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 2012
.
Year 1990 1995 2000 2005 2007
% of Labor Force 11.4% 12% 13% 14.1% 15.2%
b. The linear regression equation is y=
(Type an expression using x as the variable. Use integers or decimals for any numbers in the expression. Round to the nearest thousandth as needed.)
a. Plotting the data points, with x representing the number of years since 1990 and y representing the percentage of adults in the labor force ages 65 or older who are still working:
| Year | x | % of Labor Force | y |
|------|---|-----------------|---|
| 1990 | 0 | 11.4% | 11.4 |
| 1995 | 5 | 12% | 12 |
| 2000 | 10 | 13% | 13 |
| 2005 | 15 | 14.1% | 14.1 |
| 2007 | 17 | 15.2% | 15.2 |
b. To find the linear regression equation of the line of best fit, we can use the least squares method.
Let's calculate the slope and y-intercept:
First, we need to find the means of x and y:
mean(x) = (0 + 5 + 10 + 15 + 17) / 5 = 9.4
mean(y) = (11.4 + 12 + 13 + 14.1 + 15.2) / 5 = 13.54
Next, we calculate the deviations from the means:
xi - mean(x): -9.4, -4.4, 0.6, 5.6, 7.6
yi - mean(y): -2.14, -1.54, -0.54, 0.56, 1.66
Now we calculate the sums of the products of the deviations:
Σ((xi - mean(x)) * (yi - mean(y))) = (-9.4 * -2.14) + (-4.4 * -1.54) + (0.6 * -0.54) + (5.6 * 0.56) + (7.6 * 1.66) = 77.716
Next, we calculate the sums of the squared deviations of x:
Σ((xi - mean(x))^2) = (-9.4)^2 + (-4.4)^2 + 0.6^2 + 5.6^2 + 7.6^2 = 236.8
Now we can calculate the slope:
slope = Σ((xi - mean(x)) * (yi - mean(y))) / Σ((xi - mean(x))^2) = 77.716 / 236.8 = 0.328
Finally, let's calculate the y-intercept:
y-intercept = mean(y) - slope * mean(x) = 13.54 - 0.328 * 9.4 = 10.223
Therefore, the linear regression equation of the line of best fit is:
y = 0.328x + 10.223
c. The slope of this line indicates the rate of increase in the percentage of the labor force that is 65 years old or older per year. In this situation, for each additional year since 1990, the percentage of adults in the labor force ages 65 or older who are still working increases by approximately 0.328%.
d. To predict the percent of the labor force that is 65 years old or older in the year 2012, which is 22 years after 1990, we can use the linear regression equation:
x = 22
y = 0.328x + 10.223
y = 0.328 * 22 + 10.223
y ≈ 17.591
Therefore, we predict that the percent of the labor force that is 65 years old or older in the year 2012 is approximately 17.591%.