Q. 4.6 - Elementary Statistics

Question

Custom Homes. Hanna Properties specializes in custom home resales in the Equestrian Estates, an exclusive subdivision in Phoenix, Arizona. A random sample of nine custom homes currently listed for sale provided the following information on size and price. Here, x denotes the size, in hundreds of square feet, rounded to the nearest hundred, and y denotes price, in thousands of dollars, rounded to the nearest thousand. For part (g). predict the price of a $2600-\mathrm{sq}$. $\mathrm{ft}$. home in the Equestrian Estates.

$\begin{array}{|cccccccccc|}\hline \mathit{x}& 26& 27& 33& 29& 29& 34& 30& 40& 22\\ \mathit{y}& 540& 555& 575& 577& 606& 661& 738& 804& 496\\ \hline\end{array}$

• a. fond the regression equation for the data points.
• b. graph the regression equation and the data points.
• c. describe the apparent relationship between the two variables under consideration.
• d. interpret the slope of the regression line.
• e. identify the predictor and response variables.
• f. identify outliers and potential influential observations.
• g. predict the values of the response variable for the specified values of the predictor variable, and interpret your results.

Step-by-Step Solution

Answer

Ans:

part (a): $\hat{y}=140.08+15.8935x$

part (b): Graph representation  

part (c): Linear relationship 

part (d): The slope of the regression line is $15.8935$

part (e): The predictor variable is size and the response variable is the price of custom homes. 

part (f): All the points are near the straight line so there are no outliers in the given data 

part (g): The estimated price of a 2600 -sq. ft. home in the Equestrian Estates is $\$553.311$

Step 1. GIven information:

$\begin{array}{|cccccccccc|}\hline \mathit{x}& 26& 27& 33& 29& 29& 34& 30& 40& 22\\ \mathit{y}& 540& 555& 575& 577& 606& 661& 738& 804& 496\\ \hline\end{array}$

Step 2. Solving part (a):

$\begin{array}{|cccc|}\hline x& y& xy& x^2\\ 26& 540& 14040& 676\\ 27& 555& 14985& 729\\ 33& 575& 18975& 1089\\ 29& 577& 16733& 841\\ 29& 606& 17574& 841\\ 34& 661& 22474& 1156\\ 30& 738& 22140& 900\\ 40& 804& 32160& 1600\\ 22& 496& 10912& 484\\ \sum x=270& \sum y=5552& \sum xy=169993& \sum x^2=8316\\ \hline\end{array}$

Step 3. Continue:

The slope of the regression line is,

$$\begin{gathered} b_1=\frac{S_{xy}}{S_{xx}} \\ =\frac{\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n}{\sum x_i^2-{\left(\sum x_i\right)}^2/n} \\ =\frac{169993-\left(270\right)\left(5552\right)/9}{8316-(270{)}^2/9} \\ =\frac{3433}{216} \\ \approx 15.8935 \end{gathered}$$

The y-intercept is,

$$\begin{gathered} b_0=\frac{1}{n}\left(\sum y_i-b_1\sum x_i\right) \\ =\frac{1}{9}(5552-(15.8935\left)\right(270\left)\right) \\ =140.0838 \\ \approx 140.08 \end{gathered}$$

Therefore, the regression equation is $\hat{y}=140.08+15.8935x$

Step 4. Solving part (b):

Using Excel graph the regression equation and the data points by using the following steps:

Step 1: Import the data into the Excel Worksheet.

Step 2: Select the data to be graphed.

Step 3: On the Insert tab, click the Scatter chart.

Step 4: Click the Scatter with only the markers option.

Step 5: Right-click on the data point and click Add Trend line.

Step 6: Use the Design tab in the Chart Tools to change the Chart Title and remove the grid lines.

Step 5. Continue:

The graph representation for the equation:

Step 6. Solving part (c):

From the above graph, it can be observed that as size increases the price of the home increases.

Therefore, there is a linear relationship between the two variables.

Step 7. Solving part (d):

From part a, the slope of the regression line is $15.8935$, it means that a unit change in the size causes a change of $\$15,8935$ in the price of the custom homes.

Step 8. Solving part (e):

From the regression equation, the predictor variable is size and the response variable is the price of custom homes. 

Step 9. Solving part (f):

From the obtained regression equation graph, it can be observed that all the points are near the straight line so there are no outliers in the given data.

Step 10 . Solving part (g):

The regression line is, $\hat{y}=140.08+15.8935x$.

Substitute the value $x=26$ in the regression equation.

The predicted price of a 2600 -sq. ft. home is,

$$\begin{gathered} \hat{y}=140.08+15.8935x \\ =140.08+15.8935\left(26\right) \\ =553.311 \end{gathered}$$

Therefore, the estimated price of a 2600 -sq. ft. home in the Equestrian Estates is $\$553.311$.