Question

Consider the data.

xi	3	12	6	20	14
yi	65	35	50	15	20

The estimated regression equation for these data is

ŷ = 70 − 3x.

(a)

Compute SSE, SST, and SSR using equations

SSE = Σ(y_i − ŷ_i)²,

SST = Σ(y_i − y)²,

and

SSR = Σ(ŷ_i − y)².

SSE=SST=SSR=

(b)

Compute the coefficient of determination

r².

(Round your answer to three decimal places.)

r²

=

Comment on the goodness of fit. (For purposes of this exercise, consider a proportion large if it is at least 0.55.)

The least squares line did not provide a good fit as a small proportion of the variability in y has been explained by the least squares line.The least squares line provided a good fit as a small proportion of the variability in y has been explained by the least squares line.    The least squares line did not provide a good fit as a large proportion of the variability in y has been explained by the least squares line.The least squares line provided a good fit as a large proportion of the variability in y has been explained by the least squares line.

(c)

Compute the sample correlation coefficient. (Round your answer to three decimal places.)

Answer 1

The estimated regression equation for the above data is  .

Here, n is the total number of observations and is the sample mean of the variable y.

(a)

is the sum of squares due to error.

We know that  

For example,

Simlarly continuing for all the observations, we get the below column of  .

3	65	61	16
12	35	34	1
6	50	52	4
20	15	10	25
14	20	28	64

is the sum of squares due to regression.

is the total sum of squares.

We can see that  SST = SSR + SSE.

Note: The equations you have mentioned above have slight changes and they are wrong. I have given the correct solution.

(b)

Coefficient of determination,

The least squares line did not provide a good fit as a small proportion of the variability in y has been explained by the least squares line. ( as )

(c)

Sample correlation coefficient between X and Y is given by,

Sample correlation coefficient between X and Y is given by,

,

Where Cov (X,Y) is the sample covariance given by

  

Here, n – sample size = 5

Where and  

–

–

= - 0.968

Hence, the sample correlation coefficient is -0.968, i.e., X and Y are strongly negatively correlated.