1. In a study of caffeine and stress, college students indicate how many cups of coffee they drink per day and their stress level on a scale of 1-10. The data appear below.
Number of Cups of Coffee Stress Level
3 5
2 3
4 3
6 9
5 4
1 2
7 10
3 5
= 3.88
X= 5.13
s = 2.03 s = 2.90
2. Determine the regression equation for predicting stress level from the amount of coffee per day one consumes.
3. Predict the stress level for someone who drinks 8 cups of coffee a day
To determine the regression equation for predicting stress level from the amount of coffee consumed per day, we can use the given data.
Step 1: Calculate the mean (x̄) of the number of cups of coffee consumed per day and the mean (ȳ) of the stress level.
x̄ = (3 + 2 + 4 + 6 + 5 + 1 + 7 + 3) / 8 = 4
ȳ = (5 + 3 + 3 + 9 + 4 + 2 + 10 + 5) / 8 = 5.375
Step 2: Calculate the sum of the products of the number of cups of coffee and the stress level, as well as the sum of the squares of the number of cups of coffee.
∑(x * y) = (3 * 5) + (2 * 3) + (4 * 3) + (6 * 9) + (5 * 4) + (1 * 2) + (7 * 10) + (3 * 5) = 116
∑(x^2) = (3^2) + (2^2) + (4^2) + (6^2) + (5^2) + (1^2) + (7^2) + (3^2) = 119
Step 3: Calculate the slope (b) of the regression equation.
b = [∑(x * y) - (n * x̄ * ȳ)] / [∑(x^2) - (n * x̄^2)]
Where n is the number of data points (8 in this case).
b = [116 - (8 * 4 * 5.375)] / [119 - (8 * 4^2)]
b = [116 - (8 * 4 * 5.375)] / [119 - (8 * 16)]
b = [116 - (172)] / [119 - 128]
b = [116 - 172] / [-9]
b = -56 / -9
b ≈ 6.22
Step 4: Calculate the y-intercept (a) of the regression equation.
a = ȳ - (b * x̄)
a = 5.375 - (6.22 * 4)
a = 5.375 - 24.88
a ≈ -19.505
Therefore, the regression equation for predicting stress level (y) from the amount of coffee per day (x) is:
y ≈ -19.505 + 6.22x
To predict the stress level for someone who drinks 8 cups of coffee per day, we substitute x = 8 into the regression equation:
y ≈ -19.505 + 6.22 * 8
y ≈ -19.505 + 49.76
y ≈ 30.255
Therefore, the predicted stress level for someone who drinks 8 cups of coffee per day is approximately 30.255 on a scale of 1-10.