Question

The table to the right lists the average number of hours worked in a week and the average weekly earnings for U.S. production workers from 1967 to 1996. (The World Almanac 1998)

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. 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? c) What other variables could contribute to an increase in weekly pay?

Part 2 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 3

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.

data:

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

1.

From scatter plot we see that Weekly Earnings and Weekly hours are negatively correlated and there is a linear association between Weekly Earnings and Weekly hours exist.

(2)

Pearson correlation of Weekly Hours and Weekly Earnings = -0.949

which implies there is strong negative correlation between Weekly Earnings and Weekly hours and this is also observed from scatter plot.

(3)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.948699232
R Square	0.900030233
Adjusted R Square	0.896459885
Standard Error	31.7315281
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	1	253821.5111	253821.5111	252.0847	1.56287E-15
Residual	28	28192.91651	1006.889875
Total	29	282014.4276
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3155.066144	183.0926755	17.23207188	1.94E-16	2780.0178	3530.114488
X Variable 1	-81.60097765	5.139514981	-15.87717478	1.56E-15	-92.1287968	-71.0731585

linear regression model Predicted Weekly Earnings=3155.0661+-81.6010*Weekly Hours

Since p-value of F-test <0.05 so the regression equation is significant and R²=0.9000 i.e. 90% of total variation in Weekly Earnings is explained by this regression equation so this regression equation is well fitted model.

So when Weekly Hours= 33.8 hours, then Predicted Weekly Earnings=3155.0661-81.6010*33.8=$396.95.

(4)

a) For a production worker who wishes to increase weekly earnings, we would recommend a decrease in hours worked per week. Since the Weekly Earnings and Weekly hours is strong negatively correlated.

Part 2

(1)

From scatter plot we see that Year and Weekly hours are negatively correlated and there is a linear association between Year and Weekly hours exist.

(2)

Pearson correlation of Year and Weekly Hours = -0.948

which implies there is strong negative correlation between Year and Weekly hours and this is also observed from scatter plot.

(3)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.947736825
R Square	0.89820509
Adjusted R Square	0.894569557
Standard Error	0.372265566
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	1	34.23838042	34.23838042	247.0629	2.01528E-15
Residual	28	3.880286244	0.138581652
Total	29	38.11866667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	280.175343	15.5597006	18.00647391	6.25E-17	248.3027411	312.0479448
X Variable 1	-0.123426029	0.007852411	-15.71823365	2.02E-15	-0.13951096	-0.10734109

linear regression model Predicted Weekly Hours=280.1753--0.1234*Year

Since p-value of F-test <0.05 so the regression equation is significant and R²=0.8982 i.e. 89.82% of total variation in Weekly Hours is explained by this regression equation so this regression equation is well fitted model.

4) Since we have no idea about the relationship between these two variables outside the range so this model can't be used outside the range i.e. for future prediction.

Part 3:

(1)

From scatter plot we see that Year and Weekly Earnings are positively correlated and there is a linear association between Year and Weekly earnings exist.

(2)

Pearson correlation of Year and Weekly Earnings = 0.996

which implies there is strong positive correlation between Year and Weekly earnings and this is also observed from scatter plot.

(3)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.996324287
R Square	0.992662084
Adjusted R Square	0.992400016
Standard Error	8.596922303
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	1	279945.0295	279945.0295	3787.798	1.96786E-31
Residual	28	2069.398046	73.90707308
Total	29	282014.4276
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-21865.14999	359.3282575	-60.85007102	2.7E-31	-22601.2006	-21129.0994
X Variable 1	11.16057397	0.181339808	61.54508542	1.97E-31	10.78911621	11.53203173

Weekly Earnings=-21865.15+11.1606*Year

R²=0.9927 i.e. 99.27% of total variation of weekly earning is explained by this regression linear equation.

(4)

Since we have no idea about the relationship between these two variables outside the range so this model can't be used outside the range i.e. for future prediction.