Question

Beer and BAC How well does the number of beers a person drinks predict his or her blood alcohol content (BAC)? Sixteen volunteers aged $21$or older with an initial BAC of $0$took part in a study to find out. Each volunteer drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their BAC. A least-squares regression analysis was performed on the data using x=number of beers and y=BAC. Here is a residual plot and a histogram of the residuals. Check whether the conditions for performing inference about the regression model are met.

a. Find the critical value for a $99\%$confidence interval for the slope of the true regression line. Then calculate the confidence interval.

b. Interpret the interval from part (a).

c. Explain the meaning of “localid="1654184305701" $99\%$confident” in this context

Here is computer output from the least-squares regression analysis of the beer and blood alcohol dat

Short answer

a. The Critical value is $2.977$and the Confidence interval is $(0.0108192,0.0251088)$

b. The true slope of the population regression line is between $0.0108192$and $0.0251088$, according to $99$percent confidence.

c. The $99$percent confidence interval shows the slope of the true regression line.

Step-by-step solution

Part (a) Step 1 : Given information

We have to find the critical value and confidence interval for a $99\%$confidence interval.

Part (a) Step 2 : Simplification

We will use the following formula the boundaries of the confidence interval :

$$\begin{gathered} b-t*\times SEb1 \\ b+t*\times SEb1 \end{gathered}$$

In the row "Beers" and the column "Coefficient" of the computer output, the slope$b1$is mentioned.

$b1=0.017964$

In the row "Beers" and the column " $s.E.$Of coeff" of the mention output from the computer, the computed standard deviation of the slope $SEb1$is mentioned.

$SEb1=0.0024$

degrees of freedom :-$16-2=14$

In the student's T distribution table $df=14$and column of$c=99$percent, the t-value may be found.

$t*=2.977$

The confidence interval's bounds

$$\begin{gathered} b-t*\times SEb1=0.017964-2.977\times 0.0024=0.0108192 \\ b+t*\times SEb1=0.017964+2.977\times 0.0024=0.0251088 \end{gathered}$$

Part (b) Step 1 : Given information

We have to explain the interval from part (a).

Part (b) Step 2 : Simplification

From part (a)

$$\begin{gathered} b-t*\times SEb1=0.017964-2.977\times 0.0024=0.0108192 \\ b+t*\times SEb1=0.017964+2.977\times 0.0024=0.0251088 \end{gathered}$$

The true slope of the population regression line is between$0.0108192$ and $0.0251088$, according to $99$percent confidence.

Part (c) Step 1 : Given information

We have to explain the meaning of $99\%$confident.

Part (c) Step 2 : Simplification

The slope of the correct regression line is shown in the $99$percent confidence interval. $99$percent confidence also means that $99$percent of all samples are expected to have a $99$percent confidence interval including the true population parameter.