The percentage of adults in the labor force ages 65 or older who are still working has risen since 1990. The table below shows the data from 1990 to 2007. Let x represent the number of years since 1990.
a. Plot the data points.
b. Find the linear regression equation of the line of best fit for the data.
c. What does the slope of this line indicate in this situation?
d. If the percentage continues to increase at the same rate, predict the percent of the labor force that is 65 years old or older in the year2027
The slope of this line indicates that the number of workers over 65 is
increasing
at the rate of
per year.
a. To plot the data points, we will use the data from the table given.
| Year (x) | Percentage |
|----------|------------|
| 1990 | 12.3 |
| 1995 | 13.4 |
| 2000 | 15.2 |
| 2005 | 16.4 |
| 2007 | 16.8 |
Each data point will be plotted with the year (x) on the horizontal axis and the percentage on the vertical axis.
b. To find the linear regression equation of the line of best fit for the data, we will use the least squares method. This will allow us to find the equation of a straight line that best fits the data.
Let's calculate the slope (m) and y-intercept (b) using the formula for linear regression:
m = (n * Σxy - ΣxΣy) / (n * Σx^2 - (Σx)^2)
First, let's calculate the necessary sums:
Σx = (1990 + 1995 + 2000 + 2005 + 2007) = 9977
Σy = (12.3 + 13.4 + 15.2 + 16.4 + 16.8) = 73.1
Σxy = (1990 * 12.3 + 1995 * 13.4 + 2000 * 15.2 + 2005 * 16.4 + 2007 * 16.8) = 128,281.5
Σx^2 = (1990^2 + 1995^2 + 2000^2 + 2005^2 + 2007^2) = 19,879,763
n = 5 (number of data points)
Now, substitute these values into the formula:
m = (5 * 128,281.5 - 9977 * 73.1) / (5 * 19,879,763 - (9977)^2)
After calculating, we find that m ≈ 0.3674.
To find b (the y-intercept), we can use the formula:
b = (Σy - m * Σx) / n
Substituting the values:
b = (73.1 - 0.3674 * 9977) / 5
After calculating, we find that b ≈ -727.366.
Therefore, the linear regression equation of the line of best fit is:
y = 0.3674x - 727.366
c. The slope of this line indicates that the number of workers over 65 is increasing at a rate of approximately 0.3674% per year. This means that for every year since 1990, the percentage of workers over 65 in the labor force has been increasing by about 0.3674%.
d. To predict the percentage of the labor force that is 65 years old or older in the year 2027, we can use the linear regression equation.
Let x = 2027 - 1990 = 37 (since 2027 is 37 years after 1990).
Using the equation y = 0.3674x - 727.366, we can substitute x = 37:
y = 0.3674 * 37 - 727.366
After calculating, we find that y ≈ 0.7.
Therefore, if the percentage continues to increase at the same rate, we can predict that approximately 0.7% of the labor force will be 65 years old or older in the year 2027.