A diligent statistics student recorded the length of his faithful #2 pencil as he worked away on his homework.

Question

A diligent statistics student recorded the length of his

faithful #2 pencil as he worked away on his homework.

He discovered a strong linear relationship between the

number of hours that he worked and the length of his

pencil. Here is the regression analysis for these data.

Dependent variable: length (cm)

R² = 92.3% R²(adj)= 89.5%

coeff se t ratio p value

constant 17.047 0.128 23.58 60.0001

time (hr) -1.914 0.047 35.28 60.0001

a) Write the equation of the least square regression

line.

b) Interpret R2

in this context.

c) Interpret the equation in this context.

d) This student’s girlfriend tried out his model on a

pencil she had used for 5 hours, and found a residual

of -0.88 cm. How long was her pencil at that time?

Answer 1

A. Given that ß₀ = 17.047 and ß₁ = -1.914 and using the standard format Y = ß₀ + ß₁x the equation should be:

length(y) = 17.047cm - 1.914 cm/hr * x (cm)

B. R2 is the Correlation Coefficient and represents the proportion of the variance for a dependent variable that's explained by an independent variable. In this case 92.3%

C. The standard #2 pencil is 17.047 cm long when new and in use will get shorter at a rate of 1.914 cm per hour. So after 5 hours it is expected that the pencil will have a length of 7.477 inches.

D. If the girlfriend used a new #2 pencil for 5 hours the expected length would have been 7.477. If she measured the pencil and compared the measured length against the expected value and found there was a residual of -0.88 cm, that means that her pencil was .88cm smaller than the predicted value, meaning it was 6.597 cm long. (remember r = x - x₀ where x₀ is the predicted value and x is the measured value)