MAT 240 Project Two

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Southern New Hampshire University *

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MAT240

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Economics

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Feb 20, 2024

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Regional vs. National Housing Price Comparison Report 1 Report: Regional vs. National Housing Price Comparison Dianna Sheely Southern New Hampshire University

Regional vs. National Housing Price Comparison Report 2 Introduction Region : East North Central region, including Illinois, Indiana, Michigan, Ohio, and Wisconsin. Purpose : This project aims to determine if housing prices and square footage in the East North Central Region significantly differ from that of the national market of the United States. To analyze this appropriately, we will address the following questions: are the housing prices in the regional market lower than the national market average, is the square footage for homes in the East North Central region different from the average square footage for homes in the national market and what is the range of values for the 95% confidence interval of square footage for homes in the East North Central region? Sample: A random sample of 500 out of 1000 house sales from the East North Central region of which states of Illinois, Indiana, Michigan, Ohio, and Wisconsin from the national data provided has been taken. Data for these states containing their county, average listing prices, and square footage were given in correspondence to the specific area. We will specifically focus on this area's house listing price and square footage compared to the national average. The data used is a randomly collected sample determined using the Excel random function (=RAND), sorting the data from the lowest to the highest with a randomly generated number and selecting the first 50 of the sorts. This data will be compared to the national data regarding house listing prices and square footage.

Regional vs. National Housing Price Comparison Report 3 Questions and type of test: We will be conducting two different tests in this analysis. We will use data from the East North Central region and National markets in both tests. In both tests, we will use a significance level of 0.05. The population parameter for both tests is taken from data using the mean listing prices and square footage for 500 random homes in the East North Central area compared to the national market average. Our first hypothesis test will be a 1-Tail, left-tailed test. Are the housing prices in the East North Central region lower than the national market average? The null hypothesis for this test is that the East North Central region’s mean listing price equals the national average. With an alternative hypothesis that states the mean listing price for the North Central Region is lower than the mean listing price for the national market. The test statistic we will be using is the average house price of the national market. (µ = sample average listing price) H 0 : µ = $288,407 H 1 : µ < $288,407 α = 0.05 Our second hypothesis test will be a 2-Tailed test. Is the average square footage of the East North Central region homes different from those in the national market? The test statistic we will use in this analysis will be the average square footage of homes in the national market.

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Regional vs. National Housing Price Comparison Report 4 The null hypothesis for this test is that the East North Central region’s mean square footage equals the national average of 1,944 square feet. An alternative hypothesis states that the mean square feet for the North Central Region are different from the mean listing price for the national market. (µ = sample average square footage of homes) H 0 : µ = 1,944 H 1 : µ ≠ 1,944 α = 0.05 Level of confidence: We will use confidence intervals and estimates to help solve and support our analysis. Confidence intervals are one way we can show how good our estimates are. 1-Tail Test Hypothesis: The population parameter being tested is the mean house listing price for the East North Central region. Where µ represents the mean house listing price, we are testing that the housing prices in the East North Central region are lower than the national market average. The mean listing price for the East North Central region is less than the national average. Our sample’s mean house listing price for this region is $198,800. Null hypothesis H0: µ = $288,407 Alternative hypothesis H1: µ < $288,407 The significance level α 0.05 is used in this analysis, and the t-test is used in this left-tailed test.

Regional vs. National Housing Price Comparison Report 5 Data analysis : Graph 1 Graph 2

Regional vs. National Housing Price Comparison Report 6 The sample analyzed has produced the above histogram (chart 1), which is a right- skewed shape and is non-symmetrical, with a center of $183,896, a spread of the graph ranges between $62,800 - $395,800, and a standard deviation of $82,994.62. Our graph is similar in shape and direction to the national statistics histogram. However, the summary statistics from our sample are significantly less than the national summary statistics, as seen in Table 1 below. Overall, our conditions to test our hypothesis have been met using a random sampling of 500, more than the 30 required. Our significance level is denoted at α = 0.05, and we have a normal data distribution. n Mean Median Std. Dev. Min. Q1 Q3 Max Listin g price ($) 500 $198,799.6 0 $183,896.0 0 $82,994.6 2 $62,800.0 0 $131,853.5 7 $252,937.5 0 $490,001.4 3 Table 1 Hypothesis Test Calculations: To test our hypothesis, the following calculations must be determined. T-Statistic Using Excel, we can calculate our test statistic, where x̅ is the regional mean and µ is the national mean, by using the following formula: T Test = (x̅ - µ) / (ơ/√n) T = (198,799.60 – 288,407.00) / 3711.632 T = -24.14 Probability ( p -value) We also calculated our p-value using the = T.DIST function in Excel. = T.DIST(test statistic),(degree of freedom),True) The degree of freedom is calculated by subtracting 1 from the sample size. 500 – 1 = 499 =T.DIST(-24.14,499,TRUE) p-value = 0.000000

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Regional vs. National Housing Price Comparison Report 7 Therefore, our p-value = 0.000000 Interpretation: p -value = 0.000000 α = 0.05 The p- value helps us determine if the null hypothesis will be supported or rejected. Since our p- value of 0.00000 is less than our significance level of 0.05, we will reject the null hypothesis that the house listing prices in the East North Central region are equal to those of the national market. Evidence supports that the average listing prices in our sample for the East North Central region are lower than the national average for the United States. We are 95% confident that the average home listing price in the East North Central region is between $191,507.25 and $206,091.96. Below the national market average of $288,407. 2-Tail Test Hypotheses: The population parameter being tested is the mean square feet for homes in the East North Central region. Where µ represents the mean square feet, we are testing that the square feet of homes in the East North Central region differ from the national market average of 1,944. The mean square feet for homes in the East North Central region are less than the national average, as seen in Table 2. The mean square foot for this region is 1,814. Our null and alternative hypotheses are stated below: Null hypothesis H0: µ = 1,944 Alternative hypothesis H1: µ ≠ 1,944 The significance level α = 0.05 is used in this analysis.

Regional vs. National Housing Price Comparison Report 8 Data Analysis : Graph 3 Graph 4 Table 2 n Mean Median Std. Dev. Min. Q1 Q3 Max Square Feet 500 1,814 1,739 345.33 788 1,571.38 2,005.5 3,957

Regional vs. National Housing Price Comparison Report 9 The histogram’s shape, in Graph 3, is right skewed and non-symmetric, with a few outliers on the right side. This happens when the mean is greater than the median, which we see in our sample data for the East North Central region in Table 2. The center of the graph falls at 1,739, the sample median. We see that the overall range of the graph is from 788 sq ft. to 3,188. Compared to the national statistics graph, the histogram for our sample region is less symmetrical. Our data collection met the requirements for a two-tailed hypothesis test as the sample was collected randomly, and the sample size was normal at 500, more than 30. Hypothesis Test Calculations: To test our hypothesis, the following calculations must be determined. T-Statistic Using Excel, we can calculate our test statistic, where x̅ is the regional sample mean, and µ is the national mean for square feet of houses, by using the following formula: T Test = (x̅ - µ) / (ơ/√n) T = (1,814 – 1,944) / 345.329 T = -8.44336099 Probability ( p -value) We also calculated our p-value using the =T.DIST function in Excel. =T.DIST(test statistic),(degree of freedom),True) The degree of freedom is calculated by subtracting 1 from the sample size. 500 – 1 = 499 =T.DIST(-8.44336099,499,TRUE) = 0.0000

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Regional vs. National Housing Price Comparison Report 10 Therefore, our p-value = 0.0000. In relation, our p-value < α Interpretation: p-value = 0.0000 α = 0.05 Since our p- value is 0.000 is less than our significance level of 0.05, we will reject the null hypothesis that the square house footage for homes in the East North Central region equals those of the national market, 1,944. Evidence supports that the square footage for homes in the East North Central region differs from the United States national average. We are 95% confident that the average home square footage in the East North Central region is between 1,739 and 1,814 sq ft below the national market average of 1,944. Comparison of the Test Results: Confidence intervals help us to determine how good our estimates are. We need to determine our margin of error to calculate our confidence interval. We do this by using the following formulas: The interval = (µ - error, µ + error) First, we must determine our margin of error. E = Z α / 2 ơ √ n Or by using the =CONFIDENCE.T function in Excel. =CONFIDENCE.T(α,Standard Dev,size) =CONFIDENCE.T(0.05,345.329,500) =30.34 Lower bound = µ - error = 1,783.26 Upper bound = µ + error = 1,843.95

Regional vs. National Housing Price Comparison Report 11 = (1,783.26, 1,843.95) The national average square footage of a home in the United States is 1,944. Since this is not within the interval (1,783.26, 1,843.95), this supports the evidence that the sample mean differs from the national mean. Final Conclusions In conclusion, this project aimed to determine if housing prices and square footage in the East North Central Region significantly differ from that of the national market of the United States. Overall, the findings from our hypothesis test from the East North Central region were expected as this area is more affordable. By conducting two different tests in this analysis, a 1-tailed test for housing prices and a 2-tailed test to analyze square housing footage, we were able to determine that both the average housing prices ($198,799.60) and square footage (1,814) in this region are less than the national average of housing prices $288,407 and square footage of 1,944. Our confidence interval statistics analysis was able to help us determine the range of values for the 95% intervals in both pricing ($191,507.25 - $206,091.96) and square footage (1,783.26 – 1,843.95) and support our conclusion that we are 95% confident that both housing prices and square footage of homes in the East North Central region are lower than the national market average.

MAT 240 Project Two

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