Question

1. 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 is length (cm) r - squared = 92.3% r - squared(adj) = 89.5% a) write the equation of the least squares regression line. b) interpret ( r^{2} ) in this context. c) interpret the equation in this context. d) this students 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? e) should she have expected this model to describe the rate for her pencils? why or why not?

Explanation:

Step1: Find the regression line equation

The general form of a simple linear regression equation is \( \hat{y}=b_0 + b_1x\), where \(b_0\) is the constant (intercept) and \(b_1\) is the slope. From the table, \(b_0 = 17.047\) (constant coefficient) and \(b_1=- 1.914\) (Time (hr) coefficient). So the equation is \( \hat{Length}=17.047-1.914\times Time\).

Step2: Interpret \(R^{2}\)

\(R^{2}=92.3\%\). In regression, \(R^{2}\) represents the proportion of the variance in the dependent variable (Length of the pencil) that is predictable from the independent variable (Time). So, 92.3% of the variation in the length of the pencil can be explained by the linear relationship with the number of hours the student worked.

Step3: Interpret the regression equation

The intercept \(17.047\) (when \(Time = 0\)) represents the initial length of the pencil (in cm) before any time (\(Time=0\) hours) was spent working. The slope \(-1.914\) means that for every additional hour of work, the length of the pencil is predicted to decrease by \(1.914\) cm.

Step4: Calculate the actual length (for part d)

The formula for the residual is \(Residual = Actual - Predicted\). First, find the predicted length. Substitute \(Time = 5\) into the regression equation: \(\hat{Length}=17.047-1.914\times5=17.047 - 9.57=7.477\) cm.
Since \(Residual=Actual - Predicted\), then \(Actual=Residual + Predicted\). Given \(Residual=-0.88\) cm, \(Actual=- 0.88+7.477 = 6.597\) cm.

Step5: Evaluate the model (for part e)

No. The model was built based on the student's #2 pencil. Different pencils (even if they are the same type - #2) may have different initial lengths, hardness (which affects the rate of shortening), and other characteristics. The relationship between time and length may not be the same for her pencil.

Answer:

a) \( \hat{Length}=17.047 - 1.914\times Time\)
b) 92.3% of the variation in the length of the pencil can be explained by the linear relationship with the number of hours the student worked.
c) The initial length of the pencil is 17.047 cm, and for every additional hour of work, the length of the pencil is predicted to decrease by 1.914 cm.
d) \(6.597\) cm
e) No. Different pencils may have different initial lengths, hardness, etc., so the relationship between time and length may not hold.