Q. 4.60 - 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 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 -sq. ft. home in the Equestrian Estates.

1. find the regression equation for the data points.
2. graph the regression equation and the data points.
3. describe the apparent relationship between the two variables under consideration.
4. interpret the slope of the regression line.
5. identify the predictor and response variables.
6. identify outliers and potential influential observations.
7. predict the values of the response variable for the specified values of the predictor variable, and interpret your results.

Step-by-Step Solution

Answer

a. The following is the regression equation for predicting the price $\left(y\right)$ based on the size $\left(x\right)$: $\hat{y}=140.0839+15.8935x.$

b.

  c. The regression line slopes upward. 

 d. Increasing the size of a house by one hundred square feet increases the price by $15.8935$thousand dollars on average.

e. Size denotes a predictor variable. Price is a response variable.

f. There are no outliers or significant observations. 

g. ${\hat{y}}_{26}=553.3149$

Part (a) Step 1: Given Information

Given in the question that, 

For the data points, we must find the regression equation. 

Part (a) Step 2: Explanation

Let's use the regression beta coefficients for the calculation.  

$\hat{y}=b_0+b_{1}x$

According to the information, $n=9$

We have to find the necessary sum as below: 

$\sum x_i=26+27+\dots +22=270$

$\sum x_i^2={26}^2+{27}^2+\dots +{22}^2=8316$

$\sum y_i=540+555+\dots +496=5552$

$\sum x_i{y}_i=\left(26\right)\left(540\right)+\left(27\right)\left(555\right)+\dots +\left(22\right)\left(496\right)=169993$

Let's find $s_{xy}$ as follow:

$s_{xy}=\sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)=\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n$

$$\begin{gathered} s_{xy}=169993-\frac{\left(270\right)\left(5552\right)}{9} \\ =3433 \end{gathered}$$

Then, find $s_{xx}$ as follow:

$s_{xx}=\sum {\left(x_i-\overline{x}\right)}^2=\sum x_i^2-{\left(\sum x_i\right)}^2/n$

$$\begin{gathered} s_{xx}=8316-\frac{\left(270\right)\left(270\right)}{9} \\ =216 \end{gathered}$$

Part (a) Step 3: Calculate the parameters and averages

We can find the averages by using the given formula: 

$$\begin{gathered} \overline{x}=\frac{\sum x_i}{n} \\ \overline{y}=\frac{\sum y_i}{n} \end{gathered}$$

The value of $\overline{x}$ is:

$\overline{x}=\frac{270}{9}=30$

Then, the value of $\overline{y}$ is:

$\overline{y}=\frac{5552}{9}=616.8889$

Therefore, the parameters are:

$$\begin{gathered} b_1=\frac{s_{xy}}{s_{xx}} \\ {b}_1=\frac{3433}{216} \\ =15.8935 \end{gathered}$$

$$\begin{gathered} b_0=\overline{y}-b_1\overline{x} \\ {b}_0=616.8889-15.8935\times 30 \\ =140. 0839 \end{gathered}$$

Part (b) Step 1: Given Information

Given in the question that,

We have to graph the regression equation and the data points.  

Part(b) Step 2: Explanation

When you're six years old, the fee is:

$$\begin{gathered} {\hat{y}}_{26}=140.0839+15.8935\left(26\right) \\ =553.3149 \end{gathered}$$

The anticipated values for the provided data are also presented in the table below.

$\begin{array}{|lll|}\hline x& y& \hat{y}\\ 26& 540& 553.3149\\ 27& 555& 569.2084\\ 33& 575& 664.5694\\ 29& 577& 600.9954\\ 29& 606& 600.9954\\ 34& 661& 680.4629\\ 30& 738& 616.8889\\ 40& 804& 775.8239\\ 22& 496& 489.7409\\ \hline\end{array}$

The given points and the fitted regression line are represented in the graph below.  

Part (c) Step 1: Given Information

The apparent relationship between the two variables under investigation must be described.  

Part (c) Step 2: Explanation

It's worth noting that the regression line slopes upwards, implying that the price $y$ rises in proportion to the size $x$.

Part (d) Step 1: Given information

The apparent relationship between the two variables under investigation must be described.  

Part (d) Step 2: Explanation

The average increase in $y$ per unit increase in $x$ is represented by the slope.

The slope was determined at $15.8935$ in section (a).

Part (e) Step 1: Given Information

The predictor and response variables must be identified.  

Part (e) Step 2: Explanation

The response variable seems to be the one that will be measured.

The predictor variable is a variable that is used to estimate the output of the response variable.

The size of a house is used to determine its price. As a result, the price is the response variable, whereas the size is the predictor variable.

Part (f) Step 1: Given Information

Outliers and potentially influential observations must be identified.  

Part (f) Step 2: Explanation

An outlier is just a piece of data that is far off the regression line.

An impactful observation is one where the removal of a point causes a significant change in the regression equation. That instance, removing a point creates a significant shift in the regression line's direction.

All of the points in component (b) are closed to the regression line, indicating that there are no outliers in the dataset. There are no major changes in the orientation of the regression line when a point is removed, hence there are no influencing observations.

Part (g) Step 1: Given Information

We must forecast the response variable's values based on the values of the predictor variable and evaluate the results.  

Part (g) Step 2: Explanation

A hundred square feet is the size of the houses here. Consequently, $x=26$.

It was determined that

$\hat{y}=140.0839+15.8935x$

When $x=26$, the price of a house is,

$$\begin{gathered} {\hat{y}}_{26}=140.0839+15.8935\left(26\right) \\ =553.3149 \end{gathered}$$

A $2600$ square foot house is expected to cost $553.3149$ thousand dollars.