Question

find the correlation coefficient, r, of the data described below.
a gym franchise was considering a television marketing campaign to increase its membership. the franchises market researchers wanted to get a better sense of the television and exercise habits of the gyms target demographic. to begin, the market researchers surveyed some of the current members about how many hours they had spent watching television and exercising last month.
using the survey responses, the researchers compared the number of hours of television watched, x, to the number of hours of exercise, y, for each member.
hours of television hours of exercise
3 54
6 57
16 5
19 22
30 7
round your answer to the nearest thousandth.
r =

Explanation:

Step1: Calculate the means

Let $x$ be hours of television and $y$ be hours of exercise.
$n = 5$
$\bar{x}=\frac{3 + 6+16+19+30}{5}=\frac{74}{5}=14.8$
$\bar{y}=\frac{54 + 57+5+22+7}{5}=\frac{145}{5}=29$

Step2: Calculate the numerator of the correlation - coefficient formula

\[

$$\begin{align*} \sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(3 - 14.8)(54 - 29)+(6 - 14.8)(57 - 29)+(16 - 14.8)(5 - 29)+(19 - 14.8)(22 - 29)+(30 - 14.8)(7 - 29)\\ &=(- 11.8)\times25+(-8.8)\times28+(1.2)\times(-24)+(4.2)\times(-7)+(15.2)\times(-22)\\ &=-295-246.4 - 28.8-29.4-334.4\\ &=-934 \end{align*}$$

\]

Step3: Calculate the denominator of the correlation - coefficient formula

\[

$$\begin{align*} \sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}&=(3 - 14.8)^{2}+(6 - 14.8)^{2}+(16 - 14.8)^{2}+(19 - 14.8)^{2}+(30 - 14.8)^{2}\\ &=(-11.8)^{2}+(-8.8)^{2}+(1.2)^{2}+(4.2)^{2}+(15.2)^{2}\\ &=139.24 + 77.44+1.44 + 17.64+231.04\\ &=466.8 \end{align*}$$

\]
\[

$$\begin{align*} \sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}&=(54 - 29)^{2}+(57 - 29)^{2}+(5 - 29)^{2}+(22 - 29)^{2}+(7 - 29)^{2}\\ &=25^{2}+28^{2}+(-24)^{2}+(-7)^{2}+(-22)^{2}\\ &=625+784 + 576+49+484\\ &=2518 \end{align*}$$

\]
The denominator is $\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}=\sqrt{466.8\times2518}=\sqrt{1175402.4}\approx1084.16$

Step4: Calculate the correlation coefficient

$r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}=\frac{-934}{1084.16}\approx - 0.862$

Answer:

$-0.862$