use the given data set to determine the correlation coefficient r. age (years) height (in.) 2 31.4 3 34.2 4 37.0 5 39.8 6 42.1.937.908.999.984

Question

Accepted Answer

# Explanation:
## Step1: Calculate the means
Let $x$ be age and $y$ be height.
$\bar{x}=\frac{2 + 3+4+5+6}{5}=\frac{20}{5} = 4$
$\bar{y}=\frac{31.4+34.2+37.0+39.8+42.1}{5}=\frac{184.5}{5}=36.9$

## Step2: Calculate numerator and denominator terms
$$
\begin{align*}
\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(2 - 4)(31.4-36.9)+(3 - 4)(34.2 - 36.9)+(4 - 4)(37.0-36.9)+(5 - 4)(39.8-36.9)+(6 - 4)(42.1-36.9)\
&=(- 2)(-5.5)+(-1)(-2.7)+0\times0.1+1\times2.9+2\times5.2\
&=11 + 2.7+0+2.9+10.4\
&=27
\end{align*}
$$

$$
\begin{align*}
\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}&=(2 - 4)^{2}+(3 - 4)^{2}+(4 - 4)^{2}+(5 - 4)^{2}+(6 - 4)^{2}\
&=4+1+0+1+4\
&=10
\end{align*}
$$

$$
\begin{align*}
\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}&=(31.4-36.9)^{2}+(34.2-36.9)^{2}+(37.0-36.9)^{2}+(39.8-36.9)^{2}+(42.1-36.9)^{2}\
&=(-5.5)^{2}+(-2.7)^{2}+0.1^{2}+2.9^{2}+5.2^{2}\
&=30.25+7.29+0.01+8.41+27.04\
&=73
\end{align*}
$$

## Step3: Calculate the correlation coefficient $r$
$$r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}=\frac{27}{\sqrt{10\times73}}=\frac{27}{\sqrt{730}}\approx\frac{27}{27.02}\approx0.999$$

# Answer:
.999

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the means

Step2: Calculate numerator and denominator terms

Step3: Calculate the correlation coefficient \(r\)

Answer: