Question 1 simple linear regression (13 marks)Management of a soft-drink bottling company wants to develop a method for allocating delivery costs to customers. Although one costs clearly relates to...

Question 1:
a) The regression equation is given by
Delivery time = b0 + b1*Number of cases
Which can be solved via data analysis in excel and hence the equation with estimated coefficients is given below.
Delivery time = 24.83 + 0.14*Number of cases
b) Interpretation of b0: The average delivery time when number of cases is zero is b0 units
Interpretation of b1: The average change in delivery time for unit change in average number of cases is b1 units. Here in this case, we can say that for unit change of number of cases the delivery time would have been increased by 0.14 units.
c) The delivery time for 150 cases of soft drink using the above equation is given below.
Delivery time = 24.83 + 0.14 * 150 = 45.83
d) No, it won’t be appropriate to predict the delivery time for the number of cases being 500.
It is a case of extrapolation where the independent value is taken outside the range of the sample data used in the analysis of regression model. The regression model requires some assumptions to hold for the estimation of the parameters b0 and b1. In case of extrapolation, it may happen that the data don’t follow assumption and hence the estimated parameters may come out be biased. So, the extrapolation may cause some unreliable results and hence should not be done.
e) The coefficient of determination or R-squared may be defined as the measure of goodness of fit of the model. More precisely, the R-squared actually explains how good the data fits the model.
In our case, since the R-squared is 0.9717, we can say that almost 97% of the variance in the dependent variable, delivery time, is explained by the independent variable, number of cases.
f) The residuals are nothing but the deviations of the predicted value from the actual values. The residuals for each data point are shown below.
For the residual one should expect the points to be spread and scattered randomly. There should not be any pattern, otherwise the linearity assumption is not satisfied. In this case we can see that the points are spread out equally around the horizontal axis. Hence no pattern is visible for the residuals and thus it is good for the model itself.
    Observation
    Predicted Delivery Time
    Residuals
     1
    32.11589876
    -0.0159
    2
    33.79621441
    1.003786
    3
    35.05645114
    1.143549
    4
    36.73676679
    1.063233
    5
    38.13702983
    -0.33703
    6
    39.25724026
    0.44276
    7
    41.07758221
    -2.57758
    8
    41.77771373
    0.122286
    9
    44.85829242
    -0.65829
    10
    46.81866068
    0.281339
    11
    47.37876589
    -4.37877
    12
    50.59937089
    -1.19937
    13
    53.11984436
    4.080156
    14
    55.36026522
    1.439735
    15
    58.86092282
    1.739077
    16
    60.40121217
    0.798788
    17
    62.22155412
    -4.02155
    18
    63.34176455
    -0.24176
    19
    65.0220802
    0.57792
    20
    66.56236954
    0.73763
    
g) From the ANOVA table we can find that the p-value for the F-statistic is less than 0.05 and hence the null hypothesis that b1 not equal to zero can be rejected under 95% confidence and hence we can say that there is enough evidence that the slope coefficient is not equal to zero and hence there must be some linear relationship between the predictor and the response variable.
h) From the results obtained in a to g we can see the fact there is a strong linear relationship between number of cases and the delivery time. Hence, we can say that since a greater number of cases are associated with more delivery time, the corresponding price should be increased in those cases where there are a greater number of cases to be covered.
Question 2:
a) From the regression results we can find the predicted value of...