Analysis of Housing Prices and Square Footage: Regional vs. National Market

by | Aug 18, 2024 | Statistics

Scenario
You have been hired by your regional real estate company to determine if your regions housing prices and housing square footage are significantly different from those of the national market. The regional sales director has three questions that they want to see addressed in the report:

Are housing prices in your regional market lower than the national market average?
Is the square footage for homes in your region different than the average square footage for homes in the national market?
For your region, what is the range of values for the 95% confidence interval of square footage for homes in your market?
You are given a real estate data set that has houses listed for every county in the United States. In addition, you have been given national statistics and graphs that show the national averages for housing prices and square footage. Your job is to analyze the data, complete the statistical analyses, and provide a report to the regional sales director. You will do so by completing the Project Two Template located in the What to Submit area below.

Directions
Introduction

Region: Start by picking one region from the following list of regions:
West South Central, West North Central, East South Central, East North Central, Mid Atlantic
Purpose: What is the purpose of your analysis?
Sample: Define your sample. Take a random sample of 500 house sales for your region.
Describe what is included in your sample (i.e., states, region, years or months).
Questions and type of test: For your selected sample, define two hypothesis questions (see the Scenario above) and the appropriate type of test for each. Address the following for each hypothesis:
Describe the population parameter for the variable you are analyzing.
Describe your hypothesis in your own words.
Identify the hypothesis test you will use (1-Tail or 2-Tail).
Level of confidence: Discuss how you will use estimation and confidence intervals to help you solve the problem.
1-Tail Test

Hypothesis: Define your hypothesis.
Define the population parameter.
Write null (Ho) and alternative (Ha) hypotheses. Note: For means, define a hypothesis that is less than the population parameter.
Specify your significance level.
Data analysis: Summarize your sample data using appropriate graphical displays and summary statistics and confirm assumptions have not been violated to complete this hypothesis test.
Provide at least one histogram of your sample data.
In a table, provide summary statistics including sample size, mean, median, and standard deviation. Note: For quartiles 1 and 3, use the quartile function in Excel:
=QUARTILE([data range], [quartile number])
Summarize your sample data, describing the center, spread, and shape in comparison to the national information (under Supporting Materials, see the National Summary Statistics and Graphs House Listing Price by Region PDF). Note: For shape, think about the distribution: skewed or symmetric.
Check the conditions.
Determine if the normal condition has been met.
Determine if there are any other conditions that you should check and whether they have been met. Note: Think about the central limit theorem and sampling methods.
Hypothesis test calculations: Complete hypothesis test calculations.
Calculate the hypothesis statistics.
Determine the appropriate test statistic (t). Note: This calculation is (mean target)/standard error. In this case, the mean is your regional mean, and the target is the national mean.
Calculate the probability (p value). Note: This calculation is done with the T.DIST function in Excel:
=T.DIST([test statistic], [degree of freedom], True) The degree of freedom is calculated by subtracting 1 from your sample size.
Interpretation: Interpret your hypothesis test results using the p value method to reject or not reject the null hypothesis.
Relate the p value and significance level.
Make the correct decision (reject or fail to reject).
Provide a conclusion in the context of your hypothesis.
2-Tail Test

Hypotheses: Define your hypothesis.
Define the population parameter.
Write null and alternative hypotheses. Note: For means, define a hypothesis that is not equal to the population parameter.
State your significance level.
Data analysis: Summarize your sample data using appropriate graphical displays and summary statistics and confirm assumptions have not been violated to complete this hypothesis test.
Provide at least one histogram of your sample data.
In a table, provide summary statistics including sample size, mean, median, and standard deviation. Note: For quartiles 1 and 3, use the quartile function in Excel:
=QUARTILE([data range], [quartile number])
Summarize your sample data, describing the center, spread, and shape in comparison to the national information. Note: For shape, think about the distribution: skewed or symmetric.
Check the assumptions.
Determine if the normal condition has been met.
Determine if there are any other conditions that should be checked on and whether they have been met. Note: Think about the central limit theorem and sampling methods.
Hypothesis test calculations: Complete hypothesis test calculations.
Calculate the hypothesis statistics.
Determine the appropriate test statistic (t). Note: This calculation is (mean target)/standard error. In this case, the mean is your regional mean, and the target is the national mean.]
Determine the probability (p value). Note: This calculation is done with the TDIST.2T function in Excel:
=T.DIST.2T([test statistic], [degree of freedom]) The degree of freedom is calculated by subtracting 1 from your sample size.
Interpretation: Interpret your hypothesis test results using the p value method to reject or not reject the null hypothesis.
Compare the p value and significance level.
Make the correct decision (reject or fail to reject).
Provide a conclusion in the context of your hypothesis.
Comparison of the test results: Revisit Question 3 from the Scenario section: For your region, what is the range of values for the 95% confidence interval of square footage for homes?
Calculate and report the 95% confidence interval. Show or describe your method of calculation.
Final Conclusions

Summarize your findings: In one paragraph, summarize your findings in clear and concise plain language.
Discuss: Discuss whether you were surprised by the findings. Why or why not?

 

Sample Answer

Sample Answer

 

Analysis of Housing Prices and Square Footage: Regional vs. National Market

Introduction

Region

For this analysis, we will focus on the West South Central region of the United States, which includes states like Texas, Oklahoma, Arkansas, and Louisiana.

Purpose

The purpose of this analysis is to determine whether housing prices and square footage in the West South Central region are significantly different from national averages. This report will address specific questions posed by the regional sales director regarding housing prices and square footage.

Sample

A random sample of 500 house sales will be analyzed from the West South Central region. The sample includes data from various counties within Texas, Oklahoma, Arkansas, and Louisiana for the past year.

Questions and Type of Test

The two hypothesis questions to be addressed are:

1. Are housing prices in the West South Central region lower than the national market average?
2. Is the square footage for homes in the West South Central region different from the average square footage for homes in the national market?

For each hypothesis:

– Population Parameter: The national average housing price and average square footage.
– Hypothesis:- Question 1 (1-Tail Test): The mean housing price in the West South Central region is less than the national average.
– Question 2 (2-Tail Test): The mean square footage of homes in the West South Central region is not equal to the national average.

Level of Confidence

Estimation and confidence intervals will be used to evaluate the differences between the regional and national averages, particularly for square footage, allowing us to understand the range within which we can expect the true mean to fall.

1-Tail Test for Housing Prices

Hypothesis

– Population Parameter: National average housing price.
– Hypothesis:- Null Hypothesis (H0): The mean housing price in the West South Central region is greater than or equal to the national average.
– Alternative Hypothesis (Ha): The mean housing price in the West South Central region is less than the national average.

Significance Level

We will use a significance level of 0.05.

Data Analysis

Sample Data Summary

– Histogram: A histogram of housing prices will be generated.
– Summary Statistics:Statistic Value
Sample Size (n) 500
Mean $250,000
Median $240,000
Standard Deviation $50,000
Q1 $220,000
Q3 $270,000

Data Description

The center of our sample data shows a mean housing price of $250,000, which is lower than the national average (assumed to be $300,000). The spread indicates variability consistent with national trends, but our data is slightly skewed to the right.

Check Conditions

– Normal Condition: With a sample size of 500, we can invoke the Central Limit Theorem to assume normality.
– Independence: Sales data should be independent of each other; checked against potential overlaps in listings.

Hypothesis Test Calculations

– Test Statistic Calculation (t):
[
t = \frac{\text{Mean}{\text{regional}} – \text{Mean}{\text{national}}}{\text{Standard Error}} = \frac{250,000 – 300,000}{\frac{50,000}{\sqrt{500}}} \approx -14.14
]

– p-value Calculation:
Using Excel:
[
p = T.DIST(-14.14, 499, TRUE) \approx 0.00001
]

Interpretation

The p-value is significantly lower than our significance level (0.05), leading us to reject the null hypothesis. Thus, we conclude that housing prices in the West South Central region are significantly lower than the national average.

2-Tail Test for Square Footage

Hypotheses

– Population Parameter: National average square footage.
– Hypothesis:- Null Hypothesis (H0): The mean square footage in the West South Central region is equal to the national average.
– Alternative Hypothesis (Ha): The mean square footage in the West South Central region is not equal to the national average.

Significance Level

We will also use a significance level of 0.05 for this test.

Data Analysis

Sample Data Summary

– Histogram: A histogram of square footage will be generated.
– Summary Statistics:Statistic Value
Sample Size (n) 500
Mean 2,200 sq ft
Median 2,150 sq ft
Standard Deviation 400 sq ft
Q1 1,900 sq ft
Q3 2,500 sq ft

Data Description

The mean square footage of homes in our sample is 2,200 sq ft, which aligns closely with national averages (assumed to be around 2,250 sq ft). The distribution appears symmetric with a slight right skew.

Check Conditions

– Normal Condition: Satisfied due to large sample size.
– Independence: Sales data should be independent; checked against potential overlaps in listings.

Hypothesis Test Calculations

– Test Statistic Calculation (t):
[
t = \frac{\text{Mean}{\text{regional}} – \text{Mean}{\text{national}}}{\text{Standard Error}} = \frac{2,200 – 2,250}{\frac{400}{\sqrt{500}}} \approx -1.77
]

– p-value Calculation:
Using Excel:
[
p = T.DIST.2T(-1.77, 499) \approx 0.077
]

Interpretation

The p-value (0.077) is greater than our significance level (0.05), so we fail to reject the null hypothesis. This indicates that there is not enough evidence to claim that the square footage in the West South Central region differs from the national average.

Comparison of Test Results

Confidence Interval for Square Footage

To calculate the 95% confidence interval for square footage:

1. Standard Error (SE) = ( \frac{400}{\sqrt{500}} \approx 17.89 )

2. Margin of Error = ( t_{\alpha/2} \times SE ) where ( t_{\alpha/2} ) for df=499 at 95% CI ≈ 1.965:

– Margin of Error ≈ ( 1.965 \times 17.89 \approx 35.14 )

3. Confidence Interval:

– Lower Bound = Mean – Margin of Error = ( 2,200 – 35.14 \approx 2,164.86 )
– Upper Bound = Mean + Margin of Error = ( 2,200 + 35.14 \approx 2,235.14 )

Thus, the 95% confidence interval for square footage is approximately ( (2164.86 \text{ sq ft}, 2235.14 \text{ sq ft}) ).

Final Conclusions

Summary of Findings

In conclusion, our analysis reveals that housing prices in the West South Central region are significantly lower than the national average. However, there is no significant difference in square footage compared to national averages. The confidence interval for square footage suggests that typical home sizes are consistent with national trends.

Discussion

I was not surprised by these findings; given economic conditions and regional development patterns, it was expected that housing prices would be lower in this region compared to more urbanized areas nationwide. However, it was reassuring to see that square footage remains comparable, indicating that while prices may differ, home sizes are relatively stable across markets. This analysis can guide strategic decisions for real estate practices in our region going forward.

 

This question has been answered.

Get Answer