(a): The correlation coefficient is r = 0.788
(b): The coefficient of determination is 0.620
(c): The regression equation is: Y' = 0.102X + 1.698
(d): An estimate of the earnings for a small company with $48 million in sales is approximately 5.016 million dollars.
Here's the solution to the given problem:
Part (a): Correlation coefficient
The correlation coefficient measures the strength and direction of the linear relationship between two variables. It can range from -1 (perfect negative correlation) to 1 (perfect positive correlation), with 0 indicating no correlation.
To calculate the correlation coefficient, we can use the following formula:
r = Σ[(xi - X)(yi - Y)] / √[Σ(xi - X)² Σ(yi - y)²]
where:
xi and yi are the values of the independent (sales) and dependent (earnings) variables for each company, respectively
X and Y are the mean values of the independent and dependent variables, respectively
Plugging in the given data, we get:
r = 0.788
Part (b): Coefficient of determination
The coefficient of determination, also known as R-squared, represents the proportion of the variance in the dependent variable that can be explained by the independent variable. It ranges from 0 to 1, with 0 indicating no explanatory power and 1 indicating perfect explanatory power.
The coefficient of determination can be calculated by squaring the correlation coefficient:
R² = r² = 0.620
This means that about 62% of the variance in earnings can be explained by the variance in sales.
Part (c): Regression equation
The regression equation is a linear equation that represents the relationship between the independent and dependent variables. It can be used to predict the value of the dependent variable for a given value of the independent variable.
To find the regression equation, we use the following formulas:
b = Σ[(xi - X)(yi - Y)] / Σ(xi - X)²
a = Y - bX
where:
b is the slope of the regression line
a is the y-intercept of the regression line
Plugging in the given data, we get:
b = 0.102
a = 1.698
Therefore, the regression equation is:
Y' = 0.102X + 1.698
Part (d): Estimated earnings for a small company
For a small company with $48 million in sales, we can estimate the earnings using the regression equation:
Y' = 0.102(48) + 1.698 ≈ 5.016
Therefore, an estimate of the earnings for a small company with $48 million in sales is approximately 5.016 million dollars.