Question

A local tire dealer wants to predict the number of tires sold each month. He believes that the number of tires sold is a linear function of the amount of money invested in advertising. He randomly selects 6 months of data consisting of tire sales (in thousands of tires) and advertising expenditures (in thousands of dollars). Based on the data set with 6 observations, the simple linear regression model yielded the following results.
?X =24
?X2=124
?Y = 42
?Y2 =338
?XY =196

Calculate the coefficient of determination and the coefficient of correlation between X and Y. Interpret the coefficient of Determination. also find the slope and intercept and write the estimated Regression equation. What would the predicted sales of tires be if he spends five thousand dollars in advertising?

Answer 1

Answer:

a) r = 0.13194

b) There is weak positive relationship between X and Y.

c) Slope ($\beta}$) = 34.8333

d) Intercept ($\alpha)$ = -132.333

e) y = -132.33 + 34.83 x

f) the predicted sales of tires be if he spends five thousand dollars in advertising = 41.82 - (in thousands of tires)

Step-by-step explanation:

a) By the coefficient of determination, the formula is:

r = $\frac{n\sum{x}\sum{y} - \sum{xy}}{\sqrt{(n\sum{x}^2 - (\sum{x})^2) (n\sum{y}^2 - (\sum{y})^2})}$,

n = 6. Other values are provided too. Just substitute and obtain the value of r.

b) For correlation interpretation:

                         -1 ≤ r ≤ 1,

The closer the value to 1 the strong the relationship. If close to -1, there is strong negative relations and if close to +1, there is strong positive relationship. when the value is close to 0, there is weak positive relationship.

c) Slope is given as:

$\beta = \frac{n\sum{x}\sum{y} - \sum{xy}}{n\sum{x}^2 - (\sum{x})^2 }$, and

d) Intercept is given as:

$\alpha = \bar{y} - \beta \bar{x}, \\\bar{y} = \frac{\sum{y}}{n} \; and, \; \bar{x} = \frac{\sum{x}}{n}$

e) We obtain $\beta = 34.8333 \; and \; \alpha = -132.333$, therefore, the estimated Regression equation:

                                       y = -132.33 + 34.83 x

f) Since the values are given is thousands of dollar. Then 5000 dollars is simply 5. We substitute the value into x in the regression equation and we have:

                                    y = -132.33 + 34.83*(5) = 41.82