Question 3 (Multiple Regression, multi-collinearity) A shipping company is inter
ID: 3255730 • Letter: Q
Question
Question 3 (Multiple Regression, multi-collinearity)
A shipping company is interested in determining the factors that affect the amount of time it takes to unload freight. They consider that two of the most important factors are the size of the items being unloaded, and the weight of those items. The table below shows the time it took to unload 10 randomly selected items on one day, as well as their corresponding size and weight.
Time (Minutes):
2.5
4
3.5
7
6
4.5
6
5
2
5.5
Size (cubic Metres):
1.7
2.3
2.4
3.2
3
2.5
3.5
2.9
0.9
2.6
Weight (00's kg):
6
11.2
11.4
13.6
10.1
8
14.3
11.8
5.1
9.8
Study the correlation matrix (Figure 1) and regression output (Figure 2) based on the above data and answer the question that follows.
Figure 1 Correlation Matrix
Time (Minutes):
Size (cubic Metres):
Weight (00's kg):
Time (Minutes):
1
Size (cubic Metres):
0.919155
1
Weight (00's kg):
0.785862
0.88635
1
Figure 2 Regression Output
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.921261
R Square
0.848722
Adjusted R Square
0.8055
Standard Error
0.711125
Observations
10
ANOVA
df
SS
MS
F
Significance F
Regression
2
19.86011
9.930053
19.63628
0.001347
Residual
7
3.539894
0.505699
Total
9
23.4
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
-0.20018
0.84353
-0.23731
0.819212
-2.19481
1.794453
Size (cubic Metres):
2.211198
0.676122
3.270412
0.013667
0.612423
3.809974
Weight (00's kg):
-0.07185
0.169626
-0.42356
0.684594
-0.47295
0.329255
Based on the above regression output, interpret the regression coefficients for each independent variable and comment on their statistical significance.
Discuss fully if there is any discrepancy between the correlation matrix and regression coefficients.
Question 4
(a) (Seasonal Indices)
An analyst has been contracted to forecast the number of tourist arrivals in Australia for the June quarter 2008. The information is required for airline scheduling and hotel bookings. The quarterly data for tourist arrivals in Australia from 2004 to 2007 are given below.
Tourist Arrivals (‘000)
Year
March
June
September
December
2004
6207
5974
6712
6603
2005
6377
6299
6962
6882
2006
6740
6636
7306
7428
2007
7315
7043
7845
7822
After conducting the trend analysis, the analyst obtained the trend estimate for each quarter.
Year
Qtr
Tourist Arrivals ('000)
Trend estimate
2004
1
6207
6113.824
2
5974
6216.572
3
6712
6319.321
4
6603
6422.069
2005
1
6377
6524.818
2
6299
6627.566
3
6962
6730.315
4
6882
6833.063
2006
1
6740
6935.812
2
6636
7038.56
3
7306
7141.309
4
7428
7244.057
2007
1
7315
7346.806
2
7043
7449.554
3
7845
7552.303
4
7822
7655.051
Using the above information, calculate by hand the seasonal indices for the Tourist Arrival data and interpret the seasonal indices for the June and December quarters.
(b) (Forecast)
The following trend line and seasonal indices were calculated from four weeks of daily sales.
Trend line: yt = 200 + 1.8t
Seasonal Effect:
Time (Minutes):
2.5
4
3.5
7
6
4.5
6
5
2
5.5
Size (cubic Metres):
1.7
2.3
2.4
3.2
3
2.5
3.5
2.9
0.9
2.6
Weight (00's kg):
6
11.2
11.4
13.6
10.1
8
14.3
11.8
5.1
9.8
Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday Seasonal Index (SI 1.6 0.4 0.3 0.3 1.4 07 1.9Explanation / Answer
Answer:
Multiple questions. Q3 answered.
Question 3 (Multiple Regression, multi-collinearity)
A shipping company is interested in determining the factors that affect the amount of time it takes to unload freight. They consider that two of the most important factors are the size of the items being unloaded, and the weight of those items. The table below shows the time it took to unload 10 randomly selected items on one day, as well as their corresponding size and weight.
Time (Minutes):
2.5
4
3.5
7
6
4.5
6
5
2
5.5
Size (cubic Metres):
1.7
2.3
2.4
3.2
3
2.5
3.5
2.9
0.9
2.6
Weight (00's kg):
6
11.2
11.4
13.6
10.1
8
14.3
11.8
5.1
9.8
Study the correlation matrix (Figure 1) and regression output (Figure 2) based on the above data and answer the question that follows.
Figure 1 Correlation Matrix
Time (Minutes):
Size (cubic Metres):
Weight (00's kg):
Time (Minutes):
1
Size (cubic Metres):
0.919155
1
Weight (00's kg):
0.785862
0.88635
1
Figure 2 Regression Output
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.921261
R Square
0.848722
Adjusted R Square
0.8055
Standard Error
0.711125
Observations
10
ANOVA
df
SS
MS
F
Significance F
Regression
2
19.86011
9.930053
19.63628
0.001347
Residual
7
3.539894
0.505699
Total
9
23.4
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
-0.20018
0.84353
-0.23731
0.819212
-2.19481
1.794453
Size (cubic Metres):
2.211198
0.676122
3.270412
0.013667
0.612423
3.809974
Weight (00's kg):
-0.07185
0.169626
-0.42356
0.684594
-0.47295
0.329255
Based on the above regression output, interpret the regression coefficients for each independent variable and comment on their statistical significance.
When size increases by 1 cubic meter, the time increases by 2.211198 Minutes.
When weight increases by one unit( 100kg), the time decreases by 0.07185Minutes.
Test for size, calculated t=3.270412, P=0.013667 which is < 0.05 level. Size is significant.
Test for weight, calculated t=-0.42356, P=0.684594 which is > 0.05 level. weight is not significant.
Discuss fully if there is any discrepancy between the correlation matrix and regression coefficients.
Correlation between time and weight is positive whereas the regression coefficient of weight is negative. This is the discrepancy in the result. This may be high correlation between size and weight.
This is a problem of multi-collinearity.
Time (Minutes):
2.5
4
3.5
7
6
4.5
6
5
2
5.5
Size (cubic Metres):
1.7
2.3
2.4
3.2
3
2.5
3.5
2.9
0.9
2.6
Weight (00's kg):
6
11.2
11.4
13.6
10.1
8
14.3
11.8
5.1
9.8
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