Question

The U.S. census questionnaire defines kitchens with complete facilities as those having a sink with piped water, a range, and a refrigerator. Homes that lack complete kitchen facilities have been rare in the United States for many years. The first census for which data were tabulated on this subject was in 1970. The table shows the percentage of housing units lacking complete kitchen facilities in the western United States.

Percent of Western U.S. Homes with Incomplete
Kitchens

Year	1970	1980	1990
Homes (%)	3	2	1

(a) Use the method of least squares to find the multivariable function f with inputs a and b for the best-fitting line

y = ax + b,

where x is years since 1970.
f(a, b) =

(b) Calculate the minimum value of

f(a, b).

Explain what this minimum value indicates about the relationship between and the best-fitting line.The minimum value of

f(a, b)

is  , which indicates that the line with parameters a =  and b =  passes through each data point.
(c) Write function of the linear model that best fits the data to give the percentage of homes with incomplete kitchens in the Western United States, where x is years since 1970, with data from

0 ≤ x ≤ 20.

h(x) =

(d) In what year does the best-fitting line predict that no housing units will lack complete kitchen facilities?

Answer 1

We are given the following table

(a) Since x is starting from 1970, so for first column it will be 0, for the second it will be 10 and for the third it will be 20. So putting the value of x in the standard equation of SSE,

(b) We need to take partial derivatives w.r.t. a and b, and equate to zero

As SSE equals zero, hence the best fitting line passes through each of the given points.

(c) The function is obtained by putting values of a and b in the regression line equation

(d) We need to equate above function to zero

x is years since 1970, hence best-fitting line predict that no housing units will lack complete kitchen facilities in 2000