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
To answer questions a, b, and c, we need to calculate the slope and intercept of the line of best fit using linear regression.
The given data is as follows:
Year (x) | Percentage of Adults Still Working (y)
-------------------------------------------
1990 | 12.1
1995 | 12.9
2000 | 14.4
2005 | 16.3
2007 | 17.4
a. Plotting the data points on a graph:
Year (x) on the x-axis and Percentage of Adults Still Working (y) on the y-axis.
(1990, 12.1)
(1995, 12.9)
(2000, 14.4)
(2005, 16.3)
(2007, 17.4)
b. To find the linear regression equation of the line of best fit, we need to calculate the slope and intercept.
First, we calculate the means of x and y:
mean(x) = (1990 + 1995 + 2000 + 2005 + 2007)/5 = 1999.4
mean(y) = (12.1 + 12.9 + 14.4 + 16.3 + 17.4)/5 = 14.62
Next, we calculate the deviations of x and y from their means:
deviations of x = x - mean(x)
deviations of y = y - mean(y)
For (1990, 12.1):
deviation of x = 1990 - 1999.4 = -9.4
deviation of y = 12.1 - 14.62 = -2.52
For (1995, 12.9):
deviation of x = 1995 - 1999.4 = -4.4
deviation of y = 12.9 - 14.62 = -1.72
For (2000, 14.4):
deviation of x = 2000 - 1999.4 = 0.6
deviation of y = 14.4 - 14.62 = -0.22
For (2005, 16.3):
deviation of x = 2005 - 1999.4 = 5.6
deviation of y = 16.3 - 14.62 = 1.68
For (2007, 17.4):
deviation of x = 2007 - 1999.4 = 7.6
deviation of y = 17.4 - 14.62 = 2.78
Next, we calculate the sum of the products of the deviations:
sum of (deviations of x * deviations of y) = (-9.4 * -2.52) + (-4.4 * -1.72) + (0.6 * -0.22) + (5.6 * 1.68) + (7.6 * 2.78) = 60.48
Then, we calculate the sum of the squared deviations of x:
sum of (deviations of x)^2 = (-9.4)^2 + (-4.4)^2 + (0.6)^2 + (5.6)^2 + (7.6)^2 = 190.4
Using the formula for the slope of the line of best fit:
slope = (sum of (deviations of x * deviations of y)) / (sum of (deviations of x)^2)
slope = 60.48 / 190.4 ≈ 0.3176
c. The slope of this line indicates the average rate of change of the percentage of adults still working per year. In this situation, the slope of approximately 0.3176 suggests that the percentage of adults in the labor force ages 65 or older who are still working has been increasing by about 0.3176% per year since 1990.
To predict the percent of the labor force that is 65 years old or older in the year 2027, we can use the linear regression equation.
The equation for the line of best fit is in the form y = mx + b, where m is the slope and b is the y-intercept.
Using the values we calculated above:
m ≈ 0.3176
mean(x) = 1999.4
mean(y) = 14.62
We can substitute these values into the equation:
y = 0.3176x + b
Solving for b:
14.62 = 0.3176(1999.4) + b
14.62 = 634.328 + b
b ≈ 14.62 - 634.328
b ≈ -619.708
So, the linear regression equation of the line of best fit for the data is:
y ≈ 0.3176x - 619.708
To predict the percentage in the year 2027 (37 years since 1990):
x = 37
Using the equation, we can evaluate y:
y ≈ 0.3176(37) - 619.708
y ≈ 11.7832 - 619.708
y ≈ -607.9248
Since a negative percentage doesn't make sense in this context, it's not realistic to predict a percentage for the year 2027 using the given linear regression. We should consider other factors and possibilities in order to make a more accurate prediction.