Chapter 12, Problem R12.3RE

To determine

To explain do these data provide convincing evidence at $\alpha =0.05$ the level that taller students at this school require fewer steps to walk a fixed distance.

Answer to Problem R12.3RE

There is a convincing evidence that taller students at this school require fewer steps to walk a fixed distance.

Explanation of Solution

A study was conducted to know that do taller students require fewer steps to walk a fixed distance. The scatterplot given in the question shows the relationship between height and the number of steps require to walk the school hallway. And it has been assumed that the conditions for inferences are met. Thus, it is given that,

  $\begin{array}{l}n=36\\ \alpha =0.05\end{array}$

The slope $b_1$ is given in the row “ Height” and in the column “Coef” of the given computer output as:

  $b_1=-0.9211$

The standard error of the slope $S{E}_{b_1}$ is given in the row “Height” and in the column “SE Coef” of the given computer output as:

  $S{E}_{b_1}=0.1938$

We have to claim that the slope is negative.

Thus, let us define the null and alternative hypotheses as:

  $\begin{array}{l}{H}_0:{\beta }_1=0\\ H_1:{\beta }_1<0\end{array}$

Now, the value of test statistics is as:

  $\begin{array}{l}t=\frac{b_1-{\beta }_1}{S{E}_{b_1}}\\ =\frac{-0.9211-0}{0.1938}\\ =-4.7528\end{array}$

Thus, now we have to calculate the P-value and for this we need to find the degrees of freedom:

  $\begin{array}{l}df=n-2\\ =36-2\\ =34\end{array}$

And the P-value is as:

  $P<0.0005$

If the P-value is less than or equal to the significance level then the null hypothesis is rejected, therefore,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that there is a convincing evidence that taller students at this school require fewer steps to walk a fixed distance.