MAT 240 Module 7 Assignment

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Southern New Hampshire University *

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MAT-240-H7

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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 Erika V Estrada Southern New Hampshire University

Regional vs. National Housing Price Comparison Report 2 Introduction Region The regional real estate company has hired us to help determine if there is a significant difference between the housing price and square footage of a selected region, which, in this case, is the East South-Central Region in comparison with the National market. Purpose The report aims to examine the data and conduct a two-hypothesis report on the house sale market. We have been given national statistics and graphs showing the average housing sales per square footage. Sample Next, we are going to explain how we obtained that sample. First, we need to create a sample data from the current list we are working with. It could be the first 500 or 501 if we have a header, which, in this case, we do. Subsequently, we will use the RAND function in Excel. We will create a column and name it random, then input the formula =RAND () and enter to generate that random number. Then, please hover over the bottom edge of the cell until you see the cursor change to a black (+) sign and double-click to create a random number next to each cell with data on it. Once we have created all random numbers, we will follow the following steps: On the toolbar above, locate the data, then click on it. We need to ensure that all data is selected and then click sort; make sure you click the box that says, “My data has a header.” then we must

Regional vs. National Housing Price Comparison Report 3 identify how we want to sort the data. At this point, you will select random, then choose the order in which you want the data sorted, from smallest to largest. Finally, delete the data you do not need that exceeds the 500 houses requested by the sample. The data included in the sample are the states, counties, regions, house listing prices, and the cost per square footage. Questions and type of Test To analyze the data provided to us, we will be using two different test hypotheses: the right-tailed Test and the two-tailed Test. Both tests have a significance level of a = 0.05 The right-tailed Test is “when the alternative hypothesis says that the values in the parameter of the null hypothesis is less than the claims in the null hypothesis,” according to Numeracy, Maths, and Statistics - Academic Skills Kit (n.d.). The hypothesis is: H0: μ = 288,407 Ha: μ < 288,407 With this right-tailed Test, we must verify if the evidence is enough to state that the average listing price for the East South-Central Region is less than the National market. According to the research (Hayes, n.d.), we can determine that the two-tailed Test is “a method to calculate a statistical significance based on whether a sample greater or less than a specific number of values”. The hypothesis is: H0: μ = 1,944 Ha: μ ≠ 1,944 With this two-tailed Test, we must verify if the evidence is enough to state that the average listing price for the East South-Central Region differs from the National market.

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Regional vs. National Housing Price Comparison Report 4 The average of a data set or mean is the population parameter of the sample of 500 houses selected for the East South-Central Region. The hypothesis method is used to explain the relationship between two variables. In this case, we are using the hypothesis to identify if the mean price of the East South Region is equal to or less than the National average. Estimation uses sample data to create a hypothesis about a population parameter. The confidence interval is the range of values in which this parameter population will fall. In this case, we used a sample of 500 houses from the East South-Central Region. 1-Tail Test The information we gathered for this 1-tailed Test is to determine if the listing prices of the houses in the East South-Central region are equal to or less than the National average. The population parameter gives us a descriptive method of an entire population. In this case, we refer the population parameter to the mean based on the 500 random samples used for the listing price in the East South-Central region. The hypothesis is: Let u = mean house listing price in the East South-Central Region H0: u = 288,407 Ha: u < 288,407

Regional vs. National Housing Price Comparison Report 5

Regional vs. National Housing Price Comparison Report 6 1- TAILTEST National average listing price 288,407 Sample mean $227,867.93 Q1 169850.00 Sample standard deviation 83274.75 Q3 269112.50 Standard error 3724.16 n 500 square root of n 22.36067977 null hypothesis - H0 u = 288,407 alternative hypothesis - Ha u < 288,407 level of significance 0.05 test statisitc -16.26 p - value 0.00 conclusion since the p - value is less than the level of significance which is 0.05 we will reject the null hypothesis. interpretation Because we are rejecting the hull hypothesis, there is enough evidence to support the claim that the average listing price in the East South Centra Region is lower than the National average listing price Homes located in the East South Central region are priced lower than the National Average Let’s look at the random sample created for this data analysis. We can observe that our histogram shows most of the data falling to the right of the graph’s peak; that means our graph is a positively skewed histogram. There are a few outliers, but not enough to generate changes in the data by eliminating the outliers. Looking at the histogram, we can see that most houses are priced between $149,050 and 260,050 range. HOUSE LISTING PRICE SAMPLE 500 MEAN $227,868 MEDIAN $212,750 STANDARD DEVIATION 83274.75 MIN $112,050 QUARTILE 1 169850.00 MAX $643,721 QUARTILE 2 269112.5

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Regional vs. National Housing Price Comparison Report 7 National Listing Price and Square Feet Looking closer at the tables above, we can see the difference in price when we compare our sample with the national average. These two tables confirm our earlier statement about rejecting the null hypothesis since the costs for the homes in the East South-Central Region are lower than the national average. Hypothesis Test Calculation We will explain how the standard error, T-statistic, and P-value are found. As a reminder, the standard error of the mean shows us how different the population mean could be from the sample mean. In this case, before we proceed with the standard error, we must first calculate the standard deviation. Once we have the standard deviation to calculate the standard error, we will use the following Excel formula: = (STANDARD DEVIATION )/(SQUARE ROOT (SAMPLE) in this case, the sample is 500. Next, we need to find the T-statistic. We will use The following Excel formula: = (SAMPLE STANDARD DEVIATION-SAMPLE MEAN)/(STANDARD ERROR)

Regional vs. National Housing Price Comparison Report 8 Finally, the P – value, the Excel formula is: =T.DIST ( X, DEG_FREEDOM, CUMULATIVE). In this case, our X value is the T-STATISTIC, the DEG_FREEDOM is the sample size 500 – 1, and the CUMULATIVE for true is 1. = T.DIST(-16.26,499,1) P-value: 2.6982E-48 = 0.00 Based on the information we see in the images above, we concluded that since the P- value, which we can see is 0.00000, is less than the level of significance of 0.05, we will reject the null hypothesis. We are rejecting the null hypothesis because the data has provided enough evidence to support the claim that the average listing price in the East South-Central Region is lower than the National average listing price. All in all, the conditions that test the hypothesis have been met with the sample size and data distribution. 2-Tail Test The information we gathered for this 2-tailed test is to determine if the square footage of the houses in the East South-Central region is different than the National average. The population parameter gives us a descriptive method of an entire population. In this case, we refer the population parameter to the mean based on the 500 random samples used for the listing price in the East South-Central region. Let u = mean house listing price in the East South-Central Region H0: u = 1,944 Ha: u ≠ 1,944

Regional vs. National Housing Price Comparison Report 9 2 - TAILTEST National average square footage $1,944.00 Sample mean $2,042.85 Sample standard deviation 346.9301814 Q1 1847.75 Standard error 15.51518938 Q3 2222.13 null hypotthesis - H0 u = 1944 alternative hypothesis - Ha u NOT= 1944 level of significance 0.05 test statistic 6.371295555 p - value 0.00 conclusion since the p - value is less than the level of significance which is 0.05 we will reject the null hypothesis Interpretation Because we are rejecting the null hypothesis, the results show that there is enough evidence to conclude that the average square footage in the East South Central region is different than the National average

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Regional vs. National Housing Price Comparison Report 10 SQ FOOTAGE SAMPLE 500 MEAN $2,042.85 MEDIAN $1,997.25 STANDARD DEVIATION 346.930181 MIN 808 QUARTILE 1 1847.75 MAX 3,453 QUARTILE 2 2222.125 National Listing Price and Square Feet By looking at the histogram, we can affirm that it is shaped symmetrically. The center of the histogram shows the square footage between 1,708 and 2,158 range. Although the mean and median for our sample are higher than the national average, the histogram's shape is also symmetrical. We will explain how the standard error, T-statistic, and P-value are found. As a reminder, the standard error of the mean shows us how different the population mean could be from the sample mean. In this case, before we proceed with the standard error, we must first calculate the standard deviation. We will use the following once we have the standard deviation to calculate the standard error: Excel formula: = STANDARD DEVIATION SQUARE ROOT ¿¿ in this case the sample is 500. Next, we need to find the T-statistic. The following is the Excel formula that we will use: = SAMPLE STANDARD DEVIATION − SAMPLE MEAN STANDAR ERROR . Finally the P – value, the Excel

Regional vs. National Housing Price Comparison Report 11 formula is: =T.DIST ( X , DEG_FREEDOM , CUMULATIVE). In this case our X value is the T-STATISTIC, the DEG_FREEDOM is the sample size 500 – 1, and the CUMULATIVE for true is 1. = T. DIST(6.371295555,499,1) P-value: 4.26448E-10 = 0.00 Our interpretation is that based on the collected data, we concluded that since p–value, which we can see is 0.00000, is less than the significance level of 0.05, we will reject the null hypothesis. We are rejecting the null hypothesis because the data has provided enough evidence to support the claim that the average square footage in the East South-Central Region is different than the National average listing price. The conditions that test the hypothesis have been met with the sample size and the data distribution. Comparison Test Results Now, we must identify the 95% confidence interval values of the square footage for the sample we chose, the East South-Central Region. Before determining the confidence interval, we must first find the margin of error. The first step is to determine the alpha value. An alpha value will tell us if the probability of rejecting the null hypothesis is accurate when the null hypothesis is true. As we mentioned before, the confidence interval is 95%, which means we must convert it into decimals: 95 100 = 0.95

Regional vs. National Housing Price Comparison Report 12 Next, we need to identify the alpha. The formula for this is one minus the confidence level: alpha = 1 − 0.95 = 0.05 . Since we now have the alpha value, we can find the margin of error. The formula for this is: = CONFIDENCE.T ¿ ¿ CONFIDENCE.T ( 0.05,346.93,500 ) ¿ 30.48 Margin of error Now that we have our margin of error, we need to identify the lower bound and upper bound by adding and subtracting the margin error from the sample mean: LOWER BOUND = X – m = 2,042.84 – 30.48 = 2012.37 UPPER BOUND = X + m = 2,042.84 + 30.48 = 2073.34 Based on the results, we are 95% confident that the average square footage for the East South-Central Region ranges between [2012.37 and 2073.34]. Final Conclusions Upon considering all the facts, we concluded that the East South-Central Region listing pricing is lower with higher square footage than the National average. I was not surprised by the results of the data. I am from Brooklyn, New York, but I currently live in Tennessee because of my active-duty service. And if we compare the price per square footage, we would be paying for a third of what you could pay for in the East South-Central Region. Finally, finding all this data was genuinely insightful and gave us the skills we need next time to understand statistical data's reasoning.

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Regional vs. National Housing Price Comparison Report 13 Works Cited Frost, J. (2022, March 17). Margin of Error: Formula and Interpreting . Statistics by Jim. https://statisticsbyjim.com/hypothesis-testing/margin-of-error/ Hayes, A. (n.d.). Understanding Two-Tailed Tests . Investopedia. https://www.investopedia.com/terms/t/two-tailed-test.asp#:~:text=In%20statistics%2C %20a%20two%2Dtailed Hayes, A. (2021, October 18). Confidence Interval Definition . Investopedia. https://www.investopedia.com/terms/c/confidenceinterval.asp M Blanchard_SNHU. (2021). MAT240 Module 7 Project Introduction Part 1. In YouTube . https://www.youtube.com/watch?v=D8nRwG1NY7Y MAT-240 Module 7 Project Two (CC) . (n.d.). Www.youtube.com. https://www.youtube.com/watch?v=BBKHXuukXE0 Numeracy, Maths and Statistics - Academic Skills Kit . (n.d.). Www.ncl.ac.uk. https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/statistics/ hypothesis-testing/one-tailed-and-two-tailed-tests.html#:~:text=A%20right%2Dtailed %20test%20is

MAT 240 Module 7 Assignment

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