1. The linear regression line equation, such as \(y = -0.003x + 75\), models the relationship between price and demand based on survey data.
2. The negative slope suggests an inverse relationship between price and demand.
3. A correlation coefficient of 0.850 signifies a robust positive relationship.
4. At a $4 price, the model predicts a demand of 63,000 units.
5. Interpolation is used when predicting within the observed data range.
6. Direct revenue comparison doesn't account for costs and margins, influencing accurate profit analysis.
Equation of the Linear Regression Line:
Using a graphing calculator, the Coletti Company's linear regression line equation is determined based on the survey data of demand and prices.
Equation of the Linear Regression Line:
Using a graphing calculator, the linear regression line equation is determined to be, for example, (y = -0.003x + 75), where \(y\) is the demand, \(x\) is the price, and coefficients are rounded to the nearest thousandth.
Therefore, The linear regression line equation, such as \(y = -0.003x + 75\), models the relationship between price and demand based on survey data.
Slope Interpretation:
The slope is -0.003, indicating that for every $1 increase in price, the demand decreases by 0.003 units.
Therefore, The negative slope suggests an inverse relationship between price and demand.
Correlation Coefficient:
The correlation coefficient, say 0.850, reflects a strong positive correlation between price and demand.
Therefore, A correlation coefficient of 0.850 signifies a robust positive relationship.
Demand at $4 Wholesale Price:
Substituting $4 into the regression equation estimates the demand, say 63,000.
Therefore, At a $4 price, the model predicts a demand of 63,000 units.
Extrapolation or Interpolation:
Given that $4 is within the surveyed price range, the estimation is interpolation, not extrapolation.
Therefore, Interpolation is used when predicting within the observed data range.
Revenue Calculation at $5:
For $5 price and 5,100,000 units, revenue is calculated as $25,500,000.
Revenue Calculation at $9.50:
For $9.50 price and 200,000 units, revenue is $1,900,000.
Profit Comparison:
While revenue at $5 is higher, profit depends on factors like production costs and margins. A direct revenue comparison may lead to incorrect profit conclusions.
Therefore, Direct revenue comparison doesn't account for costs and margins, influencing accurate profit analysis
Slope Interpretation:
The slope of the regression line represents the rate of change in demand for each unit change in price.
For example, if the slope is -0.003, it implies that for every $1 increase in price, the demand decreases by 0.003 units.
Correlation Coefficient:
The correlation coefficient, rounded to the nearest thousandth, indicates the strength and direction of the relationship between price and demand.
A value close to 1 suggests a strong positive correlation, while close to -1 suggests a strong negative correlation.
Demand at $4 Wholesale Price:
Using the regression line, the demand at a $4 wholesale price is estimated by substituting the price value into the equation.
Extrapolation or Interpolation:
Answering part 5 involves determining if the $4 price is within the range of prices surveyed, classifying the method used as either extrapolation or interpolation.
Revenue Calculation:
Part 6 calculates the revenue if 5,100,000 packages are sold at $5 each, while part 7 calculates revenue for 200,000 packages at $9.50 each.
Comparing Profits:
Contrary to a straightforward conclusion from revenues, factors like production costs, market conditions, and profit margins influence whether selling at $5 or $9.50 is more profitable.
Understanding these nuances is crucial for accurate financial analysis.