(Data source: The U.S. Census Bureau and Mind On Statistics, (3rd edition), Utts and Heckard). In this example, the observations are the 50 states of the United States (Poverty data - Note: remove data from the District of Columbia). The variables are y = percentage of each state’s population living in households with income below the federally defined poverty level in the year 2002, \(x_{1}\) = birth rate for females 15 to 17 years old in 2002, calculated as births per 1000 persons in the age group, and \(x_{2}\) = birth rate for females 18 to 19 years old in 2002, calculated as births per 1000 persons in the age group.

The two x-variables are correlated (so we have multicollinearity). The correlation is about 0.95. A plot of the two x-variables is given below.

The figure below shows plots of y = poverty percentage versus each x-variable separately. Both x-variables are linear predictors of the poverty percentage.

Minitab results for the two possible simple regressions and the multiple regression are given below.

We note the following:

1. The value of the sample coefficient that multiplies a particular x-variable is not the same in the multiple regression as it is in the relevant simple regression.
2. The \(R^{2}\) for the multiple regression is not the sum of the \(R^{2}\) values for the simple regressions. An x-variable (either one) is not making an independent “add-on” in the multiple regression.
3. The 18 to 19-year-old birth rate variable is significant in the simple regression but is not in the multiple regression. This discrepancy is caused by the correlation between the two x-variables. The 15 to 17-year-old birth rate is the stronger of the two x-variables and given its presence in the equation, the 18 to 19-year-old rate does not improve \(R^{2}\) enough to be significant. More specifically, the correlation between the two x-variables has increased the standard errors of the coefficients, so we have less precise estimates of the individual slopes.