1) Construct a scatter diagram
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 | WeeklyHours | WeeklyEarnings |
|---|---|---|
| 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:
Part I Solutions
1. There is an inverse relationship between Weekly earnings andWeekly hours. From the scatter plot it can be seen that as WeeklyHours increases, the weekly earnings drop.
2. The correlation value is -0.949 which is matching with thestatement made above. It is close to -1 which confirms that asweekly 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 aworker works for 33.8 hours, we use input the value of it in theequation 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 ofR square which is 0.900. In order to confirm what is R square, itis the square of the correlation coefficient which we calculated inquestion 2. So, we can say that with 90 % confidence that thecalculated weekly earning using the regression equation is close tothe 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 inpart 1 of the question, we have proved that weekly earning andweekly hours are inversely proportional to each other. Thus, if aworker wishes to increase weekly earning, a reduction in weeklyhours would be our recommendation.
4 b)
b) Yes, a decrease in hours worked cause an increase in weeklypay. Referring to part 1 and part 2, we can see that both weeklyearning and weekly hours are inversely proportional. So, with adecrease in weekly hours there will be an increase in weeklypay.
Part II
1.
The graph shows that there is a mild but constant drop in theweekly hours of work over the years which shows an inverserelationship between the two variables.
2. The correlation between the two variables is -0.948 whichconfirms the understanding of the previous answer. The value isclose to -1 which confirms that it is almost close to perfectlynegative 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 ofR square which is 0.898. In order to confirm what is R square, itis the square of the correlation coefficient which we calculated inquestion 2. So, we can say that with 89.8 % confidence that thecalculated weekly hours using the regression equation is close tothe actual value.
4. Assuming a linear correlation between these two variables,the average weekly hours in the future will decrease. This willhappen because the trend of the past shows that with increasingtime, there has been almost a constant rise in the Weekly hours.This trend would continue indefinitely if we extend the samescatter plot for future assuming that the data is behaving the sameway as the previous years.
Part III
1.
From the above diagram, we can see that there is a positivecorrelation between Year and the Weekly earnings. This means that,as we progress from 1967 to 1996, there has been an increase in theweekly wages.
2. The correlation coefficient between Year and Weekly earningsis 0.996. This is very close to +1 which means that there is apositive correlation between Year and Weekly earnings. Thisconfirms the graphical understanding which we discussed in previousquestion.
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 ofR square which is 0.993. In order to confirm what is R square, itis the square of the correlation coefficient which we calculated inquestion 2. So, we can say that with 99.3 % confidence that thecalculated weekly earning using the regression equation is close tothe actual value.
4. Assuming a linear correlation between these two variables,the average weekly earnings in the future will increase. This willhappen because the trend of the past shows that with increasingtime, there has been almost a constant rise in the Weekly earnings.This trend would continue indefinitely if we extend the samescatter plot for future assuming that the data is behaving the sameway as the previous years.
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