Question

The marketing manager of a large supermarket chain would like to use shelf space to predict the sales of pet food. For a random sample of 12 similar stores, she gathered the following information regarding the shelf space, in feet, devoted to pet food and the weekly sales in hundreds of dollars. . Store 1 2 3 4 5 6 Shelf Space 5 5 5 10 10 10 Weekly Sales 1.6 2.2 1.4 1.9 2.4 2.6 Store 7 8 9 10 11 12 Shelf Space 15 15 15 20 20 20 Weekly Sales 2.3 2.7 2.8 2.6 2.9 3.1 Compute the value of the sample correlation coefficient between weekly sales and shelf space. A. 0.827 B. 0.308 C. 0.652 D. 0.684

Answer 1

A. 0.827

Explanation:

Correlation Coefficient(r) shows the direction and strength of relationship between two variables.

The formula used to calculate correlation is:

`Correlation(r) = (Cov(x, y))/(\sigma_(x)\sigma_(y))= (E(x-\mu_(x))(y-\mu_(y)))/(\sigma_(x)\sigma_(y))`

where, Cov(x,y) = Covariance of x and y

`\mu_(x)` = mean of x

`\mu_(y)` = mean of y

`\sigma_(x)` = standard deviation of x

`\sigma_(y)` = standard deviation of y

and, E = Expectation.

Now, we have data: Shelf Space(x) : (5,5,5,10,10,10,15,15,15,20,20,20)

Weekly Sales(y): (1.6, 2.2, 1.4, 1.9, 2.4, 2.6, 2.3, 2.7, 2.8, 2.6, 2.9, 3.1)

Mean of x = 12.5

Mean of y = 2.375

Standard deviation of x = 5.84

Standard deviation of y = 0.522

Covariance of x and y = 2.52

Putting all values, We get

Correlation Coefficient = 0.827

The correlation coefficient lies between -1 to +1.

As the increase in the value of one variable also increases the value of other variables is called positive correlation. Also, the value of positive correlation lies between 0 to 1.

And decrease in the value of one variable, increases the value of other variables is called negative correlation. Also, the value of negative correlation lies between -1 to 0.