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 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 b0 = 0 and characterize the relationship between x and y with the following model: y = b1x + e The least squares estimate of b1 for this model is bn 1 = axiyi axi 2 From the records of past flour shipments, 15 shipments were randomly chosen, and the data shown in the table below were recorded. 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 b0 = 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 = b0 + b1x + e (i.e., do not restrict b0 to equal 0). Plot this line on the same scatterplot you constructed in part

a. c. Refer to part

b. Why might bn 0 be different from 0 even though the true value of b0 is known to be 0?

d. The estimated standard error of bn 0 is equal to s B 1 n + x2 SSxx Use the t-statistic t = bn 0 - 0 s211>n2 + 1x2>SSxx2 to test the null hypothesis H0: b0 = 0 against the alternative Ha: b0 ≠ 0. Use a = .10. Should you include b0 in your model? Critical Thinking Challenges