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?
Using an online calculator, I found the following:
10 data pairs (x,y):
( 10.0 , 7.00 ); ( 8.00 , 6.00 ); ( 9.00 , 11.0 ); ( 6.00 , 4.00 ); ( 5.00 , 5.00 ); ( 3.00 , 7.00 ); ( 7.00 , 4.00 ); ( 2.00 , 5.00 ); ( 4.00 , 6.00 ); ( 1.00 , 4.00 );
Regression equation:
Predicted y = 4 + (0.345)x
For b): substitute 3 for x (mood) in the above equation and solve for predicted y (creativity).
I'll let you take it from here.
(a) To summarize the relationship between mood (X) and creativity (Y), we can calculate the correlation coefficient (r) or the coefficient of determination (r^2). Let's calculate both:
First, let's calculate the mean of mood (X) and creativity (Y):
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
Now, let's calculate the sum of the products of the deviations:
Σ((X - mean of X) * (Y - mean of Y)) = (10-5.5)(7-5.9) + (8-5.5)(6-5.9) + (9-5.5)(11-5.9) + (6-5.5)(4-5.9) + (5-5.5)(5-5.9) + (3-5.5)(7-5.9) + (7-5.5)(4-5.9) + (2-5.5)(5-5.9) + (4-5.5)(6-5.9) + (1-5.5)(4-5.9) = -7.9
Next, let's calculate the sum of the squared deviations for X and Y:
Σ(X - mean of X)^2 = (10-5.5)^2 + (8-5.5)^2 + (9-5.5)^2 + (6-5.5)^2 + (5-5.5)^2 + (3-5.5)^2 + (7-5.5)^2 + (2-5.5)^2 + (4-5.5)^2 + (1-5.5)^2 = 110
Σ(Y - mean of Y)^2 = (7-5.9)^2 + (6-5.9)^2 + (11-5.9)^2 + (4-5.9)^2 + (5-5.9)^2 + (7-5.9)^2 + (4-5.9)^2 + (5-5.9)^2 + (6-5.9)^2 + (4-5.9)^2 = 39.9
Now, let's calculate the correlation coefficient (r):
r = Σ((X - mean of X) * (Y - mean of Y)) / sqrt(Σ(X - mean of X)^2 * Σ(Y - mean of Y)^2) = -7.9 / sqrt(110 * 39.9) ≈ -0.193
And the coefficient of determination (r^2):
r^2 = 0.193^2 ≈ 0.0374
Therefore, the statistic that summarizes this relationship is the correlation coefficient (r), which is approximately -0.193.
(b) To predict the creativity score for someone scoring 3 on mood (X), we can use the regression equation derived from the data.
First, let's calculate the slope of the regression line (b):
b = Σ((X - mean of X) * (Y - mean of Y)) / Σ(X - mean of X)^2 = -7.9 / 110 ≈ -0.072
Next, let's calculate the intercept of the regression line (a):
a = mean of Y - b * mean of X = 5.9 - (-0.072 * 5.5) = 6.292
Now, we can use the regression equation to predict the creativity score (Y) for someone scoring 3 on mood (X):
Y = a + b * X = 6.292 + (-0.072 * 3) ≈ 6.076
Therefore, the predicted creativity score for someone scoring 3 on mood is approximately 6.076.
(c) If the prediction is in error, we can calculate the amount of error by subtracting the predicted value from the actual value. In this case, we do not have the actual creativity score for someone scoring 3 on mood, so we cannot determine the amount of error.
(d) If we used the overall mean creativity score as the predicted score for all participants, the error would be the difference between each participant's actual creativity score and the mean creativity score. By using the regression equation, we incorporate the relationship between mood and creativity, which reduces the overall error. Therefore, using the regression equation would result in a smaller error compared to using the overall mean creativity score.
To summarize the relationship between mood and creativity scores, we can calculate the correlation coefficient (r).
(a) To compute the correlation coefficient, we need to follow these steps:
Step 1: Calculate the mean of both mood (X) and creativity (Y).
mean(X) = (10 + 8 + 9 + 6 + 5 + 3 + 7 + 2 + 4 + 1) / 10
mean(X) = 5.5
mean(Y) = (7 + 6 + 11 + 4 + 5 + 7 + 4 + 5 + 6 + 4) / 10
mean(Y) = 5.9
Step 2: Calculate the deviations from the mean for both X and Y.
Deviation(X) = X - mean(X)
Deviation(Y) = Y - mean(Y)
For participant 1, mood = 10 and creativity = 7:
Deviation(X) = 10 - 5.5 = 4.5
Deviation(Y) = 7 - 5.9 = 1.1
Continue calculating the deviations for the remaining participants.
Step 3: Calculate the product of the deviations for each participant.
Product of Deviations = Deviation(X) * Deviation(Y)
Participant 1: Product of Deviations = 4.5 * 1.1 = 4.95
Repeat this step for all participants.
Step 4: Square the deviations for each participant.
Squared Deviation(X) = (Deviation(X))^2
Squared Deviation(Y) = (Deviation(Y))^2
Participant 1: Squared Deviation(X) = (4.5)^2 = 20.25
Repeat this step for all participants.
Step 5: Calculate the sum of the products of deviations and the sum of squared deviations for both X and Y.
Sum of Product of Deviations = ∑ Product of Deviations
Sum of Squared Deviations(X) = ∑ Squared Deviation(X)
Sum of Squared Deviations(Y) = ∑ Squared Deviation(Y)
Step 6: Calculate the correlation coefficient (r) using the following formula:
r = Sum of Product of Deviations / sqrt(Sum of Squared Deviations(X) * Sum of Squared Deviations(Y))
(b) To predict the creativity score for someone with a mood score of 3, we can use the regression equation based on the correlation coefficient obtained in part (a):
Regression Equation: Y' = a + bX
where Y' is the predicted value of Y, X is the given value of X (mood score), a is the y-intercept, and b is the slope of the regression line.
(c) To estimate the error in the prediction, we can calculate the standard error (SE) using the formula:
SE = sqrt((1 - r^2) * (Sum of Squared Deviations(Y)) / (n - 2))
where n is the number of participants.
(d) To compare the error when using the regression equation versus using the overall mean creativity score, we can calculate the standard error of the mean (SEM) using the formula:
SEM = sqrt((Sum of Squared Deviations(Y)) / n)
To compute the statistic that summarizes the relationship between mood and creativity, we can use linear regression analysis. Linear regression analysis helps us understand the relationship between two variables and predict one variable based on the other.
(a) Compute the statistic that summarizes this relationship:
To compute the statistic, we need to calculate the regression equation, which represents the best-fitting line that predicts creativity (Y) based on mood (X). The regression equation has the form: Y = a + bX, where "a" is the y-intercept and "b" is the slope of the line.
To find the regression equation, we need to perform the following steps:
Step 1: Calculate the means of the X (mood) and Y (creativity) variables:
Mean of X (mood) = (10 + 8 + 9 + 6 + 5 + 3 + 7 + 2 + 4 + 1) / 10 = 5.5
Mean of Y (creativity) = (7 + 6 + 11 + 4 + 5 + 7 + 4 + 5 + 6 + 4) / 10 = 5.9
Step 2: Calculate the sum of products of deviations:
Sum of (X - X̄) * (Y - Ŷ) = (10 - 5.5) * (7 - 5.9) + (8 - 5.5) * (6 - 5.9) + ... + (1 - 5.5) * (4 - 5.9)
Step 3: Calculate the sum of squared deviations for X:
Sum of (X - X̄)^2 = (10 - 5.5)^2 + (8 - 5.5)^2 + ... + (1 - 5.5)^2
Step 4: Calculate the slope (b):
b = Sum of (X - X̄) * (Y - Ŷ) / Sum of (X - X̄)^2
Step 5: Calculate the y-intercept (a):
a = Ȳ - b * X̄
Using these formulas, we can calculate the regression equation and the statistic that summarizes this relationship.
(b) What is the predicted creativity score for anyone scoring 3 on mood?
Once we have the regression equation, we can use it to predict the creativity score (Y) for a specific mood score (X). In this case, we want to predict the creativity score for a mood score of 3. By substituting X = 3 into the regression equation, we can find the predicted value of Y.
(c) If your prediction is in error, what is the amount of error you expect to have?
To determine the amount of error, we can calculate the difference between the predicted creativity score (from the regression equation) and the actual creativity score obtained for a specific mood score. This difference is known as the prediction error.
(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 compare the error, we can calculate the difference between the predicted creativity scores obtained using the regression equation and using the overall mean creativity score for all participants. By comparing these differences, we can determine how much smaller the error is when using the regression equation.
By following these steps, we can compute the statistic summarizing the relationship, predict creativity scores, determine prediction errors, and compare them.