Applied Statistics - Project One Submission

Introduction In this project, statistical techniques such as regression analysis were performed to address an authentic research problem. In particular, this report's purpose was to create a regression model that could predict the median housing prices for homes sold in 2019 in the United States based on its square footage. In order to perform linear regression, the relationship between the predictor and response variables should be linear, and the scatterplot should look like an approximately straight line. The predictor variable is the one that influences the response variable. Data Collection The whole data set included 978 countries all across the United States. For this assignment's purpose, 50 countries were randomly chosen using Excel (see Table 1). Table 1 County adams mclean elkhart tippecanoe kalamazoo wayne hancock sandusky eau claire waukesha madison hardin jones greene williamson navajo weld mckinley laramie plymouth median listing price $135,215 $191,054 $93,141 $314,629 $210,494 $412,547 $254,322 $110,053 $99,574 $321,961 $151,231 $266,967 $106,418 $127,516 $408,310 $106,893 $94,494 $128,071 $279,481 $157,728 median $'s per square foot $88 $107 $65 $166 $108 $181 $118 $69 $64 $136 $96 $119 $67 $84 $159 $61 $64 $82 $124 $89 median square feet 1598 1760 1591 1442 1844 2203 1936 1449 1620 2329 1577 2140 1590 1482 2529 1749 1494 1591 2173 1780 3 washington fulton richmond blair lawrence matanuska-susitna orange tulare lincoln lewis charlotte monroe bartow forsyth walker wicomico franklin orange berkeley arlington montgomery jefferson story dakota clay ward pulaski st. charles oklahoma dallas $178,639 $127,413 $121,610 $145,027 $156,788 $106,561 $138,981 $78,821 $103,858 $289,846 $130,832 $133,320 $212,572 $280,054 $448,568 $249,307 $220,555 $107,724 $198,174 $232,319 $75,309 $378,444 $322,137 $290,093 $83,253 $124,139 $274,236 $252,047 $251,713 $215,915 $90 $71 $88 $81 $82 $60 $78 $54 $71 $132 $76 $76 $98 $101 $118 $121 $95 $69 $92 $105 $53 $115 $106 $108 $57 $77 $106 $131 $114 $111 1867 1811 1440 1781 1767 1704 1803 1577 1457 2181 1651 1676 1927 2712 3700 1876 1778 1540 2117 2098 1510 3483 3098 2662 1545 1571 2641 1864 1898 1902 The predictor variable was the square footage, whereas the response variable was the price of the house. A scatterplot of the selected data set for the 50 countries is provided on the next page. As can be seen in Figure 1, the points form an approximately straight line, which means that the relationship between the two variables can be considered linear. Thus, linear regression can be used to predict the listing price based on the square feet. 4 Figure 1 $500,000 $450,000 $400,000 Listing Price $350,000 $300,000 $250,000 $200,000 $150,000 $100,000 $50,000 $0 1000 1500 2000 2500 3000 3500 4000 Square Feet Data Analysis The predictor variable is the one that influences the response variable. In this assignment, the predictor variable is the square footage, which is believed to impact the listing price. Intuitively, it can be assumed that the larger the house, the more expensive it is on the market. The sample is representative of the population because it is randomly taken from it. Figure 2 5 Figure 3 The summary statistics are given in Table 2. Table 2 median square feet median listing price Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count 197967.1 13693.91 168183.8 #N/A 96830.56 9.38E+09 -0.14089 0.801396 373258.3 75309.23 448567.5 9898353 50 Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count 1930.368 71.46914 1780.253 #N/A 505.3631 255391.9 3.549486 1.851209 2260.28 1440.214 3700.494 96518.42 50 Both the median listing price and square footage are skewed to the right (skewness of 0.8 and 1.85, respectively). The median listing price ranged from $75,309 to $448,567, with a mean 6 of $197,967 and a standard deviation of $96,831. The median square feet ranged from 1,440 to 3,700, with a mean of 1,930 and a standard deviation of 505. The Regression Model Figure 4 $500,000 $450,000 $400,000 f(x) = 155.99 x − 103151.18 R² = 0.66 Listing Price $350,000 $300,000 $250,000 $200,000 $150,000 $100,000 $50,000 $0 1000 1500 2000 2500 3000 3500 4000 Square Feet The correlation coefficient is r = 0.81. Therefore, there is a strong positive relationship between the house's square footage and its listing price. In other words, the larger the house, the more expensive it is. The Line of Best Fit As seen in Figure 2, the regression line is y = 155.99x – 103,151, in which x represents the square footage. Hence, each additional square foot increases the listing price by approximately $156. The intercept suggests that a house with 0 square feet would still cost roughly $103,151, which can be interpreted as the corresponding land price without the building itself. The coefficient of determination R2 is 0.6628, which implies that the regression model explains approximately 66.28 percent of the listing prices' observed variation. For instance, a house with 3,700 feet can be expected to cost y = 155.99(3,700) – 103,151 = $474,012. The 7 actual price for this house in the sample data set was $448,568, which is relatively close to the predicted value. Conclusions The purpose of this report was to create a regression model that could predict the median housing prices for homes sold in 2019 in the United States based on its square footage. In order to reach this goal, fifty data points were randomly sampled from the population. There was a strong positive relationship between the square footage of a house and its listing price. Furthermore, an accurate linear regression model was found that managed to explain approximately 66 percent of the observed variation. The regression equation was y = 155.99x – 103,151, in which x represents the square footage, and y represents the listing price.