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

d. -5.666 to -0.926

Step-by-step explanation:

Here a pivotal quantity is where is the true slope of the regression  equation (unknown), is its least square estimate and 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 , 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 , i.e., . Therefore, the 95% confidence interval explicitly is (-5.666, -0.926)

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