Question

Following is a simple linear regression model: yi = /alpha + /beta xi + /epsilon i The following results were obtained from some statistical software. R2 = 0.735 syx (regression standard error) = 5.137 n (total observations) = 60 Significance level = 0.05 = 5% Variable Parameter Estimate Std. Err. of Parameter Est. Interecpt 0.325 0.097 Slope of X -1.263 0.309

1. Write the fitted model. (I ALREADY KNOW THE ANSWER TO THIS. I LEFT IT INCASE IT IS NEEDED FOR THE OTHER ANSWERS.)

2. Make a prediction using the fitted model for y when x = 2.253 (I ALREADY KNOW THE ANSWER TO THIS. I LEFT IT INCASE IT IS NEEDED FOR THE OTHER ANSWERS.)

3. The intercept of the least-squares regression line is:

4. Suppose we want to test the hypotheses for the slope: H0: /beta = 0, H1:/beta not equal to 0 The value of the t statistic for this test is:

5. Suppose we want to test the hypotheses for the intercept: H0: /alpha = 0, H1:/alphanot equal to 0 The value of the t statistic for this test is:

6. A 95% confidence interval for the slope b in the simple linear regression model is:

7. A 95% confidence interval for the intercept a in the simple linear regression model is:

8. The correlation coefficient r between the x and y is:

9. What is its meaning of R2?

10. What is the meaning of the intercept in this simple linear regression model?

11. SST = ?

12. Are the intercept and slope significant at 5% significance level (You may use either hypothesis tests or confidence intervals to make your conclusions)?

Answer 1

1) Fitted model

yi = 0.325 - 1.263xi + /epsilon i

2) when x = 2.253

y = 0.325 - 1.263*2.253 = -2.52

3) The intercept of the least-squares regression line is: = 0.325

4) test the hypotheses for the slope:

H₀: = 0,

H₁: not equal to 0

test statistic t = / standard error = -1.263/0.309 = -4.087

critical value = t_0.025,59 = 2.001

Since |t| > critical value we reject null hypothesis and the result is significant.

5)

test the hypotheses for the intercept:

H₀: = 0,

H₁: not equal to 0

test statistic t = / standard error = 0.325/0.097 = 3.35

critical value = t_0.025,59 = 2.001

Since |t| > critical value we reject null hypothesis and the result is significant.

6) 95% confidence interval for the slope b

t_0.025,59 = 2.001

= (-1.263 - 2.001*0.309 , -1.263+ 2.001*0.309 ) = (-1.881, -0.645)

7) 95% confidence interval for the intercept a

= (0.325 - 2.001*0.097 , 0.325+ 2.001*0.097 ) = (-0.519, 0.131)

8. The correlation coefficient r between the x and y is: sqrt(R²) = (0.735)^0.5 = -0.857 (slope is negative)

9. What is its meaning of R²?

The model explains 73.5% variability between the data.

10. What is the meaning of the intercept in this simple linear regression model?

When the independent variable is zero then the value of dependent variable is intercept.

11. SST = ?

R^{2 =}

SST =S_yy = = 1.263 * 5.137 /0.735 = 8.83

12. From the hypothesis tests both intercept and slope are significant at 5% significance level.