Question

For the following problems, do not use statistical software such as SPSS, SAS, R and so on.

Please nish your work only using basic calculator and describe how you get your answer.

X	3	6	8	2	5	1	5
Y	4	5	6	2	5	0	3

(a) Compute the sample covariance of X and Y and explain how X and Y are related.

(b) Compute the sample correlation coecient and explain its meaning.

(c) Consider the simple linear regression model Yi = β0 + β1Xi + εi
and t the model to the data.

(d) Compute R^2 and (hat)σ2.

(e) Find a 95% CI on the slope.

Answer 1

a) Sample covariance is given by

On substitution we have

Since covariance is positive the variables are directly related ie as one increases the other one will also increase.

Standard deviation of X,S_x is

and

b) Sample correlation coefficient is given by the formula ,

Correlation coefficient measure  the strength of linear relation between the variable. Here  the variables are highly correlated. ie a high positive correlation exist between X and Y

c)The formula for linear regression of y on x is

where

Hence the regression line is

d) R_squared value =r² =0.89²=0.791