The U.S. census questionnaire defines kitchens with completefacilities as those having a sink with piped water, a range, and arefrigerator. Homes that lack complete kitchen facilities have beenrare in the United States for many years. The first census forwhich data were tabulated on this subject was in 1970. The tableshows the percentage of housing units lacking complete kitchenfacilities 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 multivariablefunction f with inputs a and b for thebest-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 relationshipbetween and the best-fitting line.The minimum value of
f(a, b)
is , which indicates that the line with parametersa = and b = passes througheach data point.
(c) Write function of the linear model that best fits the data togive the percentage of homes with incomplete kitchens in theWestern United States, where x is years since 1970, withdata from
0 ? x ? 20.
h(x) =
(d) In what year does the best-fitting line predict that no housingunits will lack complete kitchen facilities?
