Problem 54 - Precalculus with Limits

Question

Data Analysis A store manager wants to know the demand for a product as a function of the price. The table shows the daily sales \(y\) for different prices \(x\) of the product. $$ \begin{array}{|c|c|}\hline \text { Price, } & {\text { Demand }, y} \\ \hline \$ 1.00 & {45} \\ \hline \$ 1.20 & {37} \\ \hline \$ 1.50 & {23} \\\ \hline\end{array} $$ (a) Find the least squares regression line \(y=a x+b\) for the data by solving the system for \(a\) and \(b .\) $$ \left\\{\begin{array}{l}{3.00 b+3.70 a=105.00} \\ {3.70 b+4.69 a=123.90}\end{array}\right. $$ (b) Use a graphing utility to confirm the result of part (a). (c) Use the linear model from part (a) to predict the demand when the price is \(\$ 1.75 .\)

Step-by-Step Solution

Answer

After solving the system of equations, the regression line will be found. By plotting it, it confirms the validity of the model. Finally, when \(x = 1.75\), the model predicts the demand value based on the regression line.

Step 1: Solve the system of equations for a and b

Using the equations \(3.00 b+3.70 a=105.00\) and \(3.70 b+4.69 a=123.90\), we solve for \(a\) and \(b\). This can be done by substitution, elimination or any other method to solve linear equations.

Step 2: Check the solution with a graphing utility

Graph the initial given points and plot the solution line obtained from solving the equations in part (a). The line should be able to fit the given points, confirming the correctness of the solutions for \(a\) and \(b\).

Step 3: Make a prediction

Using our linear model from step 1, we substitute \(x = 1.75\) into our equation \(y = ax + b\) to get a predicted \(y\) value, symbolizing the predicted demand for the product when the price is \$1.75.

Key Concepts

Understanding Least Squares Regression

Least squares regression is a statistical method for finding the best-fitting line through a set of points. It minimizes the sum of the squares of the vertical distances between the observed values and the line. This technique is fundamental for analyzing relationships between two variables, such as price and demand in a business scenario. In our exercise, we aim to find the equation of the line that best predicts product demand based on price.
To achieve this, we express the relationship through the equation \( y = ax + b \), where \( y \) represents demand, \( x \) is the price, \( a \) is the slope, and \( b \) is the intercept. The slope \( a \) shows how much \( y \) changes for a unit change in \( x \), while \( b \) is the predicted \( y \) value when \( x \) is zero. For this specific problem, we set up a system of equations derived from minimizing the error, which is calculated by differences between the observed and predicted values.

Solving the System of Equations

In order to find our least squares line, we first need to solve a system of linear equations. The system of equations given in the exercise is:

• \(3.00b + 3.70a = 105.00\)
• \(3.70b + 4.69a = 123.90\)

We can solve this system using methods such as substitution or elimination. The goal here is to find the values of \( a \) and \( b \), which form the equation of our regression line \( y = ax + b \). Solving these equations will provide us with specific numerical values that best fit the data.
Once you have \( a \) and \( b \), use a graphing tool to plot the data points along with this newly found line. This will visually confirm whether your line correctly represents the trend of the given data points.

Applying the Model to Demand Prediction

After determining the equation of the line using least squares regression, we can use it to estimate or predict demand under new conditions.
For instance, in the exercise provided, once you've found \( a \) and \( b \), you have an equation \( y = ax + b \). To predict demand for a price of \$1.75, you substitute \( x = 1.75 \) into this equation.
This type of prediction helps businesses make informed decisions about pricing strategies:

• Adjusting prices to maximize revenue.
• Establishing a price point where demand meets supply optimally.
• Forecasting future demand under different price scenarios.

By integrating such a model into pricing strategies, businesses can respond more smartly to market dynamics and consumer behavior.