Question: Forty students are randomly assigned to two different groups. One group learns a list of scrambled letter sets with high association value (each set of letters elicits many associations), and the other group learns letter sets with low association value. The number of repetitions before an errorless recitation of the syllables is recorded for each student. The results are shown here. Compare the groups.

If you reject the null hypothesis, tell what it means in the context of the problem.

Solution: First we compute the basic descriptive statistics:

High Association Value

The sample mean is therefore

\[\bar{X}=\frac{1}{N}\sum{M_i \cdot f_i} =\frac{351}{20}={17.55}\]

The sample variance is

\[s^2 =\frac{1}{N-1}\left(\sum{M_i^2 \cdot f_i-\frac{\left(\sum{M_i \cdot f_i}\right)^2}{N}}\right) = \frac{1}{20-1}\left({6421}-\frac{{351}^2}{20 -1}\right) ={13.7342}\]

and the standard deviation is

\[s = \sqrt{13.7342}={3.706}\]

Low Association Value

The sample mean is therefore

\[\bar{X}=\frac{1}{N}\sum{M_i \cdot f_i} =\frac{489}{20}={24.45}\]

The sample variance is

\[s^2 =\frac{1}{N-1}\left(\sum{M_i^2 \cdot f_i-\frac{\left(\sum{M_i \cdot f_i}\right)^2}{N}}\right) = \frac{1}{20-1}\left({12567}-\frac{{489}^2}{20 -1}\right) ={32.1553}\]

and the standard deviation is

\[s = \sqrt{32.1553}={5.6706}\]