Question

Long strides The scatterplot shows the relationship between x = height of a student (in inches) and y = number of steps required to walk the length of a school hallway, along with the regression line y^=113.6−0.921x

a. Calculate and interpret the residual for Kiana, who is 67 inches tall and took 49 steps to walk the hallway.

b. Matthew is 10 inches taller than Samantha. About how many fewer steps do you expect Matthew to take compared to Samantha?

Short answer

Part (a) Kiana's residual is $15$ walks, which means there is a $15-$walk gap between her actual walk and the anticipated value.

Part (b) If Matthew is $10$ inches taller than Samantha, the amount of steps Matthew will take in $8$ steps will be less than Samantha's.

Step-by-step solution

Part (a) Step 1: Given information

$x:$Height of a student.

$y:$Number of steps required to walk the length of a school hallway.

Kiana took $49$ steps to walk down the corridor, despite her height of $67$ inches.

The given OLS regression line is:

$\stackrel{⏜}{y}=113.6-0.921x$

Part (a) Step 2: Explanation

Kiana's projected walk based on her height can be computed as follows:

$$\begin{gathered} \stackrel{⏜}{y}=113..60-0.921\left(67\right) \\ =51.8930 \end{gathered}$$

The residual for Kiana can be calculated as:

$$\begin{gathered} e=y-\stackrel{⏜}{y} \\ =67-51.8930 \\ =15.1070 \end{gathered}$$

The residual is the discrepancy or fluctuation between the true and anticipated value. Here, Kiana's residual is $15.1070$ or about $15$ walks, indicating that the difference between Kiana's actual and anticipated walk is $15$

Part (b) Step 1: Calculation

If Matthew is $10$ inches taller than Samantha, the number of less predicted steps that Matthew will take can be computed as follows:

Samantha ‘walk

$\stackrel{⏜}{y_1}=113.6-0.921x$

Matthew ‘walk

$$\begin{gathered} \stackrel{⏜}{y_2}=113.6-0.921\left(10x\right) \\ =113.6-9.21x \end{gathered}$$

The expected walk is:

$$\begin{gathered} E=y_1-y_2 \\ =113.6-0.921x-113.6+9.21x \\ =8.289x \\ \approx 8x \end{gathered}$$