Sometimes it is known from theoretical considerations that the straight-line relationship between two variables, x and y, passes through the origin of the xy-plane. Consider the relationship between the total weight of a shipment of 50-pound bags of flour, y, and the number of bags in the shipment, x. Because a shipment containing x = 0 bags (i.e., no shipment at all) has a total weight of y = 0, a straight-line model of the relationship between x and y should pass through the point x = 0, y = 0. In such a case, you could assume β0 = 0 and characterize the relationship between x and y with the following model:
y = β1x + ε
The least squares estimate of β1 for this model is

From the records of past flour shipments, 15 shipments were randomly chosen, and the data shown in the table below were recorded.

a. Find the least squares line for the given data under the assumption that β0 = 0. Plot the least squares line on a scatter plot of the data.
b. Find the least squares line for the given data using the model
y = β0 + β1x + ε
(i.e., do not restrict β0 to equal 0). Plot this line on the same scatter plot you constructed in part a.
c. Refer to part b. Why might 0 be different from 0 even though the true value of β0 is known to be 0?
d. The estimated standard error of 0 is equal to

Use the t-statistic

to test the null hypothesis H0: β0 = 0 against the alternative Ha: β0 ‰  0. Use α = .10. Should you include β0 in your model?

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Number of 50-Pound Bags in Shipment Weight of Shipment 5,050 10,249 20,000 7420 24.685 10,206 7325 4.958 7,162 24,000 4,900 14,501 28,000 17002 16,100 100 205 450 150 500 200 150 100 150 500 100 300 600 400 400