To determine the equation that represents the approximate line of best fit for Rob's data, we first need to perform linear regression analysis on the given data. We'll calculate the slope and intercept of the best-fit line and then compare the results with the provided options.
1. Extract the Data:
- Font Sizes ([tex]\(x\)[/tex]): 14, 12, 16, 10, 12, 14, 16, 18, 24, 22
- Word Counts ([tex]\(y\)[/tex]): 352, 461, 340, 407, 435, 381, 280, 201, 138, 114
2. Perform Linear Regression:
The linear regression will give us the slope ([tex]\(b\)[/tex]) and the intercept ([tex]\(a\)[/tex]) of the line [tex]\(y = bx + a\)[/tex].
3. Calculate the Slope and Intercept:
After performing the linear regression calculations, we find the following values:
- Slope ([tex]\(b\)[/tex]): approximately [tex]\(-26.059\)[/tex]
- Intercept ([tex]\(a\)[/tex]): approximately [tex]\(722.633\)[/tex]
4. Match Against the Given Options:
We are given four potential equations:
- [tex]\(y = -55x + 407\)[/tex]
- [tex]\(y = -41x + 814\)[/tex]
- [tex]\(y = -38x + 922\)[/tex]
- [tex]\(y = -26x + 723\)[/tex]
Comparing our calculated slope and intercept with these options:
- The closest match for the slope ([tex]\(-26.059\)[/tex] is approximately [tex]\(-26\)[/tex]) and intercept ([tex]\(722.633\)[/tex] is approximately [tex]\(723\)[/tex]) is the equation: [tex]\(y = -26x + 723\)[/tex].
Therefore, the equation that best represents the approximate line of best fit for the data is:
[tex]\[ \boxed{y = -26x + 723} \][/tex]
