63. Let x denote the number of items in an order and y denote time (min) necessary to process the order.

Processing time may be determined by various factors other than order size. So for any particular value of x, we now regard the value of total production time as a random variable Y. Consider the following data obtained by specifying various values of x and determining total production time for each one.

x 10 15 18 20 25 27 30 35 36 40 y 301 455 533 599 750 810 903 1054 1088 1196

a. Plot each observed (x, y) pair as a point on a twodimensional coordinate system with a horizontal

xi



f 1x; u2  e u x  1 11  u22ux x  0, 1, 2, . . .

axis labeled x and vertical axis labeled y. Do all points fall exactly on a line passing through (0, 0)? Do the points tend to fall close to such a line?

b. Consider the following probability model for the data. Values x1, x2, . . . , xn are speci ed, and at each xi we observe a value of the dependent variable y. Prior to observation, denote the y values by Y1, Y2, . . . , Yn , where the use of uppercase letters here is appropriate because we are regarding the y values as random variables. Assume that the Yi s are independent and normally distributed, with Yi having mean value bxi and variance s2.

That is, rather than assume that y  bx, a linear function of x passing through the origin, we are assuming that the mean value of Y is a linear function of x and that the variance of Y is the same for any particular x value. Obtain formulas for the maximum likelihood estimates of b and s2, and then calculate the estimates for the given data. How would you interpret the estimate of b?

What value of processing time would you predict when x  25? Hint: The likelihood is a product of individual normal likelihoods with different mean values and the same variance. Proceed as in the estimation via maximum likelihood of the parameters m and s2 based on a random sample from a normal population distribution (but here the data does not constitute a random sample as we have previously de ned it, since the Yi s have different mean values and therefore don t have the same distribution). Note: This model is referred to as regression through the origin.