Question

unit 2 - exploring two variable data
hw 4 - influential points and departures from linearity

1. a sample of men agreed to participate in a study to determine the relationship between several variables including height, weight, waste size, and percent body fat. a scatterplot with percent body fat on the y - axis and wrist size (in inches) on the horizontal axis revealed a positive linear association between these variables. computer output for the regression analysis is given below:

dependent variable is: %bf
r - squared = 67.8%
s = 4.713 with 250 - 2 = 248 degrees of freedom
variable coefficient se of coeff t - ratio prob
constant - 42.734 2.717 - 15.7 <.0001
waist 1.70 0.0743 22.9 <.0001
(a) write the equation of the regression line: $hat{y}=1.70x - 42.734$
(b) explain/interpret the information provided by r - squared in the context of this problem. be specific.
(c) one of the men who participated in the study had waist size 35 inches and 10% body fat. calculate the residual associated with the point for this individual.

1. scientists are trying to study the relationship between the weight of an animals heart and the length of the cavity of the hearts left ventricle. the following data was collected from various animals

length of cavity of left ventricle (in cm) heart weight (in grams)
.55 .13
1.0 .64
2.2 5.8
4.0 102
6.5 210
12.0 2030
16.0 3900
a) make a scatter plot in the space above and confirm that the relationship does not appear to be linear.

Explanation:

(a)

The regression line equation for a simple - linear regression is of the form $\hat{y}=b_0 + b_1x$, where $b_0$ is the intercept and $b_1$ is the slope. From the computer - output, the intercept (constant) is $-42.734$ and the slope for the variable 'Waist' is $1.70$. So the regression line equation is $\hat{y}=1.70x - 42.734$, where $\hat{y}$ is the predicted percent body fat and $x$ is the waist size.

(b)

The $R$-squared value, $R^{2}=67.8\%$. In the context of this problem, $R^{2}$ represents the proportion of the variability in the dependent variable (percent body fat) that is explained by the independent variable (waist size) in the regression model. So, $67.8\%$ of the variation in percent body fat can be accounted for by the linear relationship with waist size.

(c)

The residual is calculated as $e = y-\hat{y}$. First, we find the predicted value $\hat{y}$ when $x = 35$ (waist size).

Step1: Calculate the predicted value

Substitute $x = 35$ into the regression equation $\hat{y}=1.70x - 42.734$.
$\hat{y}=1.70\times35-42.734=59.5 - 42.734 = 16.766$.

Step2: Calculate the residual

The observed value of $y$ (percent body fat) is $10$. The residual $e=y - \hat{y}=10 - 16.766=-6.766$.

Answer:

(a) $\hat{y}=1.70x - 42.734$
(b) $67.8\%$ of the variation in percent body fat can be accounted for by the linear relationship with waist size.
(c) $-6.766$