Question

annual high temperatures in a certain location have been tracked for several years. let x represent the year and y the high temperature. based on the data shown below, calculate the regression line (each value to two decimal places).
y = x +
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Explanation:

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

The mean of \(x\) values: \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\)
\(\sum_{i=1}^{14}x_{i}=3 + 4+5+6+7+8+9+10+11+12+13+14+15+16=\frac{(3 + 16)\times14}{2}=133\)
\(n = 14\), so \(\bar{x}=\frac{133}{14}=9.5\)

The mean of \(y\) values: \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\)
\(\sum_{i=1}^{14}y_{i}=28.52+29.56+32.6+34.64+34.68+39.82+40.26+42.4+46.44+48.08+51.22+51.76+56.7+58.44 = 585.72\)
\(\bar{y}=\frac{585.72}{14}\approx41.84\)

Step2: Calculate the slope \(b\)

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

First, calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) and \((x_{i}-\bar{x})^{2}\) for each \(i\):
For \(x = 3,y = 28.52\): \((3 - 9.5)(28.52-41.84)=(- 6.5)\times(-13.32)=86.58\), \((3 - 9.5)^{2}=(-6.5)^{2}=42.25\)
For \(x = 4,y = 29.56\): \((4 - 9.5)(29.56 - 41.84)=(-5.5)\times(-12.28)=67.54\), \((4 - 9.5)^{2}=(-5.5)^{2}=30.25\)
For \(x = 5,y = 32.6\): \((5 - 9.5)(32.6-41.84)=(-4.5)\times(-9.24)=41.58\), \((5 - 9.5)^{2}=(-4.5)^{2}=20.25\)
For \(x = 6,y = 34.64\): \((6 - 9.5)(34.64 - 41.84)=(-3.5)\times(-7.2)=25.2\), \((6 - 9.5)^{2}=(-3.5)^{2}=12.25\)
For \(x = 7,y = 34.68\): \((7 - 9.5)(34.68 - 41.84)=(-2.5)\times(-7.16)=17.9\), \((7 - 9.5)^{2}=(-2.5)^{2}=6.25\)
For \(x = 8,y = 39.82\): \((8 - 9.5)(39.82 - 41.84)=(-1.5)\times(-2.02)=3.03\), \((8 - 9.5)^{2}=(-1.5)^{2}=2.25\)
For \(x = 9,y = 40.26\): \((9 - 9.5)(40.26 - 41.84)=(-0.5)\times(-1.58)=0.79\), \((9 - 9.5)^{2}=(-0.5)^{2}=0.25\)
For \(x = 10,y = 42.4\): \((10 - 9.5)(42.4 - 41.84)=(0.5)\times(0.56)=0.28\), \((10 - 9.5)^{2}=(0.5)^{2}=0.25\)
For \(x = 11,y = 46.44\): \((11 - 9.5)(46.44 - 41.84)=(1.5)\times(4.6)=6.9\), \((11 - 9.5)^{2}=(1.5)^{2}=2.25\)
For \(x = 12,y = 48.08\): \((12 - 9.5)(48.08 - 41.84)=(2.5)\times(6.24)=15.6\), \((12 - 9.5)^{2}=(2.5)^{2}=6.25\)
For \(x = 13,y = 51.22\): \((13 - 9.5)(51.22 - 41.84)=(3.5)\times(9.38)=32.83\), \((13 - 9.5)^{2}=(3.5)^{2}=12.25\)
For \(x = 14,y = 51.76\): \((14 - 9.5)(51.76 - 41.84)=(4.5)\times(9.92)=44.64\), \((14 - 9.5)^{2}=(4.5)^{2}=20.25\)
For \(x = 15,y = 56.7\): \((15 - 9.5)(56.7 - 41.84)=(5.5)\times(14.86)=81.73\), \((15 - 9.5)^{2}=(5.5)^{2}=30.25\)
For \(x = 16,y = 58.44\): \((16 - 9.5)(58.44 - 41.84)=(6.5)\times(16.6)=107.9\), \((16 - 9.5)^{2}=(6.5)^{2}=42.25\)

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=86.58+67.54+41.58+25.2+17.9+3.03+0.79+0.28+6.9+15.6+32.83+44.64+81.73+107.9 = 532.57\)

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=42.25+30.25+20.25+12.25+6.25+2.25+0.25+0.25+2.25+6.25+12.25+20.25+30.25+42.25=227.5\)

\(b=\frac{532.57}{227.5}\approx2.34\)

Step3: Calculate the intercept \(a\)

The formula for the intercept \(a=\bar{y}-b\bar{x}\)
\(a = 41.84-2.34\times9.5=41.84 - 22.23=19.61\)

Answer:

\(y = 2.34x+19.61\)