Find the linear correlation coefficient (r) and determine whether the relationship between average temperature and ice cream sales is strong, weak, or neutral for the following dataset of ice cream sales.

Linear Regression Problem for correlation coefficient (r)

The given data sample size = n = 12

Formulae -

$$ Linear\ Correlation\ Coefficient = r = \frac {S_{XY}}{\sqrt {S_{XX} \times S_{YY}}} $$

Where,

$$ S_{XY} = \Bigl(\sum xy\Bigr) - n. \overline x. \overline y $$

$$ S_{XX} = \Bigl(\sum x^2 \Bigr) - n. \overline x^2 $$

$$ S_{YY} = \Bigl(\sum y^2\Bigr) - n. \overline y^2 $$

Step 1

Calculate the $ \sum x$, $\sum y$, $\sum x^2$, $\sum y^2$, and $ \sum xy$

Step 2

Calculate the sample means of temperature (x) and ice cream sales (y) as follows:

$$ \overline x = \frac {\sum x}{n} = \frac {120}{12} = 10 $$

$$ \overline y = \frac {\sum y}{n} = \frac {1200}{12} = 100 $$

Step 3

Calculate the $S_{XY}$

$$ S_{XY} = \Bigl(\sum xy\Bigr) - n. \overline x. \overline y $$

$$ S_{XY} = 13362 - 12 \times 10 \times 100 $$

$$ S_{XY} = 13362 - 12000 $$

$$ S_{XY} = 1362 $$

Calculate the $S_{XX}$

$$ S_{XY} = 1362 $$

$$ S_{XX} = \Bigl(\sum x^2 \Bigr) - n. \overline x^2 $$

$$ S_{XX} = 1450 - 12 \times (10)^2 $$

$$ S_{XX} = 1450 - 1200 $$

$$ S_{XX} = 250 $$

Calculate the $S_{YY}$

$$ S_{YY} = \Bigl(\sum y^2\Bigr) - n. \overline y^2 $$

$$ S_{YY} = 127674 - 12 \times (100)^2 $$

$$ S_{YY} = 127674 - 120000 $$

$$ S_{YY} = 7674 $$

Step 4

Finally, calculate the Linear Correlation Coefficient (r)

$$ r = \frac {S_{XY}}{\sqrt {S_{XX} \times S_{YY}}} $$

$$ r = \frac {1362}{\sqrt {250 \times 7674}} $$

$$ r = \frac {1362}{1385.099274} $$

$$ r = 0.983 $$

Step 5

• Thus we get the Linear Correlation Coefficient r = 0.983

• This implies that there is a strong, POSITIVE linear relationship between average temperature and ice cream sales since the linear correlation coefficient (r) is very close to +1.