Is there a relationship between a student’s GPA and the number of pencils in his or her backpack? Jordynn and Angie decided to find out by selecting a random sample of students from their high school. Here is computer output from a least squares regression analysis using$\mathrm{x}=\mathrm{number}$pencils and $\mathrm{y}=\mathrm{GPA}:$

$$\begin{gathered} \mathrm{Predictor}\mathrm{Coef}\mathrm{SE}\mathrm{Coef}\mathrm{T}\mathrm{P} \\ \mathrm{Constant}3.24130.180917.9200.0000 \\ \mathrm{Pencils}-0.04230.0631-0.6700.5062 \\ \mathrm{S}=0.738533\mathrm{R}-\mathrm{Sq}=0.9\%\mathrm{R}-\mathrm{Sq}\left(\mathrm{adj}\right)=0.0\% \end{gathered}$$

Is there convincing evidence of a linear relationship between $\mathrm{GPA}$ and the number of pencils for students at this high school? Assume the conditions for inference are met.

We need to find convincing evidence of a linear relationship between $\mathrm{GPA}$and the number of pencils for students at this high school.

Consider:

$$\begin{gathered} \mathrm{n}=\mathrm{Sample}\mathrm{size}=\mathrm{Unknown} \\ \mathrm{\alpha }=\mathrm{Significance}\mathrm{level}=0.05 \end{gathered}$$

The estimate of the slope ${\mathrm{b}}_1$is given in the row "Pencils" and in the column "Coef" of the given computer output:

${\mathrm{b}}_1=-0.0423$

The estimated standard deviation of the slope ${\mathrm{SE}}_{\mathrm{b}1}$is given in the row "Time" and in the column "SE Coef" of the given computer output:

${\mathrm{SE}}_{\mathrm{b}1}=0.0631$

Given claim: Slope is nonzero (reduction):

The null hypothesis or the alternative hypothesis states the given claim The null hypothesis states that the slope is zero. If the given claim is the null hypothesis, then the alternative hypothesis states the opposite of the null hypothesis:

$$\begin{gathered} {\mathrm{H}}_0:{\mathrm{\beta }}_1=0 \\ {\mathrm{H}}_{\alpha }:{\mathrm{\beta }}_1<0 \end{gathered}$$

Compute the value of the test statistic

$\mathrm{t}=\frac{{\mathrm{b}}_1-{\mathrm{\beta }}_1}{{\mathrm{SE}}_{{\mathrm{b}}_1}}=\frac{-0.0423-0}{0.0631}\approx -0.6704$

The $\mathrm{P}$-Value is given in the row "Pencils" and in the column "P" of the computer output:

$\mathrm{P}=0.5062$