Problem

Regression through the origin. Sometimes it is known from theoretical considerations that...

Regression through the origin. 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 y of a shipment of 50-pound bags of flour and the number x of bags in the shipment. Since 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 that 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 following table were recorded. These data are saved in the FLOUR file.

Weight of Shipment	Number of 50 Pound Bags in Shipment
5,050	100
10,249	205
20,000	450
7,420	150
24,685	500
10,206	200
7,325	150
4,958	100
7,162	150
24,000	500
4,900	100
14,501	300
28,000	600
17,002	400
16,100	400

a. Find the least squares line for the given data under the assumption that β0 = 0. Plot the least squares line on a scatterplot 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 scatterplot you constructed in part a.
c. Refer to part b. Why might be different from 0 even though the true value of β0 is known to be 0?
d. The estimated standard error of is equal to
Use the t –statistic
to test the null hypothesis H0:β0 = 0 against the alternative H0:β0 ≠ 0 Take α =.10. Should you include β0 in your model?