question 7 of 10 what is the r - value of the following data to three decimal places? x y 1 20 3 14 5 10 9 6 16 4

Question

Accepted Answer

# Explanation:
## Step1: Calculate means
Let $n = 5$.
$\bar{x}=\frac{1 + 3+5 + 9+16}{5}=\frac{34}{5}=6.8$
$\bar{y}=\frac{20 + 14+10 + 6+4}{5}=\frac{54}{5}=10.8$

## Step2: Calculate numerator and denominators
$$
\begin{align*}
S_{xy}&=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})\
&=(1 - 6.8)(20 - 10.8)+(3 - 6.8)(14 - 10.8)+(5 - 6.8)(10 - 10.8)+(9 - 6.8)(6 - 10.8)+(16 - 6.8)(4 - 10.8)\
&=(- 5.8)\times9.2+(-3.8)\times3.2+(-1.8)\times(- 0.8)+2.2\times(-4.8)+9.2\times(-6.8)\
&=-53.36-12.16 + 1.44-10.56-62.56\
&=-137.2
\end{align*}
$$

$$
\begin{align*}
S_{xx}&=\sum_{i = 1}^{n}(x_i-\bar{x})^2\
&=(1 - 6.8)^2+(3 - 6.8)^2+(5 - 6.8)^2+(9 - 6.8)^2+(16 - 6.8)^2\
&=(-5.8)^2+(-3.8)^2+(-1.8)^2+2.2^2+9.2^2\
&=33.64 + 14.44+3.24+4.84+84.64\
&=140.8
\end{align*}
$$

$$
\begin{align*}
S_{yy}&=\sum_{i = 1}^{n}(y_i-\bar{y})^2\
&=(20 - 10.8)^2+(14 - 10.8)^2+(10 - 10.8)^2+(6 - 10.8)^2+(4 - 10.8)^2\
&=9.2^2+3.2^2+(-0.8)^2+(-4.8)^2+(-6.8)^2\
&=84.64+10.24 + 0.64+23.04+46.24\
&=164.8
\end{align*}
$$

## Step3: Calculate correlation coefficient $r$
$$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{- 137.2}{\sqrt{140.8\times164.8}}=\frac{-137.2}{\sqrt{23203.84}}\approx\frac{-137.2}{152.328}\approx - 0.901$$

# Answer:
$-0.901$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate means

Step2: Calculate numerator and denominators

Step3: Calculate correlation coefficient \(r\)

Answer: