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 year 2030.
Year % of labor force
1990 11.4%
1995 12.8%
2000 14%
2005 14.8%
2007 16.2%
To plot the data points, we will use x as the number of years since 1990 and y as the percentage of the labor force that is 65 years old or older.
Year (x) % of labor force (y)
1990 (0) 11.4
1995 (5) 12.8
2000 (10) 14
2005 (15) 14.8
2007 (17) 16.2
a. We can plot the data points on a graph with x on the x-axis and y on the y-axis.
b. To find the linear regression equation of the line of best fit, we will use the formula y = mx + b, where m is the slope and b is the y-intercept. With the data provided, we can calculate the slope and y-intercept using the least squares method.
Using the formula:
n = 5 (number of data points)
∑x = 47 (sum of x-values)
∑y = 69.2 (sum of y-values)
∑xy = 670 (sum of the product of x and y values)
∑x^2 = 1070 (sum of squared x-values)
m = (n∑xy - ∑x∑y) / (n∑x^2 - (∑x)^2)
= (5*670 - 47*69.2) / (5*1070 - (47^2))
= (3350 - 3234.4) / (5350 - 2209)
= 115.6 / 3141
≈ 0.0368
To find b, we can substitute the values back into the formula:
b = (∑y - m∑x) / n
= (69.2 - 0.0368*47) / 5
= (69.2 - 1.7296) / 5
≈ 13.894
Therefore, the linear regression equation of the line of best fit is y = 0.0368x + 13.894.
c. The slope of this line indicates that for every year since 1990, the percentage of the labor force that is 65 years old or older increases by approximately 0.0368%.
d. To predict the percent of the labor force that is 65 years old or older in the year 2030, we can substitute the value of x as 40 (2030-1990) into the linear regression equation.
y = 0.0368*40 + 13.894
= 1.472 + 13.894
≈ 15.366%
Therefore, if the percentage continues to increase at the same rate, we can predict that approximately 15.366% of the labor force will be 65 years old or older in the year 2030.