Question

find the equation for the least squares regression line of the data described below.
santiago is a psychology professor studying the effect of media violence on modern culture. as part of his research, he analyzed the content of several recent action movies.
for each movie, he noted the running time (in minutes), x, and the number of explosions, y.
running time (in minutes) explosions
86 22
88 9
94 25
95 14
99 18
round your answers to the nearest thousandth.
y = x +
save answer

Explanation:

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\).
For \(x\): \(x_1 = 86\), \(x_2=88\), \(x_3 = 94\), \(x_4=95\), \(x_5 = 99\). \(\sum_{i=1}^{5}x_{i}=86 + 88+94+95+99=462\). \(\bar{x}=\frac{462}{5}=92.4\)
For \(y\): \(y_1 = 22\), \(y_2=9\), \(y_3 = 25\), \(y_4=14\), \(y_5 = 18\). \(\sum_{i=1}^{5}y_{i}=22 + 9+25+14+18=88\). \(\bar{y}=\frac{88}{5}=17.6\)

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

The formula for the slope \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\((x_1-\bar{x})(y_1-\bar{y})=(86 - 92.4)(22-17.6)=(- 6.4)\times4.4=-28.16\)
\((x_2-\bar{x})(y_2-\bar{y})=(88 - 92.4)(9 - 17.6)=(-4.4)\times(-8.6)=37.84\)
\((x_3-\bar{x})(y_3-\bar{y})=(94 - 92.4)(25 - 17.6)=1.6\times7.4 = 11.84\)
\((x_4-\bar{x})(y_4-\bar{y})=(95 - 92.4)(14 - 17.6)=2.6\times(-3.6)=-9.36\)
\((x_5-\bar{x})(y_5-\bar{y})=(99 - 92.4)(18 - 17.6)=6.6\times0.4 = 2.64\)
\(\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})=-28.16+37.84 + 11.84-9.36+2.64=14.8\)

\((x_1-\bar{x})^{2}=(86 - 92.4)^{2}=(-6.4)^{2}=40.96\)
\((x_2-\bar{x})^{2}=(88 - 92.4)^{2}=(-4.4)^{2}=19.36\)
\((x_3-\bar{x})^{2}=(94 - 92.4)^{2}=1.6^{2}=2.56\)
\((x_4-\bar{x})^{2}=(95 - 92.4)^{2}=2.6^{2}=6.76\)
\((x_5-\bar{x})^{2}=(99 - 92.4)^{2}=6.6^{2}=43.56\)
\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=40.96+19.36+2.56+6.76+43.56=113.2\)

\(b=\frac{14.8}{113.2}\approx0.131\)

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

The formula for the \(y-\)intercept \(a=\bar{y}-b\bar{x}\)
\(a = 17.6-0.131\times92.4\)
\(a=17.6 - 12.1\)
\(a = 5.5\)

Answer:

\(y=0.131x + 5.5\)