Chapter 12, Problem R12.1RE

(a)

To determine

To describe what the scatterplot tells you about the relationship between height and the number of steps.

Answer to Problem R12.1RE

The relationship between height and the number of steps is moderately strong, negative, linear association and no outliers.

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. Thus, from the scatterplot we can see that,

The direction of the scatterplot is negative because the pattern in the scatterplot slopes downward. The form of the scatterplot is linear because there is no strong curvature present in the scatterplot. The strength of the scatterplot is moderately strong because the points in the scatterplot do not lie extremely close together but are not spread far apart either. The unusual feature is that there appear to be no outlier because there are no points that deviate strongly from the pattern in the other points.

(b)

To determine

To find out what is the equation of least square regression line.

Answer to Problem R12.1RE

  $\widehat{y}=113.57-0.9211x$ .

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. The general equation of the least square regression line is:

  $\widehat{y}=b_0+b_{1}x$

Thus, form the computer output, we have that the estimate of the constant is given in the row “Constant” and in the column “Coef” as:

  $b_0=113.57$

And the estimate of the slope is given in the row “Height” and in the column “Coef” as:

  $b_1=-0.9211$

Now we will put the above values in the equation of the regression line as:

  $\begin{array}{l}\widehat{y}=b_0+b_{1}x\\ =113.57-0.9211x\end{array}$

Where x represents the height and y represents the number of steps.

(c)

To determine

To identify the value of each of the following given from the computer output and interpret each value.

Answer to Problem R12.1RE

The values are:

  $\begin{array}{l}{b}_0=113.57\\ b_1=-0.9211\\ s=3.50429\\ S{E}_{b_1}=0.1938\end{array}$

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. Now we have to identify the value and interpret it:

1. $b_0$ : The y-intercept $b_0$ is given in the row “Constant” and in the column “Coef” of the given computer output as:

  $b_0=113.57$

The y-intercept represents the average y-value when x is zero. Thus, on average, the number of steps are $113.57$ steps when the height is zero inches.

2. $b_1$ : 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 slope represents the average increase or decrease of y per unit of x. thus, on average, the number of steps decreases by $-0.9211$ steps from the actual number of steps.

3. $s$: the standard error of the estimate s is given after “S=” in the computer output as:

  $s=3.50429$

Thus, the predicted number of steps deviates on average by $3.50429$ steps from the actual number of steps.

4. $S{E}_{b_1}$ : 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:

  $S{E}_{b_1}=0.1938$

Thus, the slope of the sample regression line varies on average by $0.1938$ from the slope of the true population regression line.