Region: East North Central
Purpose: The purpose of this report is to determine whether;
The approach will be through the 1 tail and 2 tail t t-tests. The two hypotheses are therefore;
H01 – Housing prices in the East North Central market are lower than the national market average
H02 – The square footage for homes East North Central are different compared to the average square footage for homes in the national market
The random sample will consist of 500 listings chosen from the 1000 units data set provided for the assignment
Sample: Since the study is about housing prices and square footage. That is what is included in the sample of 500 listings (randomized through MS excels ‘rand ()’ function). Housing prices are the listing cost of purchasing the property while square footage is the size of the property.
Questions and type of test: The two hypothesis questions are;
H01 – Housing prices in the East North Central market are lower than the national market average i.e. μ < 342,365
H02 – The square footage for homes East North Central are different compared to the average square footage for homes in the national market i.e. μ ≠ 2,111
The population parameter in the first case is Housing prices while in the second it is square footage. The appropriate test for the first hypothesis is the 1-tail test because there is a juxtaposition in one direction and not the other i.e. μ ≥ 342,365 (using the t test) while the appropriate test for the second hypothesis is the 2-tail test because it checks two directions i.e. μ = 2,111 (using the t test)
In the 1 tail test, point estimation will be useful while in the 2 tail test interval estimation will be critical. The t test will have a 95% confidence level.
Hypothesis: The population parameter is average Housing prices.
The null hypothesis is H01 – μ < 342,365 and the alternative hypothesis is μ ≥ 342,365. For this study the significance level is 95%
Data analysis:
Summary statistics
House listing price | |
Mean | 195,697.40 |
Standard Error | 3,736.10 |
Median | 176,275.00 |
Mode | 139,950.00 |
Standard Deviation | 83,542.60 |
Minimum | 62,800.00 |
1st quartile | 129,950.00 |
3rd quartile | 249,950.00 |
Maximum | 490,001.40 |
Count | 500 |
The median for listing price is $176,275. The spread is normal with most listings crowded at the center of the dataset as evident in the histogram hence the bell shape of the curve with a hunched center and tails in the left and right. This makes the shape symmetric on both sides. The central limit theorem states that the distribution of a sample approximates normality as the sample becomes larger. Despite the mean, median and mode not being the same in the samples of housing prices and, it is our assumption that as the sample increases in size the distribution will increasingly become normal.
Hypothesis Test Calculations:
The test statistic for the housing prices is (195,697.40 – 342,365) / 3,736.1 = – 39.26
The probability (p value, with -39.26 and 499 degrees of freedom) is 1.
Interpretation:
This p value is above the significance level of 0.05.
Since the p value is more than the significance level we must accept the null hypothesis
This means that Housing prices in the East North Central market are lower than the national market average i.e. μ < 342,365
Hypotheses: The population parameter is square footage. The null hypothesis is H02 – The square footage for homes in East North Central region are different compared to the average square footage for homes in the national market i.e. μ ≠ 2,111 and the alternative hypotheses is μ = 2,111
The significance level for this test will be 95%
Summary statistics
Square footage | |
Mean | 1,812.70 |
Standard Error | 16.2 |
Median | 1,724.80 |
Mode | 1,600.00 |
Standard Deviation | 363 |
Minimum | 1,230.50 |
1st quartile | 1,572.80 |
3rd quartile | 1,976.50 |
Maximum | 4,069.00 |
Count | 500 |
The mean, median and mode for square footage are 1,812.70, 1,724.80 and 1,600.00 respectively. In an ideal situation where this would be a perfect normal distribution, the three values would be the same. However, the spread appears normal with most listings crowded at the center of the dataset as evident in the histogram hence the bell shape of the curve with a hunched center and tails in the left and right. This makes the shape symmetric on both sides. The central limit theorem states that the distribution of a sample approximates normality as the sample becomes larger.
Hypothesis Test Calculations:
The appropriate test statistic (t) is (1,812.70 – 1,881) / 16.2 = -4.23
The probability (p value) is 0.999
Interpretation:
Since the significance level is 0.05 and the p value is 0.999 we are compelled to accept the null hypothesis. This means that The square footage for homes in East North Central region are different compared to the average square footage for homes in the national market
Comparison of the Test Results:
Mean – 1,812,
Critical value – 1.96
Standard deviation – 363
C.I. = 1,812 ± (1.96*363) = 1100.52, 2523.48
This confidence interval is the range of values within which 95% of the listed properties fall in terms of square footage
Final Conclusions
[Summarize Your Findings: Refer back to Step 1 and summarize your findings of the sample you selected.]
The purpose of this report was to determine;
The approach was through the 1 tail and 2 tail t t-tests. Both hypotheses proved correct when using a sample of 500 listed houses
The findings were not surprising due to the significant differences observable in the descriptive statistics for both the East North Central region and the national region.
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