Question

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

Answer 1

Answer:

d. -5.666 to -0.926

Step-by-step explanation:

Here a pivotal quantity is $T = \frac{\hat{\beta_{1}}-\beta_{1}}{se(\hat{\beta_{1}})}$ where $\beta_{1}$ is the true slope of the regression  equation (unknown), $\hat{\beta_{1}}$ is its least square estimate and $se(\hat{\beta_{1}})=1.097$ 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 $\hat{\beta_{1}} = -3.30$, 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 $\hat{\beta_{1}}\pm t_{0.025}se(\hat{\beta_{1}})$, i.e., $-3.30\pm (2.16)(1.097)$. Therefore, the 95% confidence interval explicitly is (-5.666, -0.926)