To determine how much the puppy weighed at birth based on the given data and the assumption that the relationship between time and weight is linear, we need to find the equation of the linear relationship between the number of weeks and the weight of the puppy.
Given data points:
- At week 4: weight = 16 lb
- At week 5: weight = 18.5 lb
- At week 6: weight = 21 lb
- At week 7: weight = 23.5 lb
Since the relationship between the number of weeks and the weight is linear, we can use the linear equation of the form:
[tex]\[ \text{weight} = m \cdot (\text{number of weeks}) + c \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept of the line, which represents the weight of the puppy at birth (i.e., week 0).
To find the slope [tex]\( m \)[/tex] and the intercept [tex]\( c \)[/tex], we can use the method of linear regression. Here's the step-by-step determination of the slope and intercept:
1. Calculating the slope [tex]\( m \)[/tex]:
The formula for the slope [tex]\( m \)[/tex] in linear regression is:
[tex]\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \][/tex]
Where [tex]\( x \)[/tex] represents the weeks and [tex]\( y \)[/tex] represents the weights, and [tex]\( N \)[/tex] is the number of data points.
2. Calculating the y-intercept [tex]\( c \)[/tex]:
The formula for the y-intercept [tex]\( c \)[/tex] is:
[tex]\[ c = \frac{\sum y - m(\sum x)}{N} \][/tex]
Given the calculations, we determine:
- Slope [tex]\( m = 2.5 \)[/tex]
- Intercept [tex]\( c = 6.0 \)[/tex]
Thus, the linear equation becomes:
[tex]\[ \text{weight} = 2.5 \cdot (\text{number of weeks}) + 6.0 \][/tex]
The intercept [tex]\( c \)[/tex] represents the weight of the puppy at birth (week 0). From these calculations, we find:
[tex]\[ \text{Weight at birth} = 6.0 \text{ pounds} \][/tex]
Conclusion:
The puppy weighed 6 pounds at birth. Therefore, the correct answer to how much the puppy weighed at birth is [tex]\( \boxed{6 \text{ pounds}} \)[/tex].
