To tackle this problem, we first need to establish the linear regression model that predicts the calories burned [tex]\( C \)[/tex] per minute as a function of body weight [tex]\( x \)[/tex].
### 1a. Linear Regression Model
The table provides the body weights and corresponding calories burned per minute when walking at 3 miles per hour. From these data points, we perform a linear regression analysis to determine the relationship between body weight and calories burned.
The linear regression model is of the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( y \)[/tex] represents the calories burned per minute,
- [tex]\( x \)[/tex] represents the body weight in pounds,
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the y-intercept.
Given the results from the analysis, the equation of the linear regression model is:
[tex]\[ C = 0.0270x - 0.0170 \][/tex]
This model indicates that "C", the number of calories burned per minute, can be estimated by multiplying the body weight [tex]\( x \)[/tex] by 0.0270 and then subtracting 0.0170.
### 1b. Meaning of the Slope
The slope [tex]\( m \)[/tex] of the linear regression model represents the rate of change of calories burned concerning body weight. Specifically, the slope tells us how many more calories are burned per minute for each additional pound of body weight.
In this scenario, the slope is [tex]\( 0.0270 \)[/tex].
Interpretation:
For every pound increase in body weight, approximately 0.0270 more calories are burned per minute when walking at a speed of 3 miles per hour. This means that heavier individuals will burn more calories per minute while walking at this speed compared to lighter individuals.
This interpretation helps us understand the positive correlation between body weight and calories burned: as body weight increases, the calories burned also increase, assuming the walking speed remains constant at 3 miles per hour.
