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
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?
○ y = -55x + 407
○ y = -41x + 814
○ y = -38x + 922
○ y = -26x + 723

Explanation:

Step1: Calculate the mean of x (font size)

First, sum all the font sizes: \(14 + 12 + 16 + 10 + 12 + 14 + 16 + 18 + 24 + 22\)
\(= (14\times2)+(12\times2)+(16\times2)+10 + 18 + 24 + 22\)
\(= 28 + 24 + 32 + 10 + 18 + 24 + 22\)
\(= 28+24 = 52; 52+32 = 84; 84+10 = 94; 94+18 = 112; 112+24 = 136; 136+22 = 158\)
Mean of x, \(\bar{x}=\frac{158}{10}=15.8\)

Step2: Calculate the mean of y (word count)

Sum all the word counts: \(352 + 461 + 340 + 407 + 435 + 381 + 280 + 201 + 138 + 114\)
\(= 352+461 = 813; 813+340 = 1153; 1153+407 = 1560; 1560+435 = 1995; 1995+381 = 2376; 2376+280 = 2656; 2656+201 = 2857; 2857+138 = 2995; 2995+114 = 3109\)
Mean of y, \(\bar{y}=\frac{3109}{10}=310.9\)

Step3: Test each equation with \(\bar{x}\) and \(\bar{y}\)

The line of best fit should pass close to the point \((\bar{x},\bar{y})=(15.8, 310.9)\)

Test Option 1: \(y = -55x + 407\)

Substitute \(x = 15.8\): \(y=-55\times15.8 + 407=-869 + 407=-462\) (Far from 310.9)

Test Option 2: \(y = -41x + 814\)

Substitute \(x = 15.8\): \(y=-41\times15.8+814=-647.8 + 814 = 166.2\) (Far from 310.9)

Test Option 3: \(y = -38x + 922\)

Substitute \(x = 15.8\): \(y=-38\times15.8+922=-600.4 + 922 = 321.6\) (Close to 310.9)

Test Option 4: \(y = -26x + 723\)

Substitute \(x = 15.8\): \(y=-26\times15.8+723=-410.8 + 723 = 312.2\) (Also close, but let's check the trend)

Wait, maybe better to check the slope. As font size (x) increases, word count (y) decreases, so negative slope. Let's check another point, say x=12, y should be around 461 or 435 (mean for x=12: (461+435)/2=448)

Test Option 3 at x=12: \(y=-38\times12 + 922=-456 + 922 = 466\) (Close to 448)

Test Option 4 at x=12: \(y=-26\times12 + 723=-312 + 723 = 411\) (Further from 448)

Another point: x=24, y=138

Option 3: \(y=-38\times24 + 922=-912 + 922 = 10\) (No, wait, miscalculation: -3824=-912, 922-912=10? No, 922-912=10? Wait 3824: 4024=960, minus 224=48, so 960-48=912. 922-912=10. But actual y=138. So maybe my mean approach was better. Wait, maybe I made a mistake in mean. Wait the data points:

Wait let's recalculate the mean of x: 14,12,16,10,12,14,16,18,24,22. Let's add again:

14+12=26; 26+16=42; 42+10=52; 52+12=64; 64+14=78; 78+16=94; 94+18=112; 112+24=136; 136+22=158. Yes, 158/10=15.8. Correct.

Mean of y: 352+461=813; +340=1153; +407=1560; +435=1995; +381=2376; +280=2656; +201=2857; +138=2995; +114=3109. 3109/10=310.9. Correct.

Now test Option 3: y=-38x + 922. At x=15.8, y=-3815.8 + 922. 3815=570, 38*0.8=30.4, so 570+30.4=600.4. 922-600.4=321.6. Close to 310.9.

Option 4: y=-26x + 723. At x=15.8, y=-2615.8 + 723. 2615=390, 26*0.8=20.8, so 390+20.8=410.8. 723-410.8=312.2. Also close. But let's check the slope magnitude. As x increases, y decreases. Let's take two points: (10,407) and (24,138). Slope: (138-407)/(24-10)=(-269)/14≈-19.2. But the options have slopes -55, -41, -38, -26. Wait, maybe my two points are not the best. Let's take (12,461) and (22,114). Slope: (114-461)/(22-12)=(-347)/10=-34.7. Close to -38. So Option 3 (y=-38x + 922) is closer.

Answer:

\(y = -38x + 922\) (Option C, assuming the options are labeled C as \(y = -38x + 922\))