The D. M. Pan National Real Estate Company

Full Answer Section

In a linear regression model, the predictor variable (x), also known as the independent variable, is the variable that is believed to influence or cause a change in the other variable. The response variable (y), also known as the dependent variable, is the variable that is being predicted or explained. In this context, square footage (x) will be selected as the predictor variable because it is a characteristic of the house that is known prior to determining the price and is commonly understood to influence a home's value. Listing price (y) will be the response variable, as it is the outcome we are trying to predict based on the square footage.

Data Collection

To perform this analysis, a random sample of 50 houses from the provided U.S. home sales dataset for 2019 was selected.

Sampling the data: To obtain a random sample of 50 houses from a larger dataset in Excel, the following steps would typically be used:

1. Add a Helper Column: Insert a new column (e.g., "Random Number") next to your existing data.
2. Generate Random Numbers: In the first cell of this new column, enter the formula =RAND() and press Enter.
3. Fill Down: Drag the fill handle (the small square at the bottom-right of the cell) down to the last row of your data to generate a random number for every house.
4. Sort Data: Select all your data (including the new "Random Number" column). Go to "Data" tab, then "Sort & Filter" group, and click "Sort." Sort your data by the "Random Number" column in ascending order.
5. Select Top 50: The first 50 rows after sorting will represent your random sample. Copy these 50 rows (including square footage and listing price) to a new sheet for analysis.

Identification of Variables:

• Predictor Variable (x): Square Footage
• Response Variable (y): Listing Price

Scatterplot of Predictor and Response Variables:

(Self-correction: Since I cannot generate an actual plot, I will describe what a suitable scatterplot would look like and what it would indicate.)

A scatterplot of Listing Price (y-axis) against Square Footage (x-axis) for the sample of 50 houses is crucial to visually assess the relationship. For a linear model to be appropriate, the scatterplot should display a general linear trend. This means that as square footage increases, the listing price should generally increase, forming an upward sloping cloud of points. There should not be a distinct curve, a random scatter, or multiple distinct clusters of points. Any clear non-linear patterns would suggest that linear regression might not be the most appropriate model.

Data Analysis

(Self-correction: I will simulate typical summary statistics and describe the expected shape/spread. You would generate these from your actual Excel sample.)

Histograms:

Histogram for Square Footage (x): (Description: A histogram for square footage would likely show a unimodal, right-skewed distribution. The majority of homes would fall within a common range of square footage, with a tail extending to the right representing larger, less common homes. The peak would represent the most frequent square footage range.)

Histogram for Listing Price (y): (Description: A histogram for listing price would also likely be unimodal and right-skewed. Similar to square footage, most homes would fall within a common price range, with a tail to the right indicating a smaller number of very expensive homes. The peak would represent the most common listing price range.)

Summary Statistics:

(Note: The values above are illustrative examples. You will populate this table with the actual statistics from your sample data.)

Interpretation of Graphs and Statistics:

For Sample House Sales (Listing Price):

• Center: The mean listing price (e.g., $320,000) represents the average price of homes in our sample. The median listing price (e.g., $305,000) indicates the middle value when all prices are ordered. Given the likely right-skewness, the mean will likely be higher than the median, pulled upwards by higher-priced homes.
• Spread: The standard deviation (e.g., $85,000) measures the typical deviation of listing prices from the mean. A larger standard deviation indicates greater variability in home prices within the sample. The range of prices (min to max) would also show the full extent of spread.
• Shape: The histogram is expected to be right-skewed (positively skewed), indicating that most homes are clustered at lower to moderate prices, with a few much more expensive homes pulling the distribution's tail to the right. This is typical for real estate data.
• Unusual Characteristics: There might be a few outliers on the higher end of the price spectrum, representing exceptionally large or luxury homes that deviate significantly from the general pricing trend.

For Sample Square Footage:

• Center: The mean square footage (e.g., 1,850 sq ft) represents the average size of homes in our sample. The median square footage (e.g., 1,780 sq ft) is the middle value. Again, a slight right-skewness would suggest the mean is a bit higher than the median.
• Spread: The standard deviation (e.g., 450 sq ft) indicates the typical variation in square footage. A larger standard deviation suggests a wider variety of home sizes in the sample.
• Shape: The histogram is also expected to be right-skewed, reflecting that most homes are of moderate size, with a smaller number of very large homes.
• Unusual Characteristics: Similar to pricing, there might be some outliers in square footage, representing unusually large or small properties.

Comparison to National Population (National Summary Statistics and Graphs House Listing Price by Region PDF):

(Self-correction: Since I don't have access to the PDF, I will describe a hypothetical comparison. You will perform this comparison with your actual data and the provided national statistics.)

To determine if the sample is representative of national housing market sales, we would compare our sample's summary statistics (mean, median, standard deviation) and histogram shapes for both listing price and square footage against the national statistics and graphs provided in the "National Summary Statistics and Graphs House Listing Price by Region PDF."

• Center Comparison: We would compare our sample's mean and median listing prices to the national average. If our sample's central tendency is significantly different from the national average, it might suggest our sample is not fully representative (e.g., if our sample mean is much higher than the national mean, our sample might disproportionately contain more expensive homes).
• Spread Comparison: We would compare our sample's standard deviation to the national standard deviation. A similar standard deviation suggests comparable variability. If our sample has a much smaller standard deviation than the national data, it might indicate our sample is less diverse in pricing.
• Shape Comparison: We would compare the shape of our histograms (right-skewed) to the national distributions. If the national graphs also show a right-skewed distribution for both variables, it supports our sample being representative in terms of the overall pattern.
• Outlier Comparison: We would note if the presence and magnitude of outliers in our sample are consistent with what's observed at the national level.

Determine Representativeness: If our sample's mean, median, and standard deviation for both square footage and listing price are reasonably close to the national figures, and the shapes of our distributions are similar to the national ones, then our sample can be considered representative of the national housing market sales. Significant deviations would suggest a potential sampling bias. For example, if our sample's average price is considerably higher than the national average, it might indicate that our random sampling inadvertently picked more expensive homes or homes from a higher-priced region.

Develop Your Regression Model

(Self-correction: I will describe the expected output and analysis steps. You will generate these with your data.)

Scatterplot with Line of Best Fit and Regression Equation:

(Description: The scatterplot would display the 50 data points, with Square Footage on the x-axis and Listing Price on the y-axis. A line of best fit (regression line) would be drawn through these points, visually representing the linear relationship. The regression equation would be displayed on the chart, typically in the form y = mx + b or Listing Price = (Slope) * Square Footage + Intercept.)

Example Regression Equation: Listing Price = 150 * Square Footage + 50,000 (This is purely illustrative)

Appropriateness of Regression Model: Based on the scatterplot, a linear regression model is appropriate if the points generally follow a straight line. If the points are scattered randomly, show a strong curve, or if there are significant outliers pulling the line in an illogical direction, then a linear model might not be the best fit. Assuming a positive linear trend, the model would be appropriate.

Discussion of Associations:

• Direction: The association between square footage and listing price is expected to be positive. This means that as the square footage of a house increases, its listing price tends to increase as well. This is intuitively sensible as larger homes generally command higher prices.
• Strength: The strength of the association would be indicated by how closely the data points cluster around the line of best fit. If the points are tightly clustered, the association is strong. If they are widely dispersed, the association is weak. We would visually estimate this from the scatterplot before calculating 'r'.
• Form: The form of the association is expected to be linear, meaning the relationship can be reasonably described by a straight line.

Possible Outliers or Influential Points and Their Effect on Correlation:

Visually inspecting the scatterplot, there might be individual data points that lie far away from the general cluster of points and the regression line. These are potential outliers. An outlier in square footage or listing price could significantly influence the slope and intercept of the regression line, potentially skewing the perception of the relationship.

An influential point is an outlier that, when removed, would significantly change the slope of the regression line. For example, a house with exceptionally large square footage but a relatively low price (or vice-versa) could pull the line towards itself, weakening the perceived positive correlation or even changing the slope.

Discuss Keeping or Removing Outlier Data Points and Their Impact: The decision to keep or remove outlier data points is critical and requires careful consideration:

• Reasons to Keep: Outliers might represent legitimate, though unusual, data points (e.g., a genuinely unique luxury home, or a fixer-upper with large square footage sold at a discount). Removing them could lead to an inaccurate representation of the full market range. If the outlier truly belongs to the population of interest, keeping it ensures the model is robust and accounts for variability.
• Reasons to Remove: Outliers might be due to data entry errors or truly anomalous situations that are not representative of the general housing market (e.g., a family transaction well below market value, or data incorrectly recorded). If they are errors, removing them improves the accuracy of the model for typical homes. If they significantly distort the linear relationship and are not representative, removing them can lead to a stronger, more accurate predictive model for the majority of cases.

Impact of Decision:

• Keeping Outliers: If significant outliers are kept, they may weaken the correlation coefficient (r), making the linear relationship appear less strong than it is for the majority of the data. They can also alter the slope and intercept of the regression line, potentially leading to less accurate price predictions for typical homes.
• Removing Outliers: If outliers are removed, the correlation coefficient might increase, indicating a stronger linear relationship. The regression line may become a better fit for the majority of the data, potentially leading to more accurate predictions for typical homes. However, the model would then be less applicable to the types of extreme cases represented by the removed outliers.

Ultimately, the decision should be justified by the context of the data and the purpose of the model. If the CEO wants a model for "typical" homes, removing extreme outliers might be justifiable, assuming they are true anomalies. If the model needs to account for the full spectrum, even unusual properties, then they might be retained.

Calculate r (Correlation Coefficient):

(Self-correction: You would calculate this in Excel using the CORREL function, e.g., =CORREL(Range_Square_Footage, Range_Listing_Price))

**Calculated r value: ** [e.g., 0.85] (This is an illustrative example; your actual value will vary)

Explanation of r value support for scatterplot: A correlation coefficient (r) of [e.g., 0.85] indicates a strong, positive linear relationship between square footage and listing price.

• Positive: The positive sign of 'r' confirms the upward-sloping trend observed in the scatterplot – as square footage increases, listing price generally increases.
• Strong: A value close to +1 (like 0.85) signifies that the data points in the scatterplot are tightly clustered around the line of best fit. This visually confirms a strong linear pattern where changes in square footage are highly associated with predictable changes in listing price. If 'r' were closer to 0, it would suggest a weaker or no linear relationship, and if it were negative, it would suggest an inverse relationship (as one increases, the other decreases).

Determine the Line of Best Fit:

(Self-correction: You would determine this using Excel's regression analysis tools or the SLOPE and INTERCEPT functions. The equation is y = mx + b)

The line of best fit, also known as the least-squares regression line, minimizes the sum of the squared vertical distances from each data point to the line.

• Slope (m): Represents the average change in the Listing Price (y) for every one-unit increase in Square Footage (x).

  • Calculation in Excel: =SLOPE(Range_Listing_Price, Range_Square_Footage)
  • Example Value: m = 150 (meaning for every additional square foot, the listing price increases by an average of $150).

Sample Answer

D. M. Pan National Real Estate Company: Predicting Housing Prices Based on Square Footage

Report Date: May 29, 2025

Introduction

This report aims to develop a predictive model for housing prices for homes sold in 2019, specifically focusing on the relationship between a home's square footage and its listing price. The D. M. Pan National Real Estate Company intends to leverage this information to assist their real estate agents in setting more accurate and competitive listing prices, using square footage as a key benchmark.

The central question this report seeks to answer is: Is the square footage of a house a good indicator for what its listing price should be?

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. It is most appropriate when there is an assumed linear relationship between the variables, meaning that as one variable increases or decreases, the other tends to increase or decrease proportionally. When using linear regression, we would expect the scatterplot to show a roughly linear pattern, where the data points tend to cluster around a straight line, exhibiting either an upward (positive) or downward (negative) trend.