Question

Sally has been running the following number of ads in the local newspaper to help attract customers into her store. She has also been keeping track of customers who have come into the store as a result of the ads, as well as the amount of money they spend. Use Excel to make our calculations, show all work and all formulas. This is the data from the last 4 weeks:

Number of ads Increased Sales
------------------------------------------
16 330
12 270
18 380
14 300
Calculate the correlation between these two variables: number of ads and increased sales.

Answer 1

The correlation between these two variables, number of ads and increased sales is equal to 0.9909.

In Mathematics, the Pearson product-moment correlation coefficient (r) can be modeled by the following mathematical equation;

`r=\frac{\sum (x - \bar{x})(y- \bar{y})}{\sqrt{\sum (x - \bar{x})^2\sum (y - \bar{y})^2} }`

Where:

• r represents the correlation coefficient.
• x is the independent variable in a sample.
• 
  `\bar{x}` is the mean of the independent variable.
• x is the dependent variable in a sample.
• 
  `\bar{y}` is the mean of the dependent variable.

Next, we would calculate the mean of the the independent variable (x-variable) and dependent variable (y-variable) as follows;

Mean,
`\bar{x}` = [∑(x)]/n

Mean,
`\bar{x}` = (16 + 12 + 18 + 14)/4

Mean,
`\bar{x}` = 15

Mean,
`\bar{y}` = [∑(y)]/n

Mean,
`\bar{y}` = (330 + 270 + 380 + 300)/4

Mean,
`\bar{y}` = 320

`\sum(x - \bar{x})(y - \bar{y}) = (16-15)(330-320) + (12-15)(270-320)+(18-15)(380-320)+(14-15)(300-320)\\\\\sum(x - \bar{x})(y - \bar{y}) = 360\\\\\\\sum(x - \bar{x})^2 = (16-15)^2 + (12-15)^2 +(18-15)^2 + (14-15)^2\\\\\sum(x - \bar{x})^2 = 20`

`\sum(x - \bar{x})^2 = (330-320)^2 + (270-320)^2 + (380-320)^2 + (300-320)^2\\\\\sum(x - \bar{x})^2 = 6600\\\\\\\sqrt{(\sum(x - \bar{x})^2\sum(y - \bar{y})^2)} = √(20 * 6600)\\\\\sqrt{(\sum(x - \bar{x})^2\sum(y - \bar{y})^2)} = √(132000)\\\\\sqrt{(\sum(x - \bar{x})^2\sum(y - \bar{y})^2)} = 363.3180`

Now, we can calculate the Pearson product-moment correlation coefficient;

Correlation coefficient, r = 360/363.3180

Correlation coefficient, r = 0.9909.