Question

Less mess? Kerry and Danielle wanted to investigate if tapping on a can of soda would reduce the amount of soda expelled after the can has been shaken. For their experiment, they vigorously shook $40$cans of soda and randomly assigned each can to be tapped for $0$seconds, $4$seconds, $8$seconds, or $12$seconds. After opening the cans and waiting for the fizzing to stop, they measured the amount expelled (in milliliters) by subtracting the amount remaining from the original amount in the can. Here are their data:

Here is some computer output from a least-squares regression analysis of these data. Construct and interpret a $95\%$confidence interval for the slope of the true regression line.

Short answer

The slope of the regression line is $95$ percent certain to be between $-2.9962298$and $-2.2737702$.

Step-by-step solution

Given information

We have to construct and interpret the $95\%$confidence interval for the slope of the true regression line.

Simplification

We will use the following formula :-

The boundaries of the confidence interval

$$\begin{gathered} b-t^{*}\times S{E}_b \\ b+t^{*}\times S{E}_b \end{gathered}$$

The slope $b_1$is calculated in the row "Tapping time" and the column "coef" of the following computer output:

$b1=-2.6350$

In the row "tapping time" and the column "SE Coef" of the given computer output, the computed standard deviation of the slope $S{E}_{b_1}$is mentioned:

$S{E}_{b_1}=0.1769$

In the T distribution table, you'll find the crucial.

$df-n-2=40-2=38$

$df=38$,as a result, it would use the nearest smaller degrees of freedom, $df=30$, in the column with $c=95\%$: $\hspace{1em}{t}^{*}=2.042$

The confidence interval's bounds

$$\begin{gathered} b-t^{*}\times S{E}_b=-2.6350-2.042\times 0.1769=-2.9962298 \\ b+t^{*}\times S{E}_b=-2.6350+2.042\times 0.1769=-2.2737702 \end{gathered}$$

There are $95\%$ confident that the slope of the regression line is between $-2.9962298$ and $-2.2737702$.