Answer to Question #292002 in Statistics and Probability for Roba Jirma

Question

Researcher is using data for a sample of 10 observations to estimate the relation 

between consumption expenditure and income. Preliminary analysis of the sample data 

produces the following data. 

 ∑xy = 700 , 1000 2 ∑x = , ∑X = 100

 ∑Y = 200 

Where i x Xi X

__

= − and 

__

y = Yi − Y

a. Use the above information to compute OLS estimates of the intercept and slope 

coefficients and interpret the result 

b. Calculate the variance of the slope parameter 

c. Compute the value R

2

 (coefficient of determination) and interpret the result 

d. Compute 95% confidence interval for the slope parameter 

e. Test the significance of the slope parameter at 5% level of confidence using t-test

Answer 1

(a) Define Y^{^} = b₀ + b₁x

where b₀ is the y intercept and b₁ is the slope or regression coefficient of the line.

b₁ =  (∑xy - (∑X ∑Y)/n ) /( ∑X² - ( ∑X )²/n )

which yields

= (700 - (100 * 200)/10) / ( 1000 - (100)²/10)

= (-1300)/ ( 0 ) which is undefined

since b₁ is undefined,it makes it impossible to handle part b, c, d and e respectively.

b) variance of the slope = MSE/ ∑( X_I - X^- )²

c) coeeficient of determination = ( coefficient of correlation)²

d) t = b₁/s_b1 , where b₁ is the slope and s_b1 is the $square\,root$ of the variance of the slope

e) if t_calculated > t_tabulated ,then we conclude that the slope is significant.