Weighted mean is calculated by the formula:
Here, are the samples sizes and are the means of respective groups.
a) Overall mean memory score for females:
Here, . Thus,
b) Overall mean memory score for males:
. Thus,
c) Overall mean memory score for volunteers receiving Treatment A:
.
d) Overall mean memory score for volunteers receiving Treatment B:
.
After puting these results in the table given in question the table becomes:
Treatment A | Treatment B | ||
Females | M = 9 n = 40 |
M = 5 n = 20 |
M = 7.7 |
Males | M = 9 n = 20 |
M = 5 n = 40 |
M = 6.3 |
M = 9.0 | M = 5.0 |
The line graph for the mean score of females and males is given below:
The lines for males and females completely overlap so only one line is visible.
Now, considering the overall mean for females and males calculated earlier suggests that females have better memories than males, as the overall mean for females is 7.7 and that for males is 6.3.
However, the line graph shows that there is no difference between memories of females and males.
This is because, in the line graph the sample size for each treatment group is not considered while comparing memories of males and females. These sample sizes act as weights while computing weighted means in order to make the two groups comparable. So, if the line graph was made for overall scores, it would have given the same conclusion as the calculation.
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