Page 240 - Matrix Analysis & Applied Linear Algebra

4.6 Classical Least Squares 235

2
4.6.2. Show that the slope of the line that passes through the origin in and
comes closest in the least squares sense to passing through the points
2
{(x 1 ,y 1 ), (x 2 ,y 2 ),..., (x n ,y n )} is given by m = x i y i / x .
i i i
4.6.3. A small company has been in business for three years and has recorded
annual profits (in thousands of dollars) as follows.
Year 1 2 3
Sales 7 4 3
Assuming that there is a linear trend in the declining profits, predict the
year and the month in which the company begins to lose money.

4.6.4. An economist hypothesizes that the change (in dollars) in the price of a
loaf of bread is primarily a linear combination of the change in the price
of a bushel of wheat and the change in the minimum wage. That is, if B
is the change in bread prices, W is the change in wheat prices, and M
is the change in the minimum wage, then B = αW +βM. Suppose that
for three consecutive years the change in bread prices, wheat prices, and
the minimum wage are as shown below.
Year 1 Year 2 Year 3
B +$1 +$1 +$1
W +$1 +$2 0$

M +$1 0$ −$1
Use the theory of least squares to estimate the change in the price of
bread in Year 4 if wheat prices and the minimum wage each fall by $1.

4.6.5. Suppose that a researcher hypothesizes that the weight loss of a pint of
ice cream during storage is primarily a linear function of time. That is,
y = α 0 + α 1 t + ε,
where y = the weight loss in grams, t = the storage time in weeks, and
ε is a random error function whose mean value is 0. Suppose that an
experiment is conducted, and the following data is obtained.

Time (t) 1 2 3 4 5 6 7 8
Loss (y) .15 .21 .30 .41 .49 .59 .72 .83
(a) Determine the least squares estimates for the parameters α 0
and α 1 .
(b) Predict the mean weight loss for a pint of ice cream that is stored
for 20 weeks.