Question

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
font size 14 12 16 10 12 14 16 18 24 22
word count 352 461 340 407 435 381 280 201 138 114
which equation represents the approximate line of best fit for data, where x represents font size and y represents the number of words on one page?
o y = -55x + 407
o y = -41x + 814
o y = -38x + 922
o y = -26x + 723

Explanation:

Step1: Recall the slope - intercept form

The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use the fact that as the font size ($x$) increases, the word count ($y$) decreases, so the slope $m$ should be negative.

Step2: Use a point - slope approximation

We can take two points from the data, say $(10,407)$ and $(24,138)$. The slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{138 - 407}{24 - 10}=\frac{- 269}{14}\approx - 19.21$. But we can also estimate the slope by looking at the general trend.
Let's use another way. We know that when $x = 10$, $y\approx400$ and when $x = 20$, $y\approx200$. The slope $m=\frac{200 - 400}{20 - 10}=\frac{-200}{10}=- 20$.
We can also use the fact that we can substitute $x$ values into the equations and see which one gives values closest to the data points.
For $y=-41x + 814$:
When $x = 10$, $y=-41\times10 + 814=-410+814 = 404$ (close to 407).
When $x = 12$, $y=-41\times12 + 814=-492+814 = 322$ (not as close as we'd like).
For $y=-38x + 922$:
When $x = 10$, $y=-38\times10+922=-380 + 922=542$ (not close).
For $y=-26x + 723$:
When $x = 10$, $y=-26\times10+723=-260 + 723 = 463$ (not close).
For $y=-55x + 407$:
When $x = 10$, $y=-55\times10+407=-550 + 407=-143$ (way off).
If we calculate the least - squares regression (a more accurate method), but for a quick estimate, we can test points.
Let's take the average of $x$ values $\bar{x}=\frac{14 + 12+16+10+12+14+16+18+24+22}{10}=\frac{168}{10}=16.8$
and the average of $y$ values $\bar{y}=\frac{352+461+340+407+435+381+280+201+138+114}{10}=\frac{3109}{10}=310.9$
We know that the line of best fit $y=mx + b$ passes through $(\bar{x},\bar{y})$ approximately.
Substituting into $y=-41x + 814$: $y=-41\times16.8+814=-688.8+814 = 125.2$ (not great).
Substituting into $y=-38x + 922$: $y=-38\times16.8+922=-638.4+922 = 283.6$ (relatively close).

Answer:

$y=-38x + 922$