The following data on the number of iron workers in the United States for the years 1978 through 2008 are provided by the U.S. Bureau of Labor Statistics. Using regression techniques discussed in this section, analyze the data for trend. Develop a scatter plot of the data and fit the trend line through the data. Discuss the strength of the model.
Year
Union members
(1000’s)
1978
17,340
1979
16,996
1980
16.975
1981
16.913
1982
17,002
1983
16,960
1984
16,740
1985
16.568
1986
16,390
1987
16,598
1988
16,748
1989
16,360
1990
16.269
1991
16,110
1992
16,211
1993
16,477
1994
16,334
1995
16,305
1996
16,145
1997
15,776
1998
15,472
1999
16,685
2000
15,359
2001
16,670
2002
16,098
2003
16,212
2004
16,316
2005
16,718
2006
16,707
2007
16,113
2008
15,128
To analyze the trend in the number of iron workers in the United States from 1978 to 2008, we can use regression techniques. Regression helps us determine the relationship between two variables and create a trend line to estimate future values.
First, we can create a scatter plot of the data to visualize the relationship between the years and the number of union members. The years will be plotted on the x-axis, and the number of union members (in thousands) will be plotted on the y-axis.
Once we have the scatter plot, we can fit a trend line through the data using regression. The trend line will provide an estimate of how the number of iron workers changed over time.
To determine the strength of the model, we can look at the correlation coefficient (r) and the coefficient of determination (r^2). The correlation coefficient measures the strength and direction of the relationship between the two variables, while the coefficient of determination represents the proportion of the variance in the dependent variable that can be explained by the independent variable(s).
Let's calculate and analyze the trend using regression techniques:
Step 1: Create a scatter plot.
Plot the years (1978-2008) on the x-axis and the number of union members (in thousands) on the y-axis. Mark the points for each year.
Step 2: Fit a trend line.
Using regression analysis, fit a trend line through the data points. This line represents the best-fit line that minimizes the distance between the trend line and the data points.
Step 3: Calculate the correlation coefficient.
Calculate the correlation coefficient (r) to determine the strength and direction of the relationship between the years and the number of union members.
Step 4: Calculate the coefficient of determination.
Calculate the coefficient of determination (r^2) to determine the proportion of the variance in the number of union members that can be explained by the years.
Step 5: Analyze the strength of the model.
Based on the correlation coefficient and coefficient of determination, assess the strength of the model. A higher correlation coefficient (>0.5) and coefficient of determination (>0.5) indicate a stronger relationship and higher predictive power of the model.
By following these steps, we can analyze the trend in the number of iron workers from 1978 to 2008 and determine the strength of the model.