# 2.2 - Another Example of Slope Inference

2.2 - Another Example of Slope Inference## Exampe 2-1

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:

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.