Question

Using the table of data at the bottom.

Part One:

1) Construct a scatter diagram and comment on the relationship, if any, between the variables Weekly Hours and Weekly Earnings.

2) Determine and interpret the correlation for hours worked and earnings. The CORREL function in Excel will be helpful. Based upon the value of the correlation, is your answer to the previous question reasonable?

3) Based upon the data given, estimate the average weekly earnings for a workweek of 33.8 hours. How confident are you in your estimate? You should use a linear regression model to make your prediction. To create the linear regression model in Excel, right-click on a data point and click Add Trendline... In the options that display on the right, click Display Equation on chart.

4) Increase/decrease in weekly hours:

a) For a production worker who wishes to increase weekly earnings, would you recommend a decrease in hours worked per week? Why or why not?

b) Does a decrease in hours worked cause an increase in weekly pay?

Part Two

Using the same table of data:

1) Construct a scatter diagram and comment on the relationship, if any, between the variables Year and Hours Worked.

2) Determine and interpret the correlation for the year and hours worked. Based upon the value of the correlation, is your answer to the previous question reasonable?

3) Based upon the data given, estimate the average weekly hours worked this year. How confident are you in your estimate? You should use a linear regression model to make your prediction.

4) Assuming a linear correlation between these two variables, what will happen to the average weekly hours worked in the future? Is it possible for this pattern to continue indefinitely? Explain.

Part Three:

1) Construct a scatter diagram and comment on the relationship, if any, between the variables Year and Weekly Earnings.

2) Determine and interpret the correlation for the year and weekly earnings. Based upon the value of the correlation, is your answer to the previous question reasonable?

3) Based upon the data given, estimate the average weekly earnings this year. How confident are you in your estimate? You should use a linear regression model to make your prediction.

4) Assuming a linear correlation between these two variables, what will happen to the average weekly earnings in the future? Is it possible for this pattern to continue indefinitely? Explain.

Year	Weekly Hours	Weekly Earnings
1967	38.0	$101.84
1968	37.8	$107.73
1969	37.7	$114.61
1970	37.1	$119.83
1971	36.9	$127.31
1972	37.0	$136.90
1973	36.9	$145.39
1974	36.5	$154.76
1975	36.1	$163.53
1976	36.1	$174.45
1977	36.0	$189.00
1978	35.8	$203.70
1979	35.7	$219.91
1980	35.3	$235.10
1981	35.2	$255.20
1982	34.8	$267.26
1983	35.0	$280.70
1984	35.2	$292.86
1985	34.9	$299.09
1986	34.8	$304.85
1987	34.8	$312.50
1988	34.7	$322.02
1989	34.6	$334.24
1990	34.5	$345.35
1991	34.3	$353.98
1992	34.4	$363.61
1993	34.5	$373.64
1994	34.7	$385.86
1995	34.5	$394.34
1996	34.4	$406.26

Answer 1

Part I Solutions

1. There is an inverse relationship between Weekly earnings and Weekly hours. From the scatter plot it can be seen that as Weekly Hours increases, the weekly earnings drop.

2. The correlation value is -0.949 which is matching with the statement made above. It is close to -1 which confirms that as weekly hours increase, weekly earnings drop.

3. The regression equation for Weekly earning is as follows:

Weekly Earning = 3155.066 - 81.601 (Weekly Hours). So, if a worker works for 33.8 hours, we use input the value of it in the equation which becomes :

Weekly Earning = 3155.066 - 81.601(33.8) = 3155.066 - 2739.8618 = $415.2042.

To report the confidence value , we have to look at the value of R square which is 0.900. In order to confirm what is R square, it is the square of the correlation coefficient which we calculated in question 2. So, we can say that with 90 % confidence that the calculated weekly earning using the regression equation is close to the actual value.

4. a)

For a production worker who wishes to increase weekly earnings, we would recommend a decrease in hours worked per week because in part 1 of the question, we have proved that weekly earning and weekly hours are inversely proportional to each other. Thus, if a worker wishes to increase weekly earning, a reduction in weekly hours would be our recommendation.

4 b)

b) Yes, a decrease in hours worked cause an increase in weekly pay. Referring to part 1 and part 2, we can see that both weekly earning and weekly hours are inversely proportional. So, with a decrease in weekly hours there will be an increase in weekly pay.

Part II

1.

The graph shows that there is a mild but constant drop in the weekly hours of work over the years which shows an inverse relationship between the two variables.

2. The correlation between the two variables is -0.948 which confirms the understanding of the previous answer. The value is close to -1 which confirms that it is almost close to perfectly negative correlation.

3. The regression equation for Weekly earning is as follows:

Weekly Hours  = 280.175 - 0.173 (Year)

To report the confidence value , we have to look at the value of R square which is 0.898. In order to confirm what is R square, it is the square of the correlation coefficient which we calculated in question 2. So, we can say that with 89.8 % confidence that the calculated weekly hours using the regression equation is close to the actual value.

4. Assuming a linear correlation between these two variables, the average weekly hours in the future will decrease. This will happen because the trend of the past shows that with increasing time, there has been almost a constant rise in the Weekly hours. This trend would continue indefinitely if we extend the same scatter plot for future assuming that the data is behaving the same way as the previous years.

Part III

1.

From the above diagram, we can see that there is a positive correlation between Year and the Weekly earnings. This means that, as we progress from 1967 to 1996, there has been an increase in the weekly wages.

2. The correlation coefficient between Year and Weekly earnings is 0.996. This is very close to +1 which means that there is a positive correlation between Year and Weekly earnings. This confirms the graphical understanding which we discussed in previous question.

3.

The regression equation for Weekly earning is as follows:

Weekly Earning = -21865.150 +11.161 (Year)

To report the confidence value , we have to look at the value of R square which is 0.993. In order to confirm what is R square, it is the square of the correlation coefficient which we calculated in question 2. So, we can say that with 99.3 % confidence that the calculated weekly earning using the regression equation is close to the actual value.

4. Assuming a linear correlation between these two variables, the average weekly earnings in the future will increase. This will happen because the trend of the past shows that with increasing time, there has been almost a constant rise in the Weekly earnings. This trend would continue indefinitely if we extend the same scatter plot for future assuming that the data is behaving the same way as the previous years.