Question

a random sample of beef hotdogs was taken and the amount of sodium (in mg) and calories were measured. the data are in the table. find the correlation coefficient and coefficient of determination and then interpret both.
calories and sodium levels in beef hotdogs

calories	sodium
140	374
168	476
176	508
163	411
174	608
117	235
136	388
171	435

the correlation coefficient is r = (round to five decimal places), which is a select an answer, select an answer correlation.
the coefficient of determination is r² = (round to four decimal places), so % of the variability in the amount of sodium in beef hotdogs is explained by the number of calories.
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Explanation:

Step1: Calculate means

Let $x$ be calories and $y$ be sodium.
$n = 9$
$\bar{x}=\frac{166 + 140+168+176+163+174+117+136+171}{9}=\frac{1311}{9}\approx145.6667$
$\bar{y}=\frac{375 + 374+476+508+411+608+235+388+435}{9}=\frac{3800}{9}\approx422.2222$

Step2: Calculate numerator and denominator components

Calculate $(x_i-\bar{x})(y_i - \bar{y})$, $(x_i-\bar{x})^2$ and $(y_i - \bar{y})^2$ for each $i$ from $1$ to $9$.
Sum of $(x_i-\bar{x})(y_i - \bar{y})$:

$$
LATEXBLOCK0
$$

Sum of $(x_i-\bar{x})^2$:

$$
LATEXBLOCK1
$$

Sum of $(y_i - \bar{y})^2$:

$$
LATEXBLOCK2
$$

Step3: Calculate correlation coefficient $r$

The formula for the correlation coefficient $r$ is $r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2\sum_{i = 1}^{n}(y_i - \bar{y})^2}}$

$$
LATEXBLOCK3
$$

Since $r\approx0.80210>0$, it is a positive correlation.

Step4: Calculate coefficient of determination $r^2$

$r^2=(0.80210)^2\approx0.6434$
The coefficient of determination $r^2$ means that approximately $64.34\%$ of the variability in the amount of sodium in beef hot - dogs is explained by the number of calories.

Answer:

The correlation coefficient is $r = 0.80210$, which is a positive correlation.
The coefficient of determination is $r^2=0.6434$, so $64.34\%$ of the variability in the amount of sodium in beef hotdogs is explained by the number of calories.