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 for the population data, use the formulas a⁰ = ((∑y)(∑x²) - (∑x)(∑xy)) / (n(∑x²) - (∑x)²) and a¹ = (n(∑xy) - (∑x)(∑y)) / (n(∑x²) - (∑x)²). Then use the equation L(x) = a⁰ + a¹x to approximate the population for a given year. To predict the population of the United States in 2025, substitute x = 65 into the equation L(x) = a⁰ + a¹x.

Step-by-step explanation:

To construct the linear least squares approximation for the population data, we will use the given data set Table 1: Population in the United States for the last seven censuses. Let x represent the number of years since 1960. To find the coefficients a⁰ and a¹, we can use the formulas:

`a⁰ = ((∑y)(∑x²) - (∑x)(∑xy)) / (n(∑x²) - (∑x)²)`

`a¹ = (n(∑xy) - (∑x)(∑y)) / (n(∑x²) - (∑x)²)`

Once we have the values of a⁰ and a¹, we can use the equation L(x) = a⁰ + a¹x to approximate the population for any given year.

To predict the population of the United States in 2025, we substitute x = 2025 - 1960 = 65 into the equation L(x) = a⁰ + a¹x.