Find the sample correlation coefficient between x and y how

Question

A CONCLUDING EXAMPLE: RELATIONSHIP BETWEEN WAGES AND PRODUCTIVITY IN THE U.S. BUSINESS SECTOR, 1959-2006

According to the marginal productivity theory of microeconomics, we would expect a positive relationship between wages and worker productivity. To see if this so, in Table we provide data on labor productivity, as measured by the index of output per hour of all persons, and wages, as measured by the index of real compensation per hour, for the business sector of the U.S. economy for the period 1959 to 2006. The base year of the index is 1992 and hourly real compensation is hourly compensation divided by the consumer price index (CPI).

Let Compensation (Y) = index of real compensation and Productivity (X) = index of output per hour of all persons. Plotting these data, we obtain the scatter diagram shown in Figure.

This figure shows a very close linear relationship between labor productivity and real wages. Therefore, we can use a (bivariate) linear regression to model the data given in Table. Using EViews, we obtain the following results:

Dependent Variable: Compensation Method: Least Squares Sample: 1959 2006
Included observations: 48
	Coefficient	Std. Error	t-Statistic	Prob.
C	33.63603	1.400085	24.02428	0.0000
Productivity	0.661444	0.015640	42.29178	0.0000
R-squared	0.974926
Adjusted R-squared	0.974381
S.E. of regression	2.571761
Sum squared resid	304.2420
Durbin-Watson stat	0.146315

Let us interpret the results. The slope coefficient of about 0.66 suggests that if the index of productivity goes up by a unit, the index of real wages will go up, on average, by 0.66 units. This coefficient is highly significant, for the t value of about 42.3 (obtained under the assumption that the true population coefficient is zero) is highly significant for the p value is almost zero. The intercept coefficient, C, is also highly significant, for the p value of obtaining a t value for this coefficient of as much as about 24 is practically zero.

The R² value of about 0.97 means that the index of productivity explains about 97 percent of the variation in the index of real compensation. This is a very high value, since an R² can at most be 1.

1. Find the sample correlation coefficient between X and Y.

2. How would you decide if a simple linear regression model is appropriate for the relationship between X and Y?

3. If a simple linear regression model is indeed appropriate for the relationship, find the estimated regression line.  And interpret this line.

4. Find the predicted compensation with the productivity of 100.  What is the residual for the observation of the base year, 1992?

5. Is there strong evidence that the compensation is linearly related to productivity? Justify your answer.

6. Find a 95% confidence interval for the slope parameter B₂. 

7. Is there strong evidence that B₂ is greater than 0.65?  Justify your answer using the level of significance α = 0.05.

8. Find an estimate for the variance of the error in the simple linear regression model.

9. Find the 90% confidence interval for the mean compensation when the productivity is 85 and interpret the C.I.

10. Find the 90% confidence interval for the compensation of a year when the productivity is 85 and interpret the C.I.

11. Check the normality assumption.