Purpose
Housing price is a key social-economic status indicator in an economy. In the recent past, the prices of houses have changed in the United States due to shifts in population sizes, and other socio-economic factors. There have been sharp oscillations in the price of houses in various states and regions in the United States, as a result of both macroeconomic and microeconomic factors. Some of the factors that have significantly affected the price of houses are the government policy and regulations, such as regulation of interest rates, changes in population sizes, economic growth, and the real income of the households.
The demand for houses depends on the household income. An increase in the economic growth rate of the country results in a rise in real income, hence, it pulls up the demand for houses. With more income, people tend to look for good houses that meet their new status and for additional comfort. Also, an increase in the interest rates leads to an increase the cost of mortgages and loans, that also tend to reduce the demand for houses. In the United States, various regions have different house-prices. For instance, the state of Hawaii has a high average house price of $587,700 compared to Mississippi valued at $103,000. The study aims to examine whether the average prices and square footage of houses in the South Atlantic region differ when compared with the national average. Random sampling is a technique used in research to help eliminate bias, because it provides equal selection opportunities to all participants in a study. In this case, the houses were randomly selected from the US population. The study has two hypotheses. Hypothesis one states that, the regional house listing price differs from the mean national price. Hypothesis two states that, the regional square footage of houses significantly differs from the national average.
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Sample
A sample is regarded as a subset or a representation of a population. A sample size of 100 houses was taken for this study, where their corresponding prices were recoded, and house square feet. The study obtained data from eight states of the South Atlantic region.
Questions and type of test
Research Question 1: Are the regional housing prices higher compared to the national average price? The study parameter is the average prices of houses. These are house prices from the 100 sampled houses in the South Atlantic Region.
The study tests the null and alternative hypothesis for the research question:
H0: The average region housing prices are equal to the national average.
H1: The average region housing prices are significantly higher than the national average.
The study will employ a one-sample t-test to compare the regional house prices with the national average. The national price average will be hypothesized to establish if it is less or equal to the regional market prices. A 5 % level of significance will be used for the test.
Research Question 2: Are the region averages of home square footages significantly different from the national average? The study parameter is the square footage for the houses. The square footage was sampled from various houses or homes in the South Atlantic region.
Hypotheses:
Null (H0): The regional market means of square footages for houses do not significantly differ from the national market average
Alternative(H1): The regional market means of square footages significantly differ from the national market average.
A one-sample t-test will be used to perform comparisons for the average square footage for houses with the national average. The national average square footage will be hypothesized to establish whether they are different from the regional market, at a 5 % significancy level for the test.
Level of significance: The study compares the p-value with the level of significance, to either reject the null hypothesis or accept it. When the p-value is smaller when compared to alpha = 0.05 ( p < 0.05), the study rejects the null hypothesis, because the test is considered statistically significant. However, if the p-value is greater than 0.05 ( p > 0.05), the study accepts the null hypothesis, implying that the test is insignificant.
1-Tail Test
Hypothesis:
Null (H0): The mean price of houses in the region is equal to the national average (µ1 = µ2).
Alternative (H1): The mean price of houses in the region is greater than the national average (µ1 ≠ µ2).
The level of significance is 5 % (0.05).
Data analysis:
Histogram
The distribution of house prices is illustrated in the histogram above, which indicates that the prices of houses are positively skewed. When the data is skewed it implies that the assumptions of normality have been violated. This means that most prices of houses in the South Atlantic region are lower, hence, less expensive. The table below summarizes the descriptive statistics.
House listing price |
|
| Mean |
292514.3264 |
| Standard Error |
15341.34083 |
| Median |
255885 |
| Mode |
225050 |
| Standard Deviation |
153413.4083 |
| Sample Variance |
23535673857 |
| Kurtosis |
3.189639895 |
| Skewness |
1.70764984 |
| Range |
765000 |
| Minimum |
97550 |
| Maximum |
862550 |
| Sum |
29251432.64 |
| Count |
100 |
The table above indicates a summary of the sampled house prices. It points out that the houses in the region cost $292,514.32 in average, and exhibit a standard deviation level of $153,413.40. The table also indicates a median price that is smaller than the mean price, implying positive skewness of the data. Hence, the data it is not normally distributed. The median price for the houses in the South Atlantic region is $255,885 and the mode is $225,050. The mean price of houses sampled from the South Atlantic region is greater than the average price of houses nationally, that stands at $288,407.
Hypothesis Test Calculations:
The one sample t-test was used in this study. The following formulae is used to compute the test statistics (t).
Where
is the mean prices in the region,
is the average price nationally,
n
= sample size and
SE
= standard error. The national average price is $ 288,407.
Hence
=
=
= 0.2677
Using Excel, the p-value =T.DIST.RT(0.2677, 99) = 0.39474359
P-value = 0.39474359
Interpretation:
Results indicate that the p-value for this test is larger than alpha =0.05, thus, the study accepts the null hypothesis. These results imply that the region’s mean of house prices equals the national house-price average. Thus, the prices of houses in the South Atlantic are the same to the national average, although the one sample t-test does not provide sufficient evidence to support the claim.
2-Tail Test
A two-tailed test was used to assess whether the regional average square footage for the homes differs significantly with the national average level. The claim will be tested by a one sample t-test technique.
Hypotheses:
H0: The mean square footage for homes in the region is not significantly different from the national average.
H1: The mean square footage for homes in the region is significantly different from the national average.
The level of significance, alpha = 0.05.
Data Analysis:
Histogram
The histogram above represents the home’s square footages in the South Atlantic region. It indicates that the square footages of homes are normally distributed. This is because the histogram is symmetrical and assumes a bell-shape. Hence, the assumption of normality is met. The histogram shows that there are no outliers in the dataset. Thus, the assumption of no extreme values or outliers in the dataset is met. The table below summarizes the descriptive statistics.
|
Square footage |
|
| Mean |
1979.965 |
| Standard Error |
33.07692325 |
| Median |
1908.5 |
| Mode |
2085 |
| Standard Deviation |
330.7692325 |
| Sample Variance |
109408.2852 |
| Kurtosis |
0.259349516 |
| Skewness |
0.572378508 |
| Range |
1721 |
| Minimum |
1311 |
| Maximum |
3032 |
| Sum |
197996.5 |
| Count |
100 |
The table above shows that the mean square footage for the homes is 1979.965 feet, with a standard deviation of 330.769 feet. The median is 1908.5 feet, with a mode of 2085 feet and range of 1721 square feet. The median is relatively equal with the mean square footage, suggesting that the data is normally distributed. The average square feet for the houses in the South Atlantic region is greater than the national mean, which averages at 1944 square feet.
Hypothesis Test Calculations:
The study employed a one sample t-test. The following formulae was used to compute the test statistics (t).
Where
is the region mean square footage for homes,
is the average square footage nationally,
n
= sample size and
SE
= standard error. The national mean square footage is 1944 square feet.
Hence
=
=
= 1.078
Using Excel, the p-value =T.DIST.RT(1.078, 99) = 0.14183
P-value = 0.14183
Interpretation:
The results above imply that the null hypothesis will be accepted, because the p-value is greater than 0.05 ( p > 0.05). the study therefore deduces that the region and national averages for the house square-footage is equal. This implies that most of the homes in South Atlantic have the same square footage with the national average, although the one sample t-test does not provide sufficient evidence to support the claim.
Comparison of the Test Results:
The confidence interval is computed as: C.I =
where
is the sample mean, Zc is the z value for confidence level,
SE
is the standard error.
Housing prices
C.I =
=
=
=
Therefore, we are 95 % confident that the housing prices lies between $ 262,445.29, and $ 322,583.35.
Square Footage
C.I =
=
=
Therefore, we will be 95 % confident that the square footage for the houses lies between 1889.663 to 2019.323 square foot.
Final Conclusions
The purpose of the paper was to perform a comparison of the prices and square footage of houses between the South Atlantic region and national average in the US. This study revealed that the mean prices of houses and their average square footages are higher than the average national levels. This implies that the regional houses are highly-priced as compared to the situation of prices at the national level. Further, the South Atlantic homes’ square footages are higher than the mean national average. However, the one-sample t-test established no evidence supporting the two claims. Therefore, the housing prices and square footage of homes in the South Atlantic region are the same as the national average levels. The finding is surprising because the South Atlantic region’s cost of living is high, as well as the cost of housing. The expectation was that the price and square footage for homes in the South Atlantic region would be significantly higher and different from the national average.
