Question

playing video games makes shannon so happy that she thinks it even helps her complete more schoolwork. shannons roommate clarence is skeptical, so over the next few days, clarence asks shannon about her progress on her daily reading.
for each day, clarence notes the number of minutes shannon spends playing video games, x. he also takes the number of pages shannon reads and divides it by the total number of pages assigned, y.
round your answers to the nearest thousandth.
y = □x + □

Explanation:

Step1: Calculate the means

Let \(x\) be the minutes playing video games and \(y\) be the percentage of reading assignment.
\(\bar{x}=\frac{107 + 112+113+115+116+117+119}{7}=\frac{809}{7}\approx115.571\)
\(\bar{y}=\frac{54.51 + 52.80+43.84+58.32+38.39+35.75+33.45}{7}=\frac{316.06}{7}\approx45.151\)

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

\(n = 7\)
Numerator: \(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\)
\((107 - 115.571)(54.51-45.151)+(112 - 115.571)(52.80 - 45.151)+(113-115.571)(43.84 - 45.151)+(115-115.571)(58.32 - 45.151)+(116-115.571)(38.39 - 45.151)+(117-115.571)(35.75 - 45.151)+(119-115.571)(33.45 - 45.151)\)
\(=(- 8.571)\times9.359+(-3.571)\times7.649+(-2.571)\times(-1.311)+(-0.571)\times13.169+0.429\times(-6.761)+1.429\times(-9.401)+3.429\times(-11.701)\)
\(=-80.239-27.326 + 3.371-7.520-2.890-13.434-40.123=-168.161\)

Denominator: \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\)
\((107 - 115.571)^{2}+(112 - 115.571)^{2}+(113-115.571)^{2}+(115-115.571)^{2}+(116-115.571)^{2}+(117-115.571)^{2}+(119-115.571)^{2}\)
\(=(-8.571)^{2}+(-3.571)^{2}+(-2.571)^{2}+(-0.571)^{2}+0.429^{2}+1.429^{2}+3.429^{2}\)
\(=73.463+12.752 + 6.610+0.326+0.184+2.042+11.758=107.135\)

\(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}=\frac{-168.161}{107.135}\approx - 1.570\)

Step3: Calculate \(a\) (intercept)

\(a=\bar{y}-b\bar{x}\)
\(a = 45.151-(-1.570)\times115.571\)
\(a=45.151 + 181.446=226.597\)

Answer:

\(y=-1.570x + 226.597\)