 # The accompanying data on y = normalized energy ddaeeric 2021-01-13 Answered

The accompanying data on y = normalized energy $$\displaystyle{\left[{\left(\frac{{J}}{{m}^{{2}}}\right)}\right]}$$ and x = intraocular pressure (mmHg) appeared in a scatterplot in the article “Evaluating the Risk of Eye Injuries: Intraocular Pressure During High Speed Projectile Impacts” (Current Eye Research, 2012: 43-49), an estimated regression function was superimposed on the plot.
$$\begin{array}{|c|c|}\hline x & 2761 & 19764 & 25713 & 3980 & 12782 & 19008 & 19028 & 14397 & 9606 & 3905 & 25731 \\ \hline y & 1553 & 14999 & 32813 & 1667 & 8741 & 16526 & 26770 & 16526 & 9868 & 6640 & 1220 & 30730 \\ \hline \end{array}$$
Here is Minitab output from fitting the simple linear regression model. Does the model appear to specify a useful relationship between the two variables?
$$\begin{array}{|c|c|}\hline \text{Predictor Coef SE Coef T P Constant} & -5090 & 2257 & -2.26 & 0.048 \\ \hline \text{Pressure} & 1.2912 & 0.1347 & 9.59 & 0.000 \\ \hline \end{array}$$
$$[S=3679.36, R-Sq = 90.2\%, R-Sq(adj)=89.2\% ]$$.

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Step 1
Given:
$\left[n=12\right]$
Let us assume:
$\left[\alpha =0.05\right]$
Given in the output:
$\left[{b}_{1}=1.2912\right]$
$S{E}_{{b}_{1}}=0.1347$
Determine the hypothesis:
${H}_{0}:{\beta }_{1}=0$
${H}_{0}:{\beta }_{1}\ne q0$
Compute the value of the test statistic:
$\left[t=\frac{{b}_{1}-{\beta }_{1}}{S{E}_{{b}_{1}}}=\frac{1.2912-0}{0.1347}\approx 9.59\right]$
The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of Table B containing the t-value in the row $\left[df=n—2=12—2=10:\right]$
$\left[P<2×0.0005=0.001\right]$
If the P-value is less than or equal to the significance level, then the null
hypothesis is rejected:
$\left[P<0.05⇒Reject{H}_{0}\right]$
There is sufficient evidence to support the claim that the slope of the population regression line is not zero, which means that the model appears to specify a useful relationship between the two variables.
Result:
Yes.