Question

A large toy company introduces a lot of new toys to its product line each year. The company wants to predict the demand as measured by y, first-year sales (in millions of dollars) using x, awareness of the product (as measured by the proportion of customers who had heard of the product by the end of the second month after its introduction). A random sample of $65$new products was taken, and a correlation of $0.96$was computed. Which of the following is a correct interpretation of this value?

(a) Ninety-six percent of the time, the least-squares regression line accurately predicts first-year sales.

(b) About $92\%$of the time, the proportion of people who have heard of the product by the end of the second month will correctly predict first-year sales.

(c) About $92\%$of first-year sales can be explained by the proportion of people who have heard of the product by the end of the second month.

(d) The least-squares regression line relating the proportion of people who have heard of the product by the end of the second month and first-year sales will have a slope of $0.96$.

(e) Ninety-two percent of the variation in first-year sales can be explained by the least-squares regression line with proportion of people who have heard of the product by the end of the second month as the explanatory variable.

Short answer

The correct interpretation of the value is that Ninety-two percent of the variation in first-year sales can be explained by the least-squares regression line with proportion of people who have heard of the product by the end of the second month as the explanatory variable i.e., option (e).

Step-by-step solution

Given Information

We are given that the company wants to predict the demand as measured by y which is first year sale using x which is awareness of the product with the value of correlation $r=0.96$and there are $65$new products and we have to find out the correct interpretation of this value.

Explanation 

Now, we will calculate the coefficient of determination $R^2$

As this is a simple linear regression so we can calculate $R^2$by just squaring the correlation value,

$R^2=r^2={(0.96)}^2$,

which gives $R^2=0.92$

this tell us that coefficient of determination is directly proportional to total variance which is stated by statement (e).

Hence, option (e) is the correct interpretation of this value.