, friends), and then measured alertness again.

**First, calculate the difference scores for each participant. Subtract the after score from the before score for each participant.

A 7-8 = -1

B 2-9 = -7

C 4-6 = -2

D 5-7 = -2

E 5-6 = -1

F 3-8 = -5

**To get the mean difference (Md), add up your difference scores and divide by N (the number of difference scores):

-1+-7+-2+-2+-1+-5 = -18

Md = -18/6 = -3

Here, the mean difference in alertness is -3 points (meaning that students scored 3 points higher at the "after" measurement than they did at the "before" measurement).

** To get the Sum of Squares (SS), subtract the Md from each difference score and add up all of those results. In this instance, we have a negative mean difference, so it's important to remember that when you subtract a negative, you are actually adding. For example, -1 - (-3) is essentially -1+3, or 2. It's also important to remember that when you square a negative, you always get a positive result.

A = (-1-(-3))² = 4

B = (-7-(-3))² = 16

C = (-2-(-3))² = 1

D = (-2-(-3))² = 1

E = (-1-(-3))² = 4

F = (-5-(-3))² = 4

SS = 4+16+1+1+4+4 = 30

Our Sum of Squares (SS) is 30.

**To get the sample variance (S²), divide the SS by N-1. We divide by N-1 because this is a sample and not a population. Dividing by N-1 gives our estimate of the population variance a little "boost" to account for the fact that smaller samples are not good representations of the amount of variance in an entire population.

S²=SS/(N-1) = 30/5 = 6

**Square root the variance to get the standard deviation.

S = √S² = √6 = 2.45

**Now, you can divide the standard deviation (S) by the square root of N to get the standard error (S_Md)

S_Md = S/(√N) = 2.45/√6 = 1.00

**Finally, we are ready to conduct our hypothesis test. We can find our critical value in the t-table using the degrees of freedom (N-1) and the alpha or p level (.05 is a safe assumption if you aren't given other information).

t_cv(5df, p<.05) = ±2.571

If our obtained t-value is lower than -2.571 or higher than +2.571, we can reject our null hypothesis.

**Calculate t!

t = (M1-M2)/S_Md = -3/1 = -3

Since -3 is lower than -2.571, we can reject the null hypothesis. In the context of the problem, this means that participants did score differently on the alertness measure after drinking an incredible amount of coffee (t(5)=-3.00, p<.05).

Happy calculating!