As the carbon content in steel increases, its ductility tends to decrease. A researcher at a steel company measures carbon content and ductility for a sample of 15 types of steel. Based on these data he obtained the following regression results.
The regression equation is
Ductility = 7.67 - 3.30 Carbon Content
Predictor Coef SE Coef T P
Constant 7.671 1.507 5.09 0.000
Carbon Content -3.296 1.097 -3.01 0.010
S = 2.36317 R-Sq = 41.0% R-Sq(adj) = 36.5%
The 95% confidence interval for the slope of the regression equation is
a. -5.456 to -1.136
b. -4.393 to -2.199
c. 6.164 to 9.178
d. -5.666 to -0.926
e. 2.581 to 12.761

Here a pivotal quantity is [tex]T = \frac{\hat{\beta_{1}}-\beta_{1}}{se(\hat{\beta_{1}})}[/tex] where [tex]\beta_{1}[/tex] is the true slope of the regression equation (unknown), [tex]\hat{\beta_{1}}[/tex] is its least square estimate and [tex]se(\hat{\beta_{1}})=1.097[/tex] is its estimated standard error. And T has t-distribution with n-2=15-2=13 degrees of freedom (n is the sample size). We have that [tex]\hat{\beta_{1}} = -3.30[/tex], and because we want the 95% confidence interval, we should use the 2.5th quantile of the t distribution with 13 df, this value is -2.16 and the 95% confidence intervale is given by [tex]\hat{\beta_{1}}\pm t_{0.025}se(\hat{\beta_{1}})[/tex], i.e., [tex]-3.30\pm (2.16)(1.097)[/tex]. Therefore, the 95% confidence interval explicitly is (-5.666, -0.926)