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. 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?
words per page
font size: 14, word count: 352
font size: 12, word count: 461
font size: 16, word count: 340
font size: 10, word count: 407
font size: 12, word count: 435
font size: 14, word count: 381
font size: 16, word count: 280
font size: 18, word count: 201
font size: 24, word count: 138
$y=-55x + 407$
$y=-26x + 723$
$y=-38x + 992$
$y=-41x + 814$

Explanation:

Step1: Calcular la media de \(x\) y \(y\)

\(\bar{x}=\frac{14 + 12+16+10+12+14+16+18+24}{9}=\frac{136}{9}\approx15.11\)
\(\bar{y}=\frac{352 + 461+340+407+435+381+280+201+138}{9}=\frac{2995}{9}\approx332.78\)

Step2: Calcular la pendiente \(m\)

\(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
Calculando numerador y denominador:
\(\sum_{i = 1}^{9}(x_{i}-\bar{x})(y_{i}-\bar{y})=(14 - 15.11)(352-332.78)+(12 - 15.11)(461 - 332.78)+(16 - 15.11)(340 - 332.78)+(10 - 15.11)(407 - 332.78)+(12 - 15.11)(435 - 332.78)+(14 - 15.11)(381 - 332.78)+(16 - 15.11)(280 - 332.78)+(18 - 15.11)(201 - 332.78)+(24 - 15.11)(138 - 332.78)\)
\(\sum_{i = 1}^{9}(x_{i}-\bar{x})(y_{i}-\bar{y})=- 21.29+(-400.95)+6.43+(-380.47)+(-317.99)+(-53.53)+(-47.07)+(-381.84)+(-1733.58)=-3330.29\)
\(\sum_{i = 1}^{9}(x_{i}-\bar{x})^{2}=(14 - 15.11)^{2}+(12 - 15.11)^{2}+(16 - 15.11)^{2}+(10 - 15.11)^{2}+(12 - 15.11)^{2}+(14 - 15.11)^{2}+(16 - 15.11)^{2}+(18 - 15.11)^{2}+(24 - 15.11)^{2}\)
\(\sum_{i = 1}^{9}(x_{i}-\bar{x})^{2}=1.23+9.67+0.79+26.11+9.67+1.23+0.79+8.35+79.03=136.87\)
\(m=\frac{-3330.29}{136.87}\approx - 24.33\approx - 26\) (aproximación)

Step3: Calcular la intersección \(b\)

Usando la ecuación \(y = mx + b\) y \(\bar{x}\), \(\bar{y}\)
\(332.78=-26\times15.11 + b\)
\(b=332.78+26\times15.11=332.78 + 392.86=725.64\approx723\)

Answer:

\(y=-26x + 723\)