Study Guide #1.
Be sure to be able to look at an example problem and determine if it relates to effect size (such as an example that says two means are very different from one another) and/or statistical significance (such as an example in which a pattern found in a sample is found to likely (beyond chance) characterize the broader population at large) – see examples on this distinction, at the end of this study guide.
Here’s an example to work with for many of the computational problems:
Here’s the story: Your dog, Momo, barks way too much and you can’t figure out why. One day, you forget to give Momo doggie biscuits and you notice that she does not bark that day. After thinking about this issue for a bit, you decide to run a study using 3 other dogs to determine whether # of doggie biscuits consumed is indeed related to the amount of barking. In this study you measure (X) the number of doggie biscuits each dog eats in a day and (Y) the number of times that dog barks, in order to see what the actual relationship is between these two variables.
X (# of doggie biscuits consumed) Y (# of barks in a day)
0 2
1 1
2 3
1. Create a frequency table for X.
2. Create a frequency histogram for X.
3. Compute the following:
Mean for X: Mean for Y:
Median for X: Median for Y:
Mode for X: Mode for Y:
SSx: SSy:
SDx2: SDy2:
SDx: SDy:
4. What is Zx when X = 0?
5. If Zy = 1.4, what would be the corresponding Y value?
6. What is the sum of the cross product of Z scores between X and Y?
7. What is the correlation (r) between X and Y?
8. Draw scatterplot of raw data showing the correlation between X and Y.
9. The scatterplot shows that the relationship between X and Y is
A. very strong and positive.
B. very strong and negative.
C. moderate and positive.
D. moderate and negative.
ANSWER KEY:
* Your frequency table for X should have nothing to do at all with the Y variable; it should reflect the fact that the values of 0, 1, and 2 all appear on one occasion (i.e., have a frequency of 1). The histogram for X should graphically tell this same story – remember, the Y axis in a histogram pertains to how frequent each value is (it has nothing to do with the Y variable).
X – M (X-M) (X-M) 2 Zx
0 – 1 -1 1 -1.22
1 – 1 0 0 0
2 – 1 1 1 1.22
___________________________________
Mx = 1 SSx = 2
SDx2 = .67
SDx = .82
Y – M (Y-M) (Y-M) 2 Zy
2 – 2 0 0 0
1 – 2 -1 1 -1.22
3 – 2 1 1 1.22
___________________________________
My = 2 SSy = 2
SDy2 = .67
SDy = .82
ZxZy
-1.22*0 = 0
0*-1.22 = 0
1.22*1.22= 1.49
___________________
S(ZxZy) = 1.49
r = 1.45/3 = .50
* the scatterplot should show a moderate and positive correlation (dots generally going up and to the right) – this point should be reflected in the multiple-choice question too.
* When X = 0, Zx = -1.22 (Zx = (x-Mx)/SDx)
* when Zy = 1.44, Y = 3.15 (Y (or a raw score) = SD(Zy) + My)
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Info on distinction between effect size and statistical significance:
If a finding relates to effect size only, then that finding is only framed in terms of how large the effect is – usually in terms of how different two means are from one another (if two means are VERY different from one another, then we say that the effect size is large).
If a finding is presented in a way that only relates to statistical significance, then that finding is being presented in terms of whether the pattern found in the data likely represent a pattern that is found in the broader population.
e.g., THIS example speaks to EFFECT SIZE: A researcher found that the average test score on the chemistry exam was MUCH LOWER than the average score on the English composition exam.
e.g., THIS example speaks to statistical significance: A researcher found evidence that people like butterflies more than they like snakes. Her statistics showed that this pattern, found in her sample, is very likely to generalize to the broader population of humans in general.