Question

gas score, x sleep time, y (in hours) xy 9.0 5.6 50.4 6.9 6.8 46.92 9.1 6.5 59.15 4.0 7.2 28.8 6.4 6.1 39.04 8.1 7.4 59.94 5.1 6.3 32.13 3.6 7.8 28.08 1.9 8.1 15.39 2.8 7.1 19.88 3.8 8.5 32.3 6.0 7.9 47.4 8.1 5.3 42.93 0.9 7.2 6.48 1.4 8.7 12.18 send data to calculator send data to excel what is the slope of the least - squares regression line for these data? carry your intermediate computations to at le decimal places and round your answer to at least two decimal places. (if necessary, consult a list of formulas.)

Explanation:

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

Let \(n = 15\) (number of data points).
\(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\)
\(\sum_{i=1}^{15}x_{i}=9.0 + 6.9+9.1 + 4.0+6.4 + 8.1+5.1 + 3.6+1.9 + 2.8+3.8 + 6.0+8.1 + 0.9+1.4=70.7\)
\(\bar{x}=\frac{70.7}{15}\approx4.713\)

\(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\)
\(\sum_{i=1}^{15}y_{i}=5.6 + 6.8+6.5 + 7.2+6.1 + 7.4+6.3 + 7.8+8.1 + 7.1+8.5 + 7.9+5.3 + 7.2+8.7 = 106.5\)
\(\bar{y}=\frac{106.5}{15}=7.1\)

Step2: Calculate the numerator and denominator for the slope formula

The formula for the slope \(b\) of the least - squares regression line is \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

First, calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) and \((x_{i}-\bar{x})^{2}\) for each \(i\):
For \(x = 9.0,y = 5.6\): \((9.0 - 4.713)(5.6 - 7.1)=4.287\times(- 1.5)=-6.4305\)
\((9.0 - 4.713)^{2}=4.287^{2}\approx18.388\)

For \(x = 6.9,y = 6.8\): \((6.9 - 4.713)(6.8 - 7.1)=2.187\times(-0.3)=-0.6561\)
\((6.9 - 4.713)^{2}=2.187^{2}\approx4.783\)

For \(x = 9.1,y = 6.5\): \((9.1 - 4.713)(6.5 - 7.1)=4.387\times(-0.6)=-2.6322\)
\((9.1 - 4.713)^{2}=4.387^{2}\approx19.246\)

For \(x = 4.0,y = 7.2\): \((4.0 - 4.713)(7.2 - 7.1)=(-0.713)\times0.1=-0.0713\)
\((4.0 - 4.713)^{2}=(-0.713)^{2}\approx0.508\)

For \(x = 6.4,y = 6.1\): \((6.4 - 4.713)(6.1 - 7.1)=1.687\times(-1)=-1.687\)
\((6.4 - 4.713)^{2}=1.687^{2}\approx2.846\)

For \(x = 8.1,y = 7.4\): \((8.1 - 4.713)(7.4 - 7.1)=3.387\times0.3 = 1.0161\)
\((8.1 - 4.713)^{2}=3.387^{2}\approx11.472\)

For \(x = 5.1,y = 6.3\): \((5.1 - 4.713)(6.3 - 7.1)=0.387\times(-0.8)=-0.3096\)
\((5.1 - 4.713)^{2}=0.387^{2}\approx0.150\)

For \(x = 3.6,y = 7.8\): \((3.6 - 4.713)(7.8 - 7.1)=(-1.113)\times0.7=-0.7791\)
\((3.6 - 4.713)^{2}=(-1.113)^{2}\approx1.239\)

For \(x = 1.9,y = 8.1\): \((1.9 - 4.713)(8.1 - 7.1)=(-2.813)\times1=-2.813\)
\((1.9 - 4.713)^{2}=(-2.813)^{2}\approx7.913\)

For \(x = 2.8,y = 7.1\): \((2.8 - 4.713)(7.1 - 7.1)=(-1.913)\times0 = 0\)
\((2.8 - 4.713)^{2}=(-1.913)^{2}\approx3.669\)

For \(x = 3.8,y = 8.5\): \((3.8 - 4.713)(8.5 - 7.1)=(-0.913)\times1.4=-1.2782\)
\((3.8 - 4.713)^{2}=(-0.913)^{2}\approx0.833\)

For \(x = 6.0,y = 7.9\): \((6.0 - 4.713)(7.9 - 7.1)=1.287\times0.8 = 1.0296\)
\((6.0 - 4.713)^{2}=1.287^{2}\approx1.656\)

For \(x = 8.1,y = 5.3\): \((8.1 - 4.713)(5.3 - 7.1)=3.387\times(-1.8)=-6.0966\)
\((8.1 - 4.713)^{2}=3.387^{2}\approx11.472\)

For \(x = 0.9,y = 7.2\): \((0.9 - 4.713)(7.2 - 7.1)=(-3.813)\times0.1=-0.3813\)
\((0.9 - 4.713)^{2}=(-3.813)^{2}\approx14.539\)

For \(x = 1.4,y = 8.7\): \((1.4 - 4.713)(8.7 - 7.1)=(-3.313)\times1.6=-5.3008\)
\((1.4 - 4.713)^{2}=(-3.313)^{2}\approx10.976\)

\(\sum_{i = 1}^{15}(x_{i}-\bar{x})(y_{i}-\bar{y})=-6.4305-0.6561-2.6322-0.0713-1.687 + 1.0161-0.3096-0.7791-2.813+0-1.2782 + 1.0296-6.0966-0.3813-5.3008=-26.757\)

\(\sum_{i = 1}^{15}(x_{i}-\bar{x})^{2}=18.388 + 4.783+19.246+0.508+2.846+11.472+0.150+1.239+7.913+3.669+0.833+1.656+11.472+14.539+10.976=119.778\)

Then \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}=\frac{-26.757}{119.778}\approx - 0.22\)

Answer:

\(-0.22\)