A researcher measures how positive a person’s mood is and how creative he or she is, obtaining the following interval scores:
Participant Mood X Creativity Y
1 10 7
2 8 6
3 9 11
4 6 4
5 5 5
6 3 7
7 7 4
8 2 5
9 4 6
10 1 4
(a) Compute the statistic that summarizes this relationship. (b) What is the predicted creativity score for anyone scoring 3 on mood? (c) If your prediction is in error, what is the amount of error you expect to have? (d) How much smaller will your error be if you use the regression equation than if you merely used the overall mean creativity score as the predicted score for all participants?
To summarize the relationship between the mood (X) and creativity (Y) scores in this dataset, we can calculate the regression equation, which will give us the statistic we need.
Step 1: Find the mean of the X and Y scores:
Mean of X = (10 + 8 + 9 + 6 + 5 + 3 + 7 + 2 + 4 + 1) / 10 = 5.5
Mean of Y = (7 + 6 + 11 + 4 + 5 + 7 + 4 + 5 + 6 + 4) / 10 = 5.9
Step 2: Calculate the deviations from the mean for each X and Y score:
X deviation: 10 - 5.5 = 4.5
Y deviation: 7 - 5.9 = 1.1
Step 3: Find the sum of the products of X and Y deviations:
Sum of (X deviation * Y deviation) = (4.5 * 1.1) + (2.5 * 0.1) + (3.5 * 5.1) + (0.5 * -1.9) + (1.5 * -0.9) + (-2.5 * 1.1) + (1.5 * -1.9) + (-3.5 * 0.1) + (-1.5 * 0.1) + (-4.5 * -1.9) = 24.59
Step 4: Calculate the sum of X deviations squared:
Sum of (X deviation)^2 = (4.5^2) + (2.5^2) + (3.5^2) + (0.5^2) + (1.5^2) + (-2.5^2) + (1.5^2) + (-3.5^2) + (-1.5^2) + (-4.5^2) = 111.5
Step 5: Calculate the regression coefficient (b):
b = Sum of (X deviation * Y deviation) / Sum of (X deviation)^2 = 24.59 / 111.5 ≈ 0.2203
So, the regression equation is: Y = 5.9 + 0.2203X
Now let's answer the specific questions:
(a) The statistic that summarizes this relationship is the regression coefficient (b) which is approximately 0.2203.
(b) To predict the creativity score for someone scoring 3 on mood, we can plug X = 3 into the regression equation:
Predicted Y = 5.9 + 0.2203(3) = 6.56
Therefore, the predicted creativity score for someone scoring 3 on mood is approximately 6.56.
(c) If the prediction is in error, we can calculate the amount of error or the residual, which is the difference between the predicted Y and the actual Y score. Let's say the actual Y score for someone with X = 3 is Y = 7. Then the error would be:
Error = Actual Y - Predicted Y = 7 - 6.56 = 0.44
So, in this case, the error is 0.44.
(d) Comparing the error when using the regression equation versus using the overall mean creativity score as the predicted score for all participants:
If we use the overall mean creativity score (5.9) as the predicted score for all participants, the error for each individual would be the difference between their actual Y score and the mean:
Error = Actual Y - Mean Y = Actual Y - 5.9
Using the regression equation will generally result in smaller errors compared to using the overall mean. However, to determine exactly how much smaller the error will be, we need to calculate the sum of squared differences for both cases:
Sum of squared differences using the regression equation = Σ(Error^2) using the regression equation
Sum of squared differences using the mean = Σ(Error^2) using the mean
The value of the regression sum of squared differences will be smaller than the value of the mean sum of squared differences, indicating that the regression equation provides better predictions than simply using the mean.