Scenario: Regression equations are created by modeling data, such as the following:
Sales = (Cost Per Item × Number of Items) – Constant Charges
In this equation, constant charges may be rent, salaries, or other fixed costs. This includes anything that you have to pay for periodically as a business owner. This value is negative because this cost must be paid each period and must be paid whether you make a sale or not.
Your company may wish to release a new e-reader device. Based on data collected from various sources, your company has come up with the following regression equation for the sales of the new e-reader:
Sales = $0.15 × number of e-readers sold – $28
Or, assuming x = the number of e-readers sold, this would be the same regression equation:
Sales = 0.15x – 28
In this case, the values are given in thousands (i.e., the cost of making an individual e-reader will be $150 [0.15 × 1,000], with $28,000 [28 * 1,000] in constant charges).
Answer the following questions based on the given regression equation:
1. Discuss the meaning of the x- and y-axis values on the graph. (Hint: Label the axis with the Text tool in the graphing program.)
2.Based on the results of the graph and the sales equation provided, discuss the relationship between sales and number of e-readers produced. (Hint: Consider the slope and y-intercept.)
3.If the company does not sell a single e-reader, how much is lost in sales? Mathematically, what is this value called in the equation?
4.If the company sells 5,000 e-readers, how much will the company make (or lose) in sales?
5.If sales must equal 100 thousand, how many e-readers will your company need to sell? (Round up to the nearest e-reader.)
6.If your company is hoping to break even, how many e-readers will need to be sold to accomplish this? (Round up to the nearest e-reader.)
1. The x-axis represents the number of e-readers sold, while the y-axis represents the sales in dollars. These values are labeled on the graphing program or chart to give a visual representation of the relationship between the number of e-readers sold and the corresponding sales amount.
2. The relationship between sales and the number of e-readers produced can be determined from the given regression equation. In this case, the slope of the equation is 0.15, which means that for every additional e-reader sold, the sales will increase by $0.15. The y-intercept of -28 signifies that even if no e-reader is sold, there will still be constant charges of $28,000. Overall, the relationship between sales and the number of e-readers produced is positive, indicating that as the number of e-readers sold increases, the sales revenue also increases, but there will always be a negative offset due to the constant charges.
3. If the company does not sell a single e-reader, the sales will be zero, but there will still be the constant charges to be paid. Mathematically, this value is called the y-intercept in the equation, which is -28 in this case. So, if no e-reader is sold, the company will still face a loss of $28,000 in constant charges.
4. To calculate the sales when the company sells 5,000 e-readers, we can substitute 5,000 as the value of x in the equation:
Sales = 0.15 * 5000 - 28
This simplifies to:
Sales = $750 - $28
Therefore, the company will make $722,000 in sales when they sell 5,000 e-readers.
5. To find out how many e-readers the company needs to sell in order to achieve sales of $100,000, we can set up the equation and solve for x:
$100,000 = 0.15x - $28,000
First, move the constant charges to the other side:
$100,000 + $28,000 = 0.15x
Simplify:
$128,000 = 0.15x
Now, solve for x by dividing both sides by 0.15:
x = $128,000 / 0.15
Calculating this value yields:
x ≈ 853,333
Rounding this up to the nearest e-reader, the company would need to sell approximately 853,333 e-readers to achieve sales of $100,000.
6. To determine the number of e-readers needed to break even, we set the sales equation equal to zero and solve for x:
0 = 0.15x - $28,000
Move the constant charges to the other side:
$28,000 = 0.15x
Now, solve for x by dividing both sides by 0.15:
x = $28,000 / 0.15
Calculating this value yields:
x ≈ 186,667
Rounding this up to the nearest e-reader, the company would need to sell approximately 186,667 e-readers to break even.