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Linear Regression Problem for Pearson product moment correlation coefficient (r).

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.

Temperature (x) 4 4 7 8 12 15 16 17 14 11 7 5
Ice Cream Sale (y) 73 57 81 94 110 124 134 139 124 103 81 80
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## 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 $\sum xy$

x y x2 y2 xy
4 73 16 5329 292
4 57 16 3249 228
7 81 49 6561 567
8 94 64 8836 752
12 110 144 12100 1320
15 124 225 15376 1860
16 134 256 17956 2144
17 139 289 19321 2363
14 124 196 15376 1736
11 103 121 10609 1133
7 81 49 6561 567
5 80 25 6400 400
120 1200 1450 127674 13362

# 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.