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 2028.
year:1990, 1995, 2000, 2005, 2007
% of labor force: 11.2%, 12%, 12.7%, 13.8%, 15.2%
a. Plotting the data points:
year: 1990, 1995, 2000, 2005, 2007
% of labor force: 11.2%, 12%, 12.7%, 13.8%, 15.2%
To plot the data points, we can use the x-axis to represent the number of years since 1990 (x), and the y-axis to represent the percentage of the labor force (y).
Year (x) | % of labor force (y)
------------------------------------------------
1990 | 11.2%
1995 | 12%
2000 | 12.7%
2005 | 13.8%
2007 | 15.2%
b. Find the linear regression equation of the line of best fit for the data.
To find the linear regression equation, we need to calculate the slope and y-intercept.
First, let's assign x and y values to the data points:
x = [0, 5, 10, 15, 17]
y = [11.2, 12, 12.7, 13.8, 15.2]
Next, we calculate the means for x (x̄) and y (ȳ):
x̄ = (0 + 5 + 10 + 15 + 17) / 5 = 9.4
ȳ = (11.2 + 12 + 12.7 + 13.8 + 15.2) / 5 = 12.98
Then, calculate the deviations for x (xi - x̄) and y (yi - ȳ):
Δx = [0 - 9.4, 5 - 9.4, 10 - 9.4, 15 - 9.4, 17 - 9.4] = [-9.4, -4.4, 0.6, 5.6, 7.6]
Δy = [11.2 - 12.98, 12 - 12.98, 12.7 - 12.98, 13.8 - 12.98, 15.2 - 12.98] = [-1.78, -0.98, -0.28, 0.82, 2.22]
Next, calculate the sum of the products of deviations (ΣΔx * Δy):
Σ(Δx * Δy) = (-9.4 * -1.78) + (-4.4 * -0.98) + (0.6 * -0.28) + (5.6 * 0.82) + (7.6 * 2.22) = 34.29
Then, calculate the sum of squared deviations for x (ΣΔx^2):
Σ(Δx^2) = (-9.4)^2 + (-4.4)^2 + (0.6)^2 + (5.6)^2 + (7.6)^2 = 201.2
Now, calculate the slope (m):
m = Σ(Δx * Δy) / Σ(Δx^2) = 34.29 / 201.2 ≈ 0.17
Next, calculate the y-intercept (b):
b = ȳ - (m * x̄) = 12.98 - (0.17 * 9.4) ≈ 11.61
Therefore, the linear regression equation is:
y = 0.17x + 11.61
c. The slope of this line indicates the rate of change in the percentage of the labor force that is 65 years old or older per year. In this situation, the slope of 0.17 means that for every year since 1990, the percentage of the labor force that is 65 years old or older increases by 0.17%.
d. To predict the percent of the labor force that is 65 years old or older in the year 2028 (38 years since 1990), we can substitute x = 38 into the linear regression equation:
y = 0.17x + 11.61
y = 0.17 * 38 + 11.61
y ≈ 18.46
Therefore, if the percentage continues to increase at the same rate, we can predict that approximately 18.46% of the labor force will be 65 years old or older in the year 2028.