Question 1 simple linear regression (13 marks)Management of a soft-drink bottling company wants to...

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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 travel times within a particular route, another variable cost reflects the time required to unload the cases of soft drink at the delivery point. A sample of 20 deliveries within a territory was selected. The delivery time and the number of cases delivered were recorded. Develop a regression model to predict delivery time based on the number of cases delivered.a) Use the least-square method to calculate the regression coefficients, b0 and b1. Write your regression equation (1 mark).b) Interpret the meaning of b0 and b1 in this problem (2 marks).c) Predict the delivery time for 150 cases of soft drink (1 mark).d) Would it be appropriate to use to model to predict the delivery time for a customer who is receiving 500 cases of soft drink? Explain why (1.5 marks).e) Determine the coefficient of determination, r2, and explain it meaning in this problem (2 marks).f) Perform a residual analysis. Is there any evidence of patterns in the residuals? Explain (2 marks).g) At the 0.05 level of significance, is there evidence of a linear relationship between delivery time and the number of cases delivered? Explain (1.5 marks).h) Explain how the results in a to g can help allocate delivery costs to customers (1 mark).Question 2 multiple linear regressionA farmer who specialises in the production of carpet wool where the sheep are shorn twice per year is seeking a 75-mm-length clip from his Tukidale sheep. He believes that the proportion of sheep at each clip meeting this standard varies according to average rainfall during the six-month growing period and whether additional hand feeding of high protein sheep nits occurs during the period (because of a shortage of grass cover in the paddocks). Hand feeding is measured as 1 and no hand feeding as 0. a) Predict the proportion at 75mm if the rainfall is 180 mm and there is no hand feeding, a construct a 95% confidence interval estimate and 95% prediction interval.b) Is there a significant relationship between the clip length proportion and the two independent variables at the 0.05 level of significance?c) Construct 95% confidence interval estimates of the population slope for the relationship between clip proportion and rainfall, and between clip proportion and hand feeding.d) Interpret the meaning of the coefficient of multiple determination.e) Calculate the coefficients of partial determination and interpret their meaning.f) Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model.Question 3 business forecastingThe data contained in  are the number of males employed full time in a professional occupation in Australia from August quarter 1996 to August quarter 2008.a) Plot the series of data.b) Calculate linear trend equation and plot the trend line.c) What are your forecasts of male full-time professional employment in November quarter 2008 and February quarter 2009?d) Do you think it is reasonable to try to forecast employment in this way? Explain.

Pritam · Accepted Answer

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:

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...

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