Hinchcliff_Megan_MAT_240_Project_Two

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Regional vs. National Housing Price Comparison Report 1 Report: Regional vs. National Housing Price Comparison Megan Hinchcliff Southern New Hampshire University

Regional vs. National Housing Price Comparison Report 2 Introduction Region : Mid-Atlantic Purpose This report aims to determine if the Mid-Atlantic region’s housing prices and square footage differ significantly from the national market. This report will answer the following three questions. First, are housing prices in the Mid-Atlantic region lower than the national market average? Second, is the square footage for homes in the Mid-Atlantic region different than the average square footage for homes in the national market? Third, for the Mid-Atlantic region, what is the range of values for the 95% confidence interval of square footage for homes? Sample To begin the analysis, a random sample of 500 homes was taken from the Mid-Atlantic region, which includes states, counties, listing prices, and square footage. To ensure the sample is random with minimal bias, an equation in excel was used. The sample was created by selecting the entire data set of homes from the Mid-Atlantic region and applying the =RAND() equation. Utilizing the copy-and-paste value feature in excel for the random sample column ensures that the calculation will not change throughout the analysis. The first 500 rows of data were then selected to represent the sample of homes in the region. Questions and type of test For this analysis, two hypotheses will be established. The first hypothesis is, are the average housing prices in the Mid-Atlantic region less than the average housing prices in the national market? The second hypothesis is, is the average square footage in the Mid-Atlantic region Not equal to the average square footage in the national market?

Regional vs. National Housing Price Comparison Report 3 A population parameter is a numerical value describing an aspect of an entire group or population. For the first hypothesis, the population parameter is the average house listing prices in the Mid-Atlantic region. The null hypothesis ( H 0 ) is a statement assumed to be true until sufficient data proves the statement to be false. In contrast, the alternative hypothesis ( H a ) contradicts the null hypothesis, which asserts that the true value of the population parameter is different from the hypothesized value. With the first hypothesis, the null hypothesis ( H 0 ) is that the average housing prices in the Mid-Atlantic region are equal to the average housing prices in the national market, which is $288,407. The alternative hypothesis is that the average housing prices in the Mid-Atlantic region are less than the average national market, which is $288,407. A left-tailed hypothesis test will be conducted to confirm or deny that the average listing price of the Mid-Atlantic is less than the national average. For the second hypothesis, the population parameter is the average square footage of homes in the Mid-Atlantic region. The null hypothesis ( H 0 ) is that the average square footage of homes in the Mid-Atlantic region is equal to the average square footage in the national market, which is 1,944. The alternative hypothesis is that the average square footage of homes in the Mid-Atlantic region is not equal to the average national market, which is 1,944. A two-tailed hypothesis test will be conducted to confirm or deny that the average square footage of the Mid-Atlantic is not equal to the national average. Level of confidence Estimation is the process of obtaining information about the population parameter that is both accurate and precise. Accuracy is measured in bias, whereas the standard error measures precision. A confidence interval indicates the probability that a population parameter will fall between a set of values a certain number of times. When an interval estimate is combined with a

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Regional vs. National Housing Price Comparison Report 4 probability or hypothesis statement, it is called the confidence level. For this analysis, the confidence level is 95%, which means there is a 5% (significance level) chance of being incorrect. 1-Tail Test Population Parameter: The average house listing prices in the Mid-Atlantic region. Null Hypothesis: The average housing prices are equal to the average housing prices in the national market, which is $288,407. (µ = $288,407) Alternative Hypothesis: The average housing prices are less than the average national market, which is $288,407. (µ < $288,407) Significance Level: 0.05 or 5% Histogram of the Listing Price

Regional vs. National Housing Price Comparison Report 5 Summary Statistics of the Listing Price Table n Mean Median Std. Dev. Min Q1 Q3 Max Listing Price 500 $252,015 $189,732 $224,571 $69,950 $146,964 $269,975 $2,262,550 Data analysis The histograms above represent the homes’ frequency based on the listing prices of the 500 homes sample. The graph is skewed to the right, with a high peak of frequency data on the left and a tale of low-frequency data on the right. The center of the graph is located on the left side in the $69,950-$165,280 bin. The data spread in the listing price is between $69,950-$2,262,550. When looking at the listing price histogram, an unusual characteristic occurs because there are multiple gaps where zero homes are not listed. The graphs are very similar when comparing the sample to the National Statistics, with a skew to the right. The table above shows the summary statistics of the 500 sample homes. The mean is used to estimate the average number of homes in the listing price, and the median estimates the middle or center of the data. The standard deviation determines the average distance between the data points and the mean. In the sample of homes, the center point or the median for the listing price is $189,732. The average home listing price is $252,015. The standard deviation for the listing price is $224571. This indicates a low standard deviation where the data points are clustered around the mean or average. The summary statistics are lower when compared to the National Statistics. To conduct a t-distribution test, the following requirements must be met; the randomness of the sample group, independence of observations not affecting other observations, normality of data with no outliers present, and substantial sample size. In this analysis, most requirements are met. There are 3 outliers in the data where homes are listed significantly higher than the rest of

Regional vs. National Housing Price Comparison Report 6 the houses, causing a large impact on the analysis. Removing the outliers will benefit the analysis by strengthening the confidence level of the test since the outliers have extreme values that do not fit into the mean house listing prices for the Mid-Atlantic region. The outliers occur in New York, NY, which has a significantly higher cost of living than the rest of the region. Hypothesis Test Calculations Equations in Excel Calculations in Excel Results Sample Mean =AVERAGE (listing price column) =AVERAGE (listing price column) 252,015 Target Sample Population Collected from the hypothesis N/A 500 Standard Error =STDEV.S(listing price column)/SQRT(sample amount) =STDEV.S(listing price column)/SQRT (500) 10043 Test Statistic = (sample mean -target)/(standard error) = (252,015 – 288,407)/(10043) -3.62 Degrees of Freedom = (sample amount-1) = (500 – 1) 499 p value left-tailed test =T. DIST (test statistic, degrees of freedom, TRUE) =T. DIST (-3.62, 499, TRUE) 0.0002 Interpretation The p value measures the probability of obtaining the observed results, assuming that the null hypothesis is true. The lower the p value is, the greater the significance level of the observed difference is. The p value for the analysis is 0.0002, which is less than the significance level of 0.05. With this information, we will reject the null hypothesis. Sufficient evidence exists that the mean average listing price is less than $288,407. In conclusion, the average listing price is less than the average listing price of the national market.

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Regional vs. National Housing Price Comparison Report 7 2-Tail Test Population Parameter: The average square footage of homes in the Mid-Atlantic region. Null Hypothesis: The average square footage of homes is equal to the average square footage in the national market, which is 1,944. (µ = 1,944) Alternative Hypothesis: The average square footage of homes is not equal to the average national market, which is 1,944. (µ ≠ 1,944) Significance Level: 0.05 or 5% Histogram of Square Footage Summary Statistics of Square Footage Table n Mean Median Std. Dev. Min Q1 Q3 Max SqFt 500 1,687 1,719 338 181 1,560 1,864 2,707 Data Analysis The histograms above represent the homes’ frequency based on the square footage of the 500 homes sample. The graph is symmetric; both sides of the center point of the graph are nearly

Regional vs. National Housing Price Comparison Report 8 equal. The center of the graph is located slightly to the right side in the 1,681, $1,831 bin. The data spread in the square footage is between 181-2,707. When looking at the square footage histogram, an unusual characteristic occurs because there is a gap where zero homes are listed with square footage on the tail ends of the histogram. The table above shows the summary statistics of the 500 sample homes. The mean is used to estimate the average number of homes in square feet, and the median estimates the middle or center of the data. The standard deviation determines the average distance between the data points and the mean. In the sample of homes, the square footage’s center point or median is 1,719. The average home square footage is 1,687. The standard deviation for the square footage is 338. This indicates a low standard deviation where the data points are clustered around the mean or average. The summary statistics are significantly lower than the National Statistics, with an expectation for the standard deviation where the two data sets are close in range. To conduct a t-distribution test, the following requirements must be met; the randomness of the sample group, independence of observations not affecting other observations, normality of data with no outliers present, and substantial sample size. In this analysis, all requirements are met, therefor there is sufficient information to proceed with the t-distribution test. Although there are few outliers, the destitution of data shows a normal distribution. Hypothesis Test Calculations Equations in Excel Calculations in Excel Results Sample Mean =AVERAGE (SqFt column) =AVERAGE (SqFt column) 1,687 Target Sample Population Collected from the hypothesis N/A 500 Standard Error =STDEV.S(SqFt column)/SQRT (sample amount) =STDEV.S(SqFt column)/SQRT (500) 15 Test Statistic = (sample mean -target)/ = (1,687 – 1,944)/ (15) -17.00

Regional vs. National Housing Price Comparison Report 9 (standard error) Degrees of Freedom = (sample amount - 1) = (500 – 1) 499 p value Two-tailed test =T. DIST.2T (test statistic, degrees of freedom) =T. DIST.2T (-17.00, 499) 1.833E- 51 Interpretation The p value measures the probability of obtaining the observed results, assuming that the null hypothesis is true. The lower the p value is, the greater the significance level of the observed difference is. The p value for the analysis is 1.833E-51, which is more than the significance level of 0.05. With this information, we can fail to reject the null hypothesis. Sufficient evidence does not exist that the mean square footage is less than 1,944. In conclusion, the hypothesis test did not identify an equal relationship between the Mid-Atlantic region and the national market. Comparison of the Test Results Confidence intervals provide information regarding the range in which a value lies with a degree of probability. The sample mean or critical value is 1,687 SqFt for this analysis, and the standard error is 15. The confidence interval for the Mid-Atlantic region suggests there is a 95% confidence level that the square footage of homes is between 1,672, 1,702. Since the true population parameter is not in the confidence interval, the estimator is inaccurate and fails to reject the null hypothesis. Confidence Interval Calculation (Sample Mean) - (Standard Error), (Sample Mean) + (Standard Error) Two-Tailed Hypothesis: 1,687 -15 = 1,672 1,687 + 15 = 1,702

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Regional vs. National Housing Price Comparison Report 10 Final Conclusions I conducted a hypothesis test to determine if the Mid-Atlantic region’s housing prices and square footage differ significantly from the national market. When analyzing the data of the Mid- Atlantic region, the conclusions were to reject the null hypothesis that the listing price is less than the national market and failed to reject the null hypothesis that the square footage is different from the national market. The confidence interval implies there is a 95% confidence that the true mean square footage of homes in the Mid-Atlantic region is between 1,672 and 1,702. Since the population parameter, the National Statistics mean for the square footage of homes (1,944), is not in the confidence interval, the assessment is inaccurate and fails to reject the null hypothesis. After conducting the analysis, the results of the first hypothesis test were as expected. In contrast, the test results for the second hypothesis test were unexpected. When analyzing the Mid-Atlantic region, I expected to reject both hypothesis tests. Instead, I could not reject the average square footage for homes compared to the National Statistics. After conducting the Two- Tailed hypothesis test, I reviewed the data. I discovered that a few homes with significantly lower square footage were in higher-end substantial cities, which may have caused skewed results. Although the data analysis showed that the average square footage of the Mid-Atlantic region was less than the National Statistics, the p -value did not confirm my test results. After determining the p -value of 1.833E-51, I recalculated all equations for the test multiple times to determine if my calculations were wrong. Every time I came to the same conclusion that the p- value was greater than the significance level. This led to my analysis being unable to accept the alternative hypothesis that homes in the Mid-Atlantic region are not equal to the National Statistics.

Hinchcliff_Megan_MAT_240_Project_Two

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