2.2 - Another Example of Slope Inference

Exampe 2-1 Section

Is there a positive relationship between sales of leaded gasoline and lead burden in the bodies of newborn infants? Researchers (Rabinowitz, et al, 1984) who were interested in answering this research question compiled data (Lead Cord data) on the monthly gasoline lead sales (in metric tons) in Massachusetts and mean lead concentrations (µl/dl) in umbilical-cord blood of babies born at a major Boston hospital over 14 months in 1980-1981.

Analyzing their data, the researchers obtained the following Minitab fitted line plot:

Fitted line plot

and standard regression analysis output:

Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value
Regression 1 3.7783 3.7783 9.95 0.008
Residual Error 12 4.5560 0.3797    
Total 13 8.3343      
Model Summary
S R-sq R-sq(adj)
0.616170 45.3% 40.8%
Coefficients
Predictor Coef SE Coef T-Value P-Value
Constant 4.1082 0.6088 6.75 0.000
Sold 0.014885 0.004719 3.15 0.008
Regression Equation

Cord = 4.11 + 0.0149 Sold

Minitab reports that the P-value for testing \(H_{0} \colon \beta_{1} = 0\) against the alternative hypothesis \(H_{A} \colon \beta_{1} ≠ 0\) is 0.008. Therefore, since the test statistic is positive, the P-value for testing \(H_{0} \colon \beta_{1}= 0\) against the alternative hypothesis \(H_{A} \colon \beta_{1} > 0\) is 0.008 ÷ 2 = 0.004. The P-value is less than 0.05. There is sufficient statistical evidence, at the 0.05 level, to conclude that \(\beta_{1} > 0\).

Furthermore, since the 95% t-multiplier is \(t_{\left(0.025, 12 \right)} = 2.1788\), a 95% confidence interval for \(\beta_{1}\) is:

0.014885 ± 2.1788(0.004719) or (0.0046, 0.0252).

The researchers can be 95% confident that the mean lead concentrations in umbilical-cord blood of Massachusetts babies increases between 0.0046 and 0.0252 µl/dl for every one-metric ton increase in monthly gasoline lead sales in Massachusetts. It is up to the researchers to debate whether or not this is a meaningful increase.