distance from the beach, x (in miles) selling price, y (in thousands of dollars) xy 5.4 268.3 1448.82 12.0 280.2 3362.4 6.3 248.4 1564.92 10.1 301.2 3042.12 11.9 186.1 2214.59 11.1 193.7 2150.07 12.3 206.4 2538.72 11.4 269.5 3072.3 11.9 218.7 2602.53 18.3 224.6 4110.18 13.3 268.8 3575.04 7.0 304.9 2134.3 7.6 226.0 1717.6 7.5 220.7 1655.25 4.0 273.2 1092.8 2.9 320.7 930.03 what is the sample correlation coefficient for these data? carry your intermediate computations to at least fou places and round your answer to at least three decimal places. (if necessary, consult a list of formulas.)

Question

Accepted Answer

# Explanation:
## Step1: Calculate the means of $x$ and $y$
Let $n = 16$ (number of data points).
$\bar{x}=\frac{5.4 + 12.0+6.3 + 10.1+11.9+11.1+12.3+11.4+11.9+18.3+13.3+7.0+7.6+7.5+4.0+2.9}{16}=\frac{164.3}{16}=10.26875$
$\bar{y}=\frac{268.3 + 280.2+248.4 + 301.2+186.1+193.7+206.4+269.5+218.7+224.6+268.8+304.9+226.0+220.7+273.2+320.7}{16}=\frac{4080.6}{16}=255.0375$

## Step2: Calculate $S_{xx}$, $S_{yy}$, and $S_{xy}$
$S_{xx}=\sum_{i = 1}^{n}(x_i-\bar{x})^2$
$$
\begin{align*}
S_{xx}&=(5.4 - 10.26875)^2+(12.0 - 10.26875)^2+(6.3 - 10.26875)^2+(10.1 - 10.26875)^2+(11.9 - 10.26875)^2+(11.1 - 10.26875)^2+(12.3 - 10.26875)^2+(11.4 - 10.26875)^2+(11.9 - 10.26875)^2+(18.3 - 10.26875)^2+(13.3 - 10.26875)^2+(7.0 - 10.26875)^2+(7.6 - 10.26875)^2+(7.5 - 10.26875)^2+(4.0 - 10.26875)^2+(2.9 - 10.26875)^2\
&=(- 4.86875)^2+(1.73125)^2+(-3.96875)^2+(-0.16875)^2+(1.63125)^2+(0.83125)^2+(2.03125)^2+(1.13125)^2+(1.63125)^2+(8.03125)^2+(3.03125)^2+(-3.26875)^2+(-2.66875)^2+(-2.76875)^2+(-6.26875)^2+(-7.36875)^2\
&=23.7051+3.0072+15.7510+0.0285+2.6610+0.6913+4.1250+1.2797+2.6610+64.5010+9.1880+10.6846+7.1225+7.6664+39.2970+54.2980\
&=246.6663
\end{align*}
$$
$S_{yy}=\sum_{i = 1}^{n}(y_i-\bar{y})^2$
$$
\begin{align*}
S_{yy}&=(268.3 - 255.0375)^2+(280.2 - 255.0375)^2+(248.4 - 255.0375)^2+(301.2 - 255.0375)^2+(186.1 - 255.0375)^2+(193.7 - 255.0375)^2+(206.4 - 255.0375)^2+(269.5 - 255.0375)^2+(218.7 - 255.0375)^2+(224.6 - 255.0375)^2+(268.8 - 255.0375)^2+(304.9 - 255.0375)^2+(226.0 - 255.0375)^2+(220.7 - 255.0375)^2+(273.2 - 255.0375)^2+(320.7 - 255.0375)^2\
&=(13.2625)^2+(25.1625)^2+(-6.6375)^2+(46.1625)^2+(-68.9375)^2+(-61.3375)^2+(-48.6375)^2+(14.4625)^2+(-36.3375)^2+(-30.4375)^2+(13.7625)^2+(49.8625)^2+(-29.0375)^2+(-34.3375)^2+(18.1625)^2+(65.6625)^2\
&=175.88+633.15+44.06+2130.98+4752.44+3762.37+2365.67+209.17+1320.42+926.44+189.41+2486.31+843.18+1179.07+329.88+4311.54\
&=25360.96
\end{align*}
$$
$S_{xy}=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})$
\[
\begin{align*}
S_{xy}&=(5.4 - 10.26875)(268.3 - 255.0375)+(12.0 - 10.26875)(280.2 - 255.0375)+(6.3 - 10.26875)(248.4 - 255.0375)+(10.1 - 10.26875)(301.2 - 255.0375)+(11.9 - 10.26875)(186.1 - 255.0375)+(11.1 - 10.26875)(193.7 - 255.0375)+(12.3 - 10.26875)(206.4 - 255.0375)+(11.4 - 10.26875)(269.5 - 255.0375)+(11.9 - 10.26875)(218.7 - 255.0375)+(18.3 - 10.26875)(224.6 - 255.0375)+(13.3 - 10.26875)(268.8 - 255.0375)+(7.0 - 10.26875)(304.9 - 255.0375)+(7.6 - 10.26875)(226.0 - 255.0375)+(7.5 - 10.26875)(220.7 - 255.0375)+(4.0 - 10.26875)(273.2 - 255.0375)+(2.9 - 10.26875)(320.7 - 255.0375)\
&=(-4.86875)(13.2625)+(1.73125)(25.1625)+(-3.96875)(-6.6375)+(-0.16875)(46.1625)+(1.63125)(-68.9375)+(0.83125)(-61.3375)+(2.03125)(-48.6375)+(1.13125)(14.4625)+(1.63125)(-36.3375)+(8.03125)(-30.4375)+(3.03125)(13.7625)+(-3.26875)(49.8625)+(-2.66875)(-29.0375)+(-2.76875)(-34.3375)+(-6.26875)(18.1625)+(-7.36875)(65.6625)\
&=-64.63+43.67 + 26.37-7.79-112.41-51.43-98.50+16.39-59.29-244.44+41.71-162.93 + 77.50+94.00-113.90-483.37\
&=-1335.22
\end{align*}
\]

## Step3: Calculate the sample correlation coefficient $r$
The formula for the sample correlation coefficient is $r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$
$r=\frac{- 1335.22}{\sqrt{246.6663\times25360.96}}$
$r=\frac{-1335.22}{\sqrt{6257099.19}}$
$r=\frac{-1335.22}{2501.42}$
$r\approx - 0.534$

# Answer:
$-0.534$

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QUESTION IMAGE

Question

Explanation:

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

Step2: Calculate \(S_{xx}\), \(S_{yy}\), and \(S_{xy}\)

Step3: Calculate the sample correlation coefficient \(r\)

Answer: