Rob is investigating the effects of font size on the number of words that fit on a page. He changes the font size on an essay and records the number of words on one page of the essay. The table shows his data.

Words per Page

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline \begin{tabular}{c}
Font \\
Size
\end{tabular} & 14 & 12 & 16 & 10 & 12 & 14 & 16 & 18 & 24 & 22 \\
\hline \begin{tabular}{c}
Word \\
Count
\end{tabular} & 352 & 461 & 340 & 407 & 435 & 381 & 280 & 201 & 138 & 114 \\
\hline
\end{tabular}

Which equation represents the approximate line of best fit for the data, where [tex]$x$[/tex] represents font size and [tex]$y$[/tex] represents the number of words on one page?

A. [tex]$y = -55x + 407$[/tex]
B. [tex]$y = -41x + 814$[/tex]
C. [tex]$y = -38x + 922$[/tex]
D. [tex]$y = -26x + 723$[/tex]



Answer :

To find the equation that represents the approximate line of best fit for the given data, we'll use the least-squares regression method. The regression line can be represented by the equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the intercept.

Given the data points:

- 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

After calculating the line of best fit using the least-squares method, we find the slope [tex]\( m \)[/tex] and the intercept [tex]\( b \)[/tex]:

- The slope [tex]\( m \)[/tex] is approximately [tex]\( -26.059 \)[/tex]
- The intercept [tex]\( b \)[/tex] is approximately [tex]\( 722.633 \)[/tex]

Now, let us compare the given options with our calculated values:

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 option that matches closest to our calculated line of best fit ([tex]\( y = -26.059x + 722.633 \)[/tex]) is clearly:

[tex]\( y = -26x + 723 \)[/tex]

Thus, the equation that best represents the approximate line of best fit for the data is:

[tex]\[ y = -26x + 723 \][/tex]