Question

The following data represent the square footage and rents (dollars per month) for apartments in the La Jolla area of San Diego, California. $$ \begin{array}{|cc|} \hline \text { Square Footage, } x & \text { Rent per Month, } R \\ \hline 520 & \$ 1630 \\ 625 & \$ 1820 \\ 710 & \$ 1860 \\ 765 & \$ 1975 \\ 855 & \$ 1985 \\ 925 & \$ 2200 \\ 1040 & \$ 2360 \\ \hline \end{array} $$ (a) Using a graphing utility, draw a scatter plot of the data treating square footage as the independent variable. What type of relation appears to exist between square footage and rent? (b) Based on your response to part (a), find either a linear or a quadratic model that describes the relation between square footage and rent. (c) Use your model to predict the rent for an apartment in San Diego that is 875 square feet.

Short answer

Use a scatter plot to determine the relationship and choose a model accordingly. Then, use the model to predict the rent for 875 square feet.

Step-by-step solution

Create a Scatter Plot

Plot the given data points on a graph where the x-axis represents the square footage and the y-axis represents the rent per month. Use a graphing utility or software to visualize this. The data points are: (520, 1630), (625, 1820), (710, 1860), (765, 1975), (855, 1985), (925, 2200), (1040, 2360).

Analyze the Scatter Plot

Examine the scatter plot to determine the type of relationship between the square footage and rent. Look for a pattern or trend in the plotted points. If they seem to form a straight line, a linear relationship is likely present. If they form a curve, a quadratic or other non-linear relationship might be present.

Select a Model

Based on the visual inspection of the scatter plot, choose either a linear or a quadratic model. For instance, if the points tend to follow a straight line, a linear model is appropriate. Conversely, if the points form a curve, consider a quadratic model.

Determine the Linear Model

If a linear relationship is observed, use linear regression to find the best-fitting line. The linear equation will be of the form \[ R = a + bx \] where \( R \) is the rent and \( x \) is the square footage. Using statistical software or a calculator, compute the values of \( a \) and \( b \).

Determine the Quadratic Model

If a quadratic relationship is observed, use quadratic regression to find the best-fitting curve. The quadratic equation will be of the form \[ R = ax^2 + bx + c \] where \( R \) is the rent and \( x \) is the square footage. Using statistical software or a calculator, compute the values of \( a \), \( b \), and \( c \).

Predict the Rent for 875 Square Feet

Using the selected model, plug in the value \( x = 875 \) to predict the rent. If using a linear model, substitute 875 in place of \( x \) in the linear equation. If using a quadratic model, substitute 875 in place of \( x \) in the quadratic equation and solve for \( R \).

Key concepts

Scatter Plot

A scatter plot is a type of graph used to visualize the relationship between two variables. To create a scatter plot, you plot individual data points on a two-dimensional plane.
For this exercise, the x-axis represents the square footage of apartments, and the y-axis represents the rent per month.
Here are the steps to create a scatter plot:
1. **Plot the Data Points:** For each apartment, mark a point where the square footage value on the x-axis meets the rent value on the y-axis. The points for our data would be (520, 1630), (625, 1820), (710, 1860), (765, 1975), (855, 1985), (925, 2200), and (1040, 2360).
2. **Visualize the Pattern:** Look at how the points are distributed. Do they form a straight line, a curve, or seem random?
In our plot, the pattern helps determine which type of model (linear or quadratic) describes the relationship best. Understanding scatter plots is crucial for identifying underlying trends and making predictions based on data.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and an independent variable. In this context, it helps us find the best-fitting straight line that describes the relationship between square footage and rent.
The equation of a linear model is: \[ R = a + bx \] where:
- \( R \): Rent per month.
- \( x \): Square footage.
- \( a \): Y-intercept.
- \( b \): Slope of the line.
To perform linear regression:
1. **Examine the Scatter Plot:** If the points form a line-like pattern, a linear model may be appropriate.
2. **Use Tools:** Employ statistical software or a calculator to compute \( a \) and \( b \). These tools use methods like the least squares method to minimize the distance between data points and the line.
3. **Formulate the Equation:** With \( a \) and \( b \), create your linear equation. For example, if \( a = 1000 \) and \( b = 1.5 \), the equation becomes \( R = 1000 + 1.5x \).
With this equation, you can predict the rent for any given square footage by plugging in the value of \( x \).

Quadratic Model

A quadratic model is used when the relationship between two variables forms a curved pattern, resembling a parabola.
The equation of a quadratic model is: \[ R = ax^2 + bx + c \] where:
- \( R \): Rent per month.
- \( x \): Square footage.
- \( a \) and \( b \): Coefficients of the model.
- \( c \): Constant term.
To develop a quadratic model:
1. **Inspect the Scatter Plot:** If the points form a curved pattern, then a quadratic model might be suitable.
2. **Use Statistical Tools:** Utilize software to perform quadratic regression. The tool will calculate the values of \( a \), \( b \), and \( c \) that best fit the data.
3. **Formulate the Equation:** With the computed coefficients, establish your quadratic equation. For instance, if \( a = 0.002 \), \( b = 3 \), and \( c = 1000 \), the equation is \( R = 0.002x^2 + 3x + 1000 \).
This model is useful for making predictions for any square footage, especially when the relationship is not linear but curved.
By plugging in the desired square footage, such as 875, into the quadratic model, you can find the predicted rent for that specification.