Question

aubrey is an editor reviewing short story submissions for a literary magazine. she collected data showing that shes an efficient reviewer and wants to leverage this to get a raise.
for a week, aubrey recorded the number of words in each short story submission, x, and how long it took her to review that story (in minutes), y.
round your answers to the nearest thousandth.
y = x +

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

Let \(x\) be the number of words and \(y\) be the time in minutes.
\(\bar{x}=\frac{2827 + 2869+3450+3938+4766+4775+5301+6502}{8}=\frac{34428}{8} = 4303.5\)
\(\bar{y}=\frac{32.37+42.57+37.78+31.72+35.40+55.44+56.01+55.96}{8}=\frac{347.25}{8}=43.40625\)

Step2: Calculate the numerator and denominator for the slope \(m\)

The formula for the slope \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\((x_1-\bar{x})(y_1 - \bar{y})=(2827 - 4303.5)(32.37-43.40625)=(- 1476.5)\times(-11.03625)=16295.45625\)
\((x_2-\bar{x})(y_2 - \bar{y})=(2869 - 4303.5)(42.57-43.40625)=(-1434.5)\times(-0.83625)=1200.195625\)
\((x_3-\bar{x})(y_3 - \bar{y})=(3450 - 4303.5)(37.78-43.40625)=(-853.5)\times(-5.62625)=4800.90625\)
\((x_4-\bar{x})(y_4 - \bar{y})=(3938 - 4303.5)(31.72-43.40625)=(-365.5)\times(-11.68625)=4271.22375\)
\((x_5-\bar{x})(y_5 - \bar{y})=(4766 - 4303.5)(35.40-43.40625)=(462.5)\times(-8.00625)=-3703.828125\)
\((x_6-\bar{x})(y_6 - \bar{y})=(4775 - 4303.5)(55.44-43.40625)=(471.5)\times(12.03375)=5673.916875\)
\((x_7-\bar{x})(y_7 - \bar{y})=(5301 - 4303.5)(56.01-43.40625)=(997.5)\times(12.60375)=12572.246875\)
\((x_8-\bar{x})(y_8 - \bar{y})=(6502 - 4303.5)(55.96-43.40625)=(2198.5)\times(12.55375)=27609.416875\)
\(\sum_{i = 1}^{8}(x_{i}-\bar{x})(y_{i}-\bar{y})=16295.45625+1200.195625 + 4800.90625+4271.22375-3703.828125+5673.916875+12572.246875+27609.416875=68719.535\)

\((x_1-\bar{x})^2=(2827 - 4303.5)^2=(-1476.5)^2 = 2180052.25\)
\((x_2-\bar{x})^2=(2869 - 4303.5)^2=(-1434.5)^2=2057890.25\)
\((x_3-\bar{x})^2=(3450 - 4303.5)^2=(-853.5)^2=728462.25\)
\((x_4-\bar{x})^2=(3938 - 4303.5)^2=(-365.5)^2=133590.25\)
\((x_5-\bar{x})^2=(4766 - 4303.5)^2=(462.5)^2=213906.25\)
\((x_6-\bar{x})^2=(4775 - 4303.5)^2=(471.5)^2=222312.25\)
\((x_7-\bar{x})^2=(5301 - 4303.5)^2=(997.5)^2=995006.25\)
\((x_8-\bar{x})^2=(6502 - 4303.5)^2=(2198.5)^2=4833402.25\)
\(\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=2180052.25+2057890.25+728462.25+133590.25+213906.25+222312.25+995006.25+4833402.25=11364622\)

\(m=\frac{68719.535}{11364622}\approx0.006\)

Step3: Calculate the \(y -\)intercept \(b\)

Using the formula \(y=mx + b\), we substitute \(m\), \(\bar{x}\), and \(\bar{y}\)
\(b=\bar{y}-m\bar{x}\)
\(b = 43.40625-0.006\times4303.5\)
\(b=43.40625 - 25.821=17.585\)

Answer:

\(y = 0.006x+17.585\)