Question

Consider the data set Table 1: Population in the United States, in thousands, for the last seven censuses.

(a) Construct the linear least squares approximation L(x)=a₀ +a₁ x for the population data. It is convenient let x equal the number of years since 1960 , so that 1960 corresponds to x=0,1970 corresponds to x=10, and so on.
(b) Use L(x) to predict the population of the United States in 2025 (in thousands).

Answer 1

To construct the linear least squares approximation, we need to find the equation of the line that best fits the given population data. We can use the method of least squares to minimize the sum of the squared differences between the actual data points and the predicted values on the line.

Step-by-step explanation:

To construct the linear least squares approximation, we need to find the equation of the line that best fits the given population data. We can use the method of least squares to minimize the sum of the squared differences between the actual data points and the predicted values on the line.

(a) To find the equation of the line, we need to calculate the slope and the y-intercept. The slope is given by: b = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2) and the y-intercept is given by: a = ȳ - bẍ, where ȳ is the mean of the y-values and ẍ is the mean of the x-values. We can then substitute these values into the equation of the line: L(x) = a + bx.

(b) To predict the population of the United States in 2025, we can substitute x = 65 (since 1960 corresponds to x = 0 and every 10 years corresponds to an increase of x by 1) into the equation obtained in part (a).