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RANDYF1234_WRITINGGOODPROGRESS_001.docxMedian Housing Price Model for D. M. Pan National Real Estate Company 1
Median Housing Price Model for D. M. Pan National Real Estate Company 3
Median Housing Price Model for D. M. Pan National Real Estate Company
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Report:Median Housing Price Prediction Model for D.M Pan National Real Estate Company
Introduction
Linear regression analysis is a technique that models the relationship between two variables; the predictor, and response variables, by fitting the observed data in linear equation. It is usually performed after establishing there is significant linear relationship between the variables of interest. A scatterplot is employed to check for linear association or linear pattern. Markedly, linear regression can be employed by the real estate companies in sales price appraising because prices are based on the predictable from the factors such as size, location among others. he response variable is the variable of interest in an experiment or whose value is predicted based on another variable. The predictor variable is the variable utilized to explain the differences in the outcome variable.
This report is based on the results of linear regression on observed sample data. It presents linear model for association between the median square feet of homes and the median listing prices., that can used by D. M. Pan National Real Estate Company to estimate the median listing prices. The report also highlights the strength of the model and provides an answer to our research question; “is the size of homes ,in square footage, a benchmark for the listing price?”.
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Data Collection
Random Sample:
Random sample is a subset of n items drawn from a statistical population of N items whereby each member is chosen by chance and has same probability of being selected. In scientific studies, it is imperative to ensure randomness in the sample being used for statistical refernce. Randomness ensures that results obtained accurately approximates what would have been obtained of the population was measured (Shadish, Cook & Campbell , 2002). In other words, randomization accurately reflects and represents the target population because it prevents bias, and controls sampling error (Sekaran & Bougie, 2016). To extract random sample of 50 counties that is truly representative of the source target population, I used the Excel RAND () function. In this scenario, the RAND () function randomly assigns numbers to all the 978 and ensures that each county equally likely to be included in the sample. The steps employed to perform a random selection are;
i.). Adjacent to column name “median square feet” in the spreadsheet of the national data, I added another column and titled it “Random”.
ii.). In the cell of column” Random”, or G2, I typed “= RAND()”, and pressed enter. Then, I copied cell G2 to G979.
iv) Next , I copied G2 to G979 , and then from paste special I clicked on values(123).
v.). Next, I selected the national data plus the random column, then sorted by Random; from the smallest to largest random number.
Vi.) From the result of sorting, I then chose the first 50 data points. This formed the random sample which will be utilized in performing the analysis.
The predictor variable is median square feet. The response variable in the generated sample is median listing prices. Median listing price for homes is what is being predicted or whose outcome is being measured by manipulating the median square feet.
Scatterplot
A scatter plot aids to visually observe the relationship two variables. According to the above scatter plot, there is an upward sloping pattern and linear relationship between the two variables. Therefore, it can be said that the median square feet and median listing prices are moderately, and positively linear related.
Data Analysis
Linear regression analysis models the relationship between a single outcome variable and one predictor variable. In particular, it assesses whether the predictor explains the variation in the response variable and generates a linear model that quantifies the statistical relationship between the two variables. (Sekaran & Bougie, 2016). Of importance to note is that linear regression is anchored on some suppositions; linearity, independence, homoscedasticity, and multivariate normality (Sekaran & Bougie, 2016).
i.). The linearity assumption stipulates that the relationship between the independent and dependent variables being linear. A scatter graph visually indicates whether there exists linear relationship exist between the two variables. It also assesses presence of outliers because linear regression is sensitive to the effects of outliers (Sekaran & Bougie, 2016).
ii.).The independence assumption postulates that the variables are not multicollinearity related. In other words, the variables should be independent of each other. Some of the techniques to check for multicollinearity or independence encompasses of correlation matrix, tolerance, and variance inflation factor.
iii.).The multivariate normality premise states that the residuals or errors should be normally distributed. For any fixed value of independent variable, the response will be normally distributed. A Q-Q plot, histogram and Kolmogorov-Smirnoff test are employed to check for normality
v.). The homoscedasticity premise stipulates that the variance of errors should be the same for each value of the independent variable; constant variance of errors. In other words, equal variance of residuals for every value of X. This can be assessed by scatter graph of residuals versus predicted values. A random spread depicts a constant variance.
Besides these assumptions the two variables should be continuous.
Histograms:
According to the histograms below, the sample data values for both median square feet and median listing price variables are positively skewed. This is because more data values are concentrated on the left side and there is a longer right tail of the distribution.
Summary statistics
|
Summary Statistics |
||
|
Statistic |
Median square feet |
Median listing price |
|
Mean |
1923 |
$242,178 |
|
Median |
1841 |
$242,007 |
|
Standard deviation |
357 |
$89,913 |
Interpretation for the graphs and statistics
i.).Descriptive Statistics
The summary statistics; median, mode and mean are classified the measures of central tendency. They parsimoniously describe a set of data by identifying the central position within the data. The mean of median square feet is 1923 while that of median listing price is $242,178. The former depicts the average size of homes, in square footage, while the latter shows the average recommended price of those homes. The sample mean for both the size and retail price for homes are less than that in the national data. This means that the national data contains higher proportion of counties that have a higher median square feet and median listing prices.
The median of the median square feet shows the middle size of the homes in the sample while the median of the listing price variable shows the middle price of the homes in the sample. In other words, the median for the median square feet variable of 1842 indicates that 50% of the median square feet of homes in the 50 counties is greater more than 1841 square feet. Also, the median for the median listing price variable of $242,007 indicates that 50% of the median listing price of homes in the 50 counties is greater or less than $242,007.
Also, the median, mode and median can help us decipher the distribution for data. In this scenario, the sample means for both the median listing price and median square feet variables are greater than their corresponding sample medians. According to Sekaran and Bougie (2016), if the median is less than the median the distribution is right or positively skewed. Therefore, sample data values for both median listing price and median square feet variables have a positively skewed distribution. This is similarly depicted by the histograms as it can be see that more values are bunched up to the left and with a tail stretching toward the right hence a positively skewed distribution
On the other hand, the standard deviation is a measure of dispersion. It is a bellwether of scatteredness or spread of the data about a central value; the mean. The sample standard deviation for median square feet is 357 which is lower than of national data; it is just less 10 square feet than that of the national data. The sample standard deviation for median listing price is $89,913 which is less than of the national data. The sample data is less dispersed as compared to that of national data.
ii.).Scatter Plot : From the scatter plot , there are more than one points that are farthest from the regression line; outliers. Outliers significantly impact on the slope of the regression line.
iii.). Is the sample a representative of the national housing market sales
Since the sample in this scenario is extracted through random selection, I would expect the sample statistics to accurately reflect the characteristics of the entire population. In particular, the sample findings should be generalized to the source population. Nevertheless, not all random samples provide a true picture of the characteristics of the target population. A comparison between the sample and national statistics helps determine whether the sample is a representative of the population. According to the histograms, the data values of the two variables are positively skewed within both the sample and national data. In addition, the sample statistics for the median square feet variable are almost similar or close to that the same variable in the national data. However, the sample statistics for the median listing prices variable are different from those of the same variable within the data national data. This disparities in the sample statistics and that of the national data indicates that the sample is not geographically representative. In a nutshell, the sample doesn’t accurately reflect the national housing market.
|
Summary Statistics |
||
|
Statistic |
Median square feet |
Median listing price |
|
Mean |
1923 |
$242,178 |
|
Median |
1841 |
$242,007 |
|
Standard deviation |
357 |
$89,913 |
Regression Model
Scatterplot and associations
The median square feet is the independent variable while median listing price is the response variable. According to the scatter graph, the size of homes and their recommended retail price are positively and linearly related. In other words, there is direct association because regression line is upward sloping; as the size of homes increases, their retail price also increases. The linear relationship can be said to be moderate because good number of points in the scatter plot appear to lie close to the best line-of-fit.
Correlation Coefficient
Correlational coefficient is a statistical measure of strength and direction of linear association between two variables on a scatterplot (Rumsey,n.d). The magnitude of the correlation coefficient indicates the strength of the linear relationship whereas the sign of the coefficient depicts the direction. In this case, the correlation coefficient, r, is 0.633374628. According to the fuzzy-firm linear rule, correlation coefficient values between 0.3 and 0.7 depicts a moderate positive linear relationship (). Therefore, in scenario, there is a positive moderate linear relationship between the median square feet and the median listing price because r ,0.633374628, lies between 0.3 and 0.7. This finding is consistent with the interpretation deduced by visually observing the monotonic association between the variables on the scatterplot.
The Line of Best Fit
Regression equation: ŷ = 159.46x – 64541.15
In this case, the independent variable (x) is the median square feet while the outcome variable (y) is the median listing price. Thus, the equation of the line of best fit can be written as; median listing price = 159.46*(median square feet) – 64541.15
Interpretation of regression equation:
For simple linear regression, there is one predictor variable and the general form of the regression equation is Ŷ =b0+b1X where Ŷ is the predicted value of the dependent variable, and b0 and b1 are the slope and intercept coefficients, respective. In this scenario, the slope ,b0 =159.46 while the intercept is -64541.15.The slope coefficient b1 depicts the mean change in the outcome variable given one unit change in the predictor variable (Frost, 2020). In this scenario, the slope of 159.46 indicates that the recommended price of house will rise by $159.46 for each one square foot increment in the size of the homes. Also, the sign of the regression coefficient b1 is a bellwether of the direction of the association between the response and predictor variables (Frost, 2020). A positive sign indicates a direct relationship while a negative sign shows an inverse relationship. In this context, the positive sign of the slope coefficient,159.46, depicts a direct relationship between the size and the retail price of homes.
On the other hand, the intercept is the predicted value of the dependent variable when the value of explanatory variable is zero, x =0. The intercept b0, in this scenario, is -64541.15. The -$64541.15 is the value of the land, when no house has been erected. The value of -$64541.15 does not make sense because the value of land cannot be less than zero or negative.
Strength of the Equation
Predictions based on the Regression Equation
Based on the linear regression model, what would be the listing price for a 1,650 square feet house?
The regression equation; y= 159.46x – 64541.15
For x=15600, y= (159.46*1650) - 25496
y= 263,109- 64541.15
=$198,567.85
Thus, the recommended retail price for a 16500 square feet house is $198,567.85
Conclusion
References
Fernando, J. (2020). Correlation coefficient. Investopedia.
https://www.investopedia.com/terms/c/correlationcoefficient.asp
Jaggia, S., & Kelly, A. (2012). Business Statistics: Communicating with numbers.
McGraw-Hill Education.
Ratner, B. (2009). The correlation coefficient: Its values range between +1/−1, or do they?
Journal of Targeting, Measurement and Analysis for Marketing, 17(2), 139-142. https://doi.org/10.1057/jt.2009.5
Sekaran, U., & Bougie, R. (2016). Research methods for business: A skill building approach.
John Wiley & Sons.
Warner, R. M. (2012). Applied statistics: From bivariate through multivariate techniques.SAGE
https://onlinelibrary.wiley.com/doi/pdf/10.1111/ceo.12358
https://www.investopedia.com/ask/answers/042915/whats-difference-between-representative-sample-and-random-sample.asp
scatter plot
sc
y = 159.46x - 64541 R² = 0.4012
1674.6309523333332 1591.0714285833335 1745.8809523333332 2019.3035714166665 2338.8690475833332 1697.4025973999999 1817.1249999166666 1788.5595237500002 2052.8333333333335 1696.2261904999998 1981.0833333333333 2059.3035714166667 1571.0654761666667 1089.8273810000001 1816.7500000833334 1539.23809525 1861.8095238333333 1841.8452380833332 2074.8571429166664 2205.0476190833333 1913.5654761666667 1726.2738094166668 2280.5416666666665 3101.0654761666665 2545.7559523333334 1840.6190476666668 2125.8392857499998 2001.4285713333331 1798.0595238333335 1897.7321428333332 2271.2678570833336 1730.0297619166668 1892.7202381666666 2488.6190476666666 1384.9345239166666 1646.7440476666668 2002.36309525 1709.2083333333333 1653.9226190833333 1790.0595238333335 1620.4107143333333 1786.5952381166665 1724.48809525 1442.3273809166667 2255.9642857500003 2273.4761904166667 2130.8392857499998 2207.8214284999999 1654 2813.6130953333327 256303.80951666666 116585.11904999999 186136.30952500002 289479.40475833334 372494.64284999995 173567.5974 388485.78571666667 193292.85713333334 280044.23809166666 186989.88094999999 189357.73809999999 209871.42856666667 124139.03570833332 169160.26785 165050.70239166668 136560.30952500002 271549.40475833329 286183.52976666664 246079.16666666666 298733.92857499997 200486.30952500002 167376.19047500004 321513.35714166664 398894.40476666664 377670.92857500003 355840.86310000002 210744.571425 262830.64286666666 203697.25594999999 200816.32738333335 304790.68452499999 136254.16666666666 269974.63689999998 444914.76785 75779.761905000007 134916.54761666668 291311.03571666667 105964.880945 227819.60119166668 139955.05952500002 99574.404761666665 318602.79166666669 166125.86309999999 314629.42262500001 237934.40475833334 258870.48809999999 332796.42857500003 373639.76190833328 351701. 51785 283417.26190833334
Median Square Feet
Median listing price
scatter plot
sc 1674.6309523333332 1591.0714285833335 1745.8809523333332 2019.3035714166665 2338.8690475833332 1697.4025973999999 1817.1249999166666 1788.5595237500002 2052.8333333333335 1696.2261904999998 1981.0833333333333 2059.3035714166667 1571.0654761666667 1089.8273810000001 1816.7500000833334 1539.23809525 1861.8095238333333 1841.8452380833332 2074.8571429166664 2205.0476190833333 1913.5654761666667 1726.2738094166668 2280.5416666666665 3101.0654761666665 2545.7559523333334 1840.6190476666668 2125.8392857499998 2001.4285713333331 1798.0595238333335 1897.7321428333332 2271.2678570833336 1730.0297619166668 1892.7202381666666 2488.6190476666666 1384.9345239166666 1646.7440476666668 2002.36309525 1709.2083333333333 1653.9226190833333 1790.0595238333335 1620.4107143333333 1786.5952381166665 1724.48809525 1442.3273809166667 2255.9642857500003 2273.4761904166667 2130.8392857499998 2207.8214284999999 1654 2813.6130953333327 256303.80951666666 116585.11904999999 186136.30952500002 289479.40475833334 372494.64284999995 173567.5974 388485.78571666667 193292.85713333334 280044.23809166666 186989.88094999999 189357.73809999999 209871.42856666667 124139.03570833332 169160.26785 165050.70239166668 136560.30952500002 271549.40475833329 286183.52976666664 246079.16666666666 298733.92857499997 200486.30952500002 167376.19047500004 321513.35714166664 398894.40476666664 377670.92857500003 355840.86310000002 210744.571425 262830.64286666666 203697.25594999999 200816.32738333335 304790.68452499999 136254.16666666666 269974.63689999998 444914.76785 75779.761905000007 134916.54761666668 291311.03571666667 105964.880945 227819.60119166668 139955.05952500002 99574.404761666665 318602.79166666669 166125.86309999999 314629.42262500001 237934.40475833334 258870.48809999999 332796.42857500003 373639.76190833328 351701.51785 283417.26190833334
Median Square Feet
Median listing price
