Do NOT use a computer for this problem. A student is studying for Exam P (the first Actuarial Beer Consumption Science exam). He claims that drinking beer has no effect on the amount of time it takes for him to solve a practice problem. The following data show how many seconds he took to solve a problem after consuming various quantities of beer, measured in ounces:
x 0 12 24 36 48 60 72
y 141 127 141 163 145 179 161
Σx = 252, Σy = 1,057, Σx^2 = 13,104, Σy^2 = 161,447, Σ(x- x)2=4032, Σ(y- y)^2=1,840, Σ(x- x)(y- y)=Σ(x- x)y=2,016.
Consider the model Yi = β0 + β1 xi+ εi, where εi's are i.i.d. N(0, σ2 ).
a) Find the equation of the least-squares regression line.
b) Calculate the residuals ei. Does the sum of the residuals equal zero?
c) Give an estimate for σ, the standard deviation of the observations about the true regression line?
d) What proportion of observed variation in time needed to solve a practice problem is explained by a straight-line relationship with the amount of beer consumed?
e) How much time would you expect the student to need to solve a practice problem after consuming 156 ounces of beer.
f) Explain why it may be dangerous to predict the time needed to solve a practice problem for the amount of beer consumed in part (e).