Q. 26 - The Practice of Statistics for AP

Question

26.  Sisters and brothers (3.1, 3.2) How strongly do physical characteristics of sisters and brothers correlate? Here are data on the heights (in inches)
of 11 adult pairs:

Brother	71	68	66	67	70	71	70	73	72	65	66
Sister	69	64	65	63	65	62	65	64	66	59	62

(a) Construct a scatterplot using brother’s height as the explanatory variable. Describe what you see.
(b) Use your calculator to compute the least-squares regression line for predicting sister’s height from brother’s height. Interpret the slope in context.
(c) Damien is $70$ inches tall. Predict the height of his sister Tonya.
(d) Do you expect your prediction in (c) to be very accurate? Give appropriate evidence to support your answer.

Step-by-Step Solution

Answer

(a) A scatterplot using brother’s height as the explanatory variable is weak positive correlation.

(b) The least square regression line equation is $\hat{y}=27.635+0.527x$.

(c) The height of Tonya is $64.525$ inches.

(d) The coefficient of determination is low. So, the model isn't very accurate.

Part (a) Step 1: Given information

To construct a scatter plot using brother’s height as the explanatory variable with the given data.

Brother	71	68	66	67	70	71	70	73	72	65	66
Sister	69	64	65	63	65	62	65	64	66	59	62

Part (a) Step 2: Explanation

The horizontal axis represents the brother, while the vertical axis represents the sister. The scatter plot  can be made as follows:

Because the scatterplot slopes upwards, the association is positive. Because the points appear to be far apart, the association is weak. So, there is a weak positive correlation between to variables.

Part (b) Step 1: Given information

To use the calculator to compute the least-squares regression line for predicting sister’s height from brother’s height. And interpret the slope in context.

Part (b) Step 2: Explanation

The following is the output from a Ti-83 calculator:

As a result, the equation of the regression line is $\hat{y}=27.635+0.527x$.

Part (c) Step 1: Given information

To predict the height of his sister Tonya. Since, Damien is $70$ inches tall.

Part (c) Step 2: Explanation

Tonya's height can be estimated as follows:
$\hat{y}=27.635+0.527x$
$=27.635+0.527\left(70\right)$
$=64.525$
Asa a result, the height of Tonya's is about $64.525$ inches.

Part (d) Step 1: Given information

To determine the prediction in part (c) to be very accurate or not.

Part (d) Step 2: Explanation

Using a $Ti-83$ calculator, the coefficient of determination can be calculated as follows:

Because the coefficient of determination is low.
As a result, the model isn't very accurate.