Have you ever noticed that adults like the taste of some bitter foods that adolescents just cannot tolerate the taste of? Things like dark chocolate, grapefruit, coffee, etc. Ms. Wilson is interested in whether bitter flavors become more tolerable as people get older. She sets up a study in which subjects taste cups of coffee that are different strengths and picks their favorite one.
Ms. Wilson records the age of each subject, x, and the strength of the coffee in grams, y, that the subject liked the best.
Age of Subject
17
27
33
37
40
Strength of Coffee in grams
48
44
49
54
57
r =
y =
x +
If someone was 45 years old, what would the regression line predict for the strength of coffee they would prefer?
First, we need to calculate the regression line using the given data points:
Calculating the mean of x and y:
mean of x = (17 + 27 + 33 + 37 + 40) / 5 = 154 / 5 = 30.8
mean of y = (48 + 44 + 49 + 54 + 57) / 5 = 252 / 5 = 50.4
Calculating the covariance and variance of x and y:
covariance = Σ((xi - mean of x)(yi - mean of y)) / n = ((17-30.8)*(48-50.4) + (27-30.8)*(44-50.4) + (33-30.8)*(49-50.4) + (37-30.8)*(54-50.4) + (40-30.8)*(57-50.4)) / 5
= (-27.2*-2.4 + -3.8*-6.4 + 2.2*-1.4 + 6.2*3.6 + 9.2*6.6) / 5
= (65.28 + 24.32 - 3.08 + 22.32 + 60.72) / 5
= 169.56 / 5
= 33.912
variance of x = Σ(xi - mean of x)^2 / n = ((17-30.8)^2 + (27-30.8)^2 + (33-30.8)^2 + (37-30.8)^2 + (40-30.8)^2) / 5
= (196.8 + 11.52 + 6.12 + 41.8 + 75.12) / 5
= 331.36 / 5
= 66.272
variance of y = Σ(yi - mean of y)^2 / n = ((48-50.4)^2 + (44-50.4)^2 + (49-50.4)^2 + (54-50.4)^2 + (57-50.4)^2) / 5
= (6.24 + 40.96 + 2.16 + 12.96 + 43.56) / 5
= 105.88 / 5
= 21.176
Calculating the slope of the regression line:
slope = covariance / variance of x = 33.912 / 66.272 = 0.511
Calculating the intercept of the regression line:
intercept = mean of y - (slope * mean of x) = 50.4 - (0.511 * 30.8) = 50.4 - 15.729 = 34.671
Therefore, the regression line equation is:
y = 0.511x + 34.671
Now, if someone was 45 years old, we can plug in x = 45 into the regression line equation to find the predicted strength of coffee they would prefer:
y = 0.511(45) + 34.671 = 22.995 + 34.671 = 57.666
Thus, the regression line predicts that a 45-year-old subject would prefer a strength of coffee around 57.67 grams.