Question

gabriel is a math teacher at a large school. he wonders if his test problems are too wordy. gabriel is curious whether the wordiness is affecting student performance.
for the last several tests, gabriel computes the average number of words in each question, x, as well as the average percentage scores on the tests, y.
round your answers to the nearest thousandth.
y = x +

Explanation:

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

Let \(x\) be the average question length and \(y\) be the average student score.
\(\bar{x}=\frac{13.99 + 17.12+21.03+26.15+26.75+30.09+35.71+43.47}{8}=\frac{214.31}{8}=26.78875\)
\(\bar{y}=\frac{89.35 + 81.30+80.73+82.29+85.27+81.40+79.02+75.30}{8}=\frac{654.66}{8}=81.8325\)

Step2: Calculate the numerator and denominator for the slope \(b_1\)

The formula for the slope \(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\)
\((x_1-\bar{x})(y_1-\bar{y})=(13.99 - 26.78875)(89.35 - 81.8325)=(- 12.79875)\times7.5175=-96.277\)
\((x_2-\bar{x})(y_2-\bar{y})=(17.12 - 26.78875)(81.30 - 81.8325)=(-9.66875)\times(- 0.5325)=5.151\)
\((x_3-\bar{x})(y_3-\bar{y})=(21.03 - 26.78875)(80.73 - 81.8325)=(-5.75875)\times(-1.1025)=6.35\)
\((x_4-\bar{x})(y_4-\bar{y})=(26.15 - 26.78875)(82.29 - 81.8325)=(-0.63875)\times0.4575=-0.292\)
\((x_5-\bar{x})(y_5-\bar{y})=(26.75 - 26.78875)(85.27 - 81.8325)=(-0.03875)\times3.4375=-0.133\)
\((x_6-\bar{x})(y_6-\bar{y})=(30.09 - 26.78875)(81.40 - 81.8325)=(3.30125)\times(-0.4325)=-1.428\)
\((x_7-\bar{x})(y_7-\bar{y})=(35.71 - 26.78875)(79.02 - 81.8325)=(8.92125)\times(-2.8125)=-25.193\)
\((x_8-\bar{x})(y_8-\bar{y})=(43.47 - 26.78875)(75.30 - 81.8325)=(16.68125)\times(-6.5325)=-109.009\)
\(\sum_{i = 1}^{8}(x_i-\bar{x})(y_i - \bar{y})=-96.277+5.151 + 6.35-0.292-0.133-1.428-25.193-109.009=-220.831\)

\((x_1-\bar{x})^2=(13.99 - 26.78875)^2=(-12.79875)^2 = 163.827\)
\((x_2-\bar{x})^2=(17.12 - 26.78875)^2=(-9.66875)^2=93.575\)
\((x_3-\bar{x})^2=(21.03 - 26.78875)^2=(-5.75875)^2=33.154\)
\((x_4-\bar{x})^2=(26.15 - 26.78875)^2=(-0.63875)^2=0.408\)
\((x_5-\bar{x})^2=(26.75 - 26.78875)^2=(-0.03875)^2=0.002\)
\((x_6-\bar{x})^2=(30.09 - 26.78875)^2=(3.30125)^2 = 10.898\)
\((x_7-\bar{x})^2=(35.71 - 26.78875)^2=(8.92125)^2=79.599\)
\((x_8-\bar{x})^2=(43.47 - 26.78875)^2=(16.68125)^2=278.253\)
\(\sum_{i=1}^{8}(x_i-\bar{x})^2=163.827+93.575+33.154+0.408+0.002+10.898+79.599+278.253=669.716\)

\(b_1=\frac{-220.831}{669.716}\approx - 0.330\)

Step3: Calculate the intercept \(b_0\)

The formula for the intercept \(b_0=\bar{y}-b_1\bar{x}\)
\(b_0 = 81.8325-(-0.330)\times26.78875=81.8325 + 8.840=90.673\)

Answer:

\(y=-0.330x + 90.673\)