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
137	343
124	275
158	477
181	493
173	412
149	429
156	407
148	345

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. download csv > next question

Explanation:

Step1: Calculate means

Let $x$ be calories and $y$ be sodium.
$n = 9$
$\bar{x}=\frac{183 + 137+124+158+181+173+149+156+148}{9}=\frac{1309}{9}\approx145.4444$
$\bar{y}=\frac{465 + 343+275+477+493+412+429+407+345}{9}=\frac{3646}{9}\approx405.1111$

Step2: Calculate numerator and denominators for $r$

$S_{xy}=\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})$
$S_{xx}=\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}$
$S_{yy}=\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}$
After calculation:
$S_{xy}=(183 - 145.4444)(465 - 405.1111)+(137 - 145.4444)(343 - 405.1111)+\cdots+(148 - 145.4444)(345 - 405.1111)\approx1934.2222$
$S_{xx}=(183 - 145.4444)^{2}+(137 - 145.4444)^{2}+\cdots+(148 - 145.4444)^{2}\approx1064.2222$
$S_{yy}=(465 - 405.1111)^{2}+(343 - 405.1111)^{2}+\cdots+(345 - 405.1111)^{2}\approx45791.5556$
$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{1934.2222}{\sqrt{1064.2222\times45791.5556}}\approx0.27670$

Step3: Calculate coefficient of determination

$r^{2}=(0.27670)^{2}\approx0.0766$

Answer:

The correlation coefficient is $r = 0.27670$, which is a weak positive correlation.
The coefficient of determination is $r^{2}=0.0766$, so $7.66\%$ of the variability in the amount of sodium in beef hot - dogs is explained by the number of calories.