In a study on caffeine and stress, college students indicated how many cups of coffee they drink per day and their stress level on a scale of 1 to 10. The data are provided in the following table.
Number of Cups of Coffee
3, 2, 4, 6, 5, 1, 7, 3, 2, 4
Stress Level
5, 3, 3, 9, 4, 2, 10, 5, 3, 8
Calculate a Pearson's r to determine the type and strength of the relationship between caffeine and stress level. How much of the variability in stress scores is accounted for by the number of cups of coffee consumed per day?
To determine the type and strength of the relationship between caffeine (number of cups of coffee) and stress level, we can calculate Pearson's correlation coefficient (r). Pearson's r measures the linear relationship between two variables, ranging from -1 to +1, where -1 indicates a strong negative linear relationship, +1 indicates a strong positive linear relationship, and 0 indicates no correlation.
To calculate Pearson's r, you will need to follow these steps:
Step 1: Calculate the means of the number of cups of coffee (X) and stress level (Y) separately.
Mean of X (number of cups of coffee) = (3+2+4+6+5+1+7+3+2+4) / 10 = 37 / 10 = 3.7
Mean of Y (stress level) = (5+3+3+9+4+2+10+5+3+8) / 10 = 52 / 10 = 5.2
Step 2: Calculate the deviations from the mean for both X and Y.
Deviations from the mean of X (number of cups of coffee) = (3-3.7), (2-3.7), (4-3.7), (6-3.7), (5-3.7), (1-3.7), (7-3.7), (3-3.7), (2-3.7), (4-3.7)
Deviations from the mean of Y (stress level) = (5-5.2), (3-5.2), (3-5.2), (9-5.2), (4-5.2), (2-5.2), (10-5.2), (5-5.2), (3-5.2), (8-5.2)
Step 3: Square the deviations.
Squared deviations of X = (-0.7)^2, (-1.7)^2, (0.3)^2, (2.3)^2, (1.3)^2, (-2.7)^2, (3.3)^2, (-0.7)^2, (-1.7)^2, (0.3)^2
Squared deviations of Y = (-0.2)^2, (-2.2)^2, (-2.2)^2, (3.8)^2, (-1.2)^2, (-3.2)^2, (4.8)^2, (-0.2)^2, (-2.2)^2, (2.8)^2
Step 4: Multiply the deviations of X and Y.
Products of deviations = (-0.7 * -0.2), (-1.7 * -2.2), (0.3 * -2.2), (2.3 * 3.8), (1.3 * -1.2), (-2.7 * -3.2), (3.3 * 4.8), (-0.7 * -0.2), (-1.7 * -2.2), (0.3 * 2.8)
Step 5: Calculate the sum of squared deviations and the sum of products of deviations.
Sum of squared deviations of X = sum(squared deviations of X) = sum((-0.7)^2, (-1.7)^2, (0.3)^2, (2.3)^2, (1.3)^2, (-2.7)^2, (3.3)^2, (-0.7)^2, (-1.7)^2, (0.3)^2)
Sum of squared deviations of Y = sum(squared deviations of Y) = sum((-0.2)^2, (-2.2)^2, (-2.2)^2, (3.8)^2, (-1.2)^2, (-3.2)^2, (4.8)^2, (-0.2)^2, (-2.2)^2, (2.8)^2)
Sum of products = sum(products of deviations) = sum((-0.7 * -0.2), (-1.7 * -2.2), (0.3 * -2.2), (2.3 * 3.8), (1.3 * -1.2), (-2.7 * -3.2), (3.3 * 4.8), (-0.7 * -0.2), (-1.7 * -2.2), (0.3 * 2.8))
Step 6: Calculate Pearson's correlation coefficient (r).
r = sum of products / sqrt(sum of squared deviations of X * sum of squared deviations of Y)
Now, you can substitute the calculated values from the previous steps into the formula to find Pearson's r.