Let's solve this step-by-step:
1. Record the Given Data:
The weight (in pounds) and height (in inches) are given in the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline Weight~(x) & 8 & 12 & 18 & 24 & 30 & 32 & 35 & 37 & 40 \\
\hline Height~(y) & 22 & 23 & 26 & 30 & 32 & 33 & 35 & 36 & 38 \\
\hline
\end{array}
\][/tex]
2. Determine the Linear Regression Line:
By using appropriate technology, we perform a linear regression analysis on the data. The linear regression line is given by the equation:
[tex]\[
y = mx + b
\][/tex]
where [tex]\(m\)[/tex] is the slope of the line and [tex]\(b\)[/tex] is the y-intercept.
3. Find the Slope (m) and Intercept (b):
Using the given data and performing the regression analysis, we get the y-intercept ([tex]\(b\)[/tex]).
4. Identify the y-Intercept:
The y-intercept ([tex]\(b\)[/tex]) is a specific value that the technology calculates. This value tells us where the line crosses the y-axis, i.e., the value of [tex]\(y\)[/tex] when [tex]\(x = 0\)[/tex].
5. Interpret the y-Intercept:
In the context of this problem, when [tex]\(x = 0\)[/tex] (weight is 0 pounds), the height [tex]\(y\)[/tex] will be equal to the y-intercept value.
6. Result and Interpretation:
The result from our regression analysis gives us a y-intercept of 17.37.
Therefore, when the weight is 0 pounds, the height will be 17.37 inches.
Given the options provided:
- The [tex]$y$[/tex]-intercept is 17.37. When the weight is 0 pounds, the height will be 17.37 inches.
- The [tex]$y$[/tex]-intercept is -34.13. When the weight is 0 pounds, the height will be -34.13 inches.
- The [tex]$y$[/tex]-intercept is 17.37. When the weight is 0 pounds, it does not make sense to interpret the height.
- The [tex]$y$[/tex]-intercept is -34.13. When the weight is 0 pounds, the height is predicted to be -34.13 inches.
The correct answer is:
The [tex]$y$[/tex]-intercept is 17.37. When the weight is 0 pounds, the height will be 17.37 inches.
