To determine the equation that represents the approximate line of best fit for the data, we need to find the relationship between the font size (x) and the word count (y). The given data pairs are:
[tex]\[
\begin{array}{c|cccccccccc}
\text{Font Size (x)} & 14 & 12 & 16 & 10 & 12 & 14 & 16 & 18 & 24 & 22 \\
\hline
\text{Word Count (y)} & 352 & 461 & 340 & 407 & 435 & 381 & 280 & 201 & 138 & 114 \\
\end{array}
\][/tex]
To find the line of best fit, we calculate the slope (m) and the y-intercept (b) for the line in the form of [tex]\( y = mx + b \)[/tex].
1. Calculating Slope (m):
Using the least squares method, the slope (m) represents the change in the word count for a one-unit change in the font size.
2. Calculating Y-intercept (b):
The y-intercept (b) is the value of y when x is 0.
By performing these calculations, we obtain:
- The slope of the best fit line: [tex]\( m ≈ -26.059 \)[/tex]
- The y-intercept of the best fit line: [tex]\( b ≈ 722.633 \)[/tex]
Therefore, the equation of the line of best fit is approximately:
[tex]\[ y = -26.059 x + 722.633 \][/tex]
Given the possible equations provided, we compare the calculated slope and y-intercept with each option:
1. [tex]\( y = -55x + 407 \)[/tex]
2. [tex]\( y = -41x + 814 \)[/tex]
3. [tex]\( y = -38x + 922 \)[/tex]
4. [tex]\( y = -26x + 723 \)[/tex]
The equation [tex]\( y = -26x + 723 \)[/tex] is the closest to our calculated values where the slope is approximately -26 and the intercept is approximately 723.
Hence, the equation that best represents the approximate line of best fit for the data is:
[tex]\[ y = -26x + 723 \][/tex]
