Question

Do taller students require fewer steps to walk a fixed distance? The scatterplot shows the relationship between $x=$height (in inches) and $y=$number of steps required to walk the length of a school hallway for a random sample of 36 students at a high school.

A least-squares regression analysis was performed on the data. Here is some computer output from the analysis

Long legs Do these data provide convincing evidence at the $\alpha =0.05$level that taller students at this school require fewer steps to walk a fixed distance? Assume that the conditions for inference are met.

Short answer

We get to the conclusion that taller kids at this school take fewer steps to travel a certain distance.

Step-by-step solution

Given information

The given data is

$\alpha =0.05$

Explanation

A study was done to see if taller students needed fewer steps to go a certain distance. The scatterplot in the question depicts the link between height and the number of steps needed to walk down the school corridor. And it's been assumed that the inference requirements are met. As a result, it is assumed that

$n=36$

$\alpha =0.05$

In the row " Height" and the column "Coef" of the given computer output, the slope $b_1$is presented as:

$b_1=-0.9211$

In the row "Height" and the column "SE Coef" of the given computer output, the standard error of the slope$SE{b}_1$is presented as:

$SE{b}_1=0.1938$

It is necessary to assert that the slope is negative.

As an example, let's define the null and alternative hypotheses as follows:

$$\begin{gathered} H_0:\beta =0 \\ {H}_1:\beta <0 \end{gathered}$$

The value of test statistics is now as follows:

$t=\frac{b_1-{\beta }_1}{S{E}_{b_1}}$

Substituting the values

$t=\frac{-0.9211-0}{0.1938}$

$=-4.7528$

Now we must calculate the P-value, for which we must first determine the degrees of freedom:

$df=n-2$

$$\begin{gathered} =36-2 \\ =34 \end{gathered}$$

The P-value is as follows:

$P<0.0005$

The null hypothesis is rejected if the P-value is less than or equal to the significance level.

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

As a result, we get to the conclusion that taller kids at this school take fewer steps to travel a certain distance.