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 1900
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 2027
Year 1990 1995 2000 2005 2007
% of Labor Force 10.9% 11.5% 12.1% 13.1% 14
a. Choose the correct plot below.
[-5,20,1] by [8, 15, 1]
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.)
We can plot the data points on a scatter plot to analyze the trend:
Year (x) | % of Labor Force (y)
-----------------------
1990 | 10.9%
1995 | 11.5%
2000 | 12.1%
2005 | 13.1%
2007 | 14%
(a) The correct plot would be:
x-axis: [1990, 1995, 2000, 2005, 2007]
y-axis: [10.9, 11.5, 12.1, 13.1, 14]
The points on the graph would be:
(1990, 10.9)
(1995, 11.5)
(2000, 12.1)
(2005, 13.1)
(2007, 14)
(b) To find the linear regression equation, we need to find the line of best fit for the data points. Using the least squares method, we can calculate the equation y = mx + b, where m is the slope and b is the y-intercept.
To find the slope:
m = [(n∑(xy)) - (∑x∑y)] / [(n∑(x^2)) - (∑x)^2]
n is the number of data points (5 in this case)
∑(xy) is the sum of all products of x and y coordinates
∑x is the sum of all x coordinates
∑y is the sum of all y coordinates
∑(x^2) is the sum of all x squared coordinates
Calculating the slope:
∑(xy) = (1990*10.9) + (1995*11.5) + (2000*12.1) + (2005*13.1) + (2007*14)
= 21738.3 + 22942.5 + 24200 + 26278.5 + 28098
= 123257.3
∑x = 1990 + 1995 + 2000 + 2005 + 2007
= 9997
∑y = 10.9 + 11.5 + 12.1 + 13.1 + 14
= 61.6
∑(x^2) = (1990^2) + (1995^2) + (2000^2) + (2005^2) + (2007^2)
= 3960100 + 3980025 + 4000000 + 4010025 + 4028049
= 19978199
m = [(5*123257.3) - (9997*61.6)] / [(5*19978199) - (9997^2)]
= [616286.5 - 615833.2] / [99890995 - 99940009]
= 453.3 / -49014
≈ -0.00924453
To find the y-intercept, b:
b = (∑y - m∑x) / n
b = (61.6 - (-0.00924453 * 9997)) / 5
= (61.6 + 92.40518941) / 5
≈ 30.40103788 / 5
≈ 6.08
Therefore, the linear regression equation is y = -0.009x + 6.08.
(c) The slope of the line indicates the rate at which the percentage of the labor force ages 65 or older is increasing or decreasing. In this case, since the slope is negative (-0.009), it means that the percentage is decreasing over time.
(d) To predict the percentage of the labor force that is 65 years old or older in the year 2027, we need to substitute x = 2027 - 1900 = 127 into the linear regression equation.
y = -0.009x + 6.08
y = -0.009 * 127 + 6.08
y ≈ -1.143 + 6.08
y ≈ 4.937
Therefore, if the percentage continues to increase at the same rate, we can predict that approximately 4.937% of the labor force will be 65 years old or older in the year 2027.