Descriptive Statistics

This report explains the meaning behind the statistical data that was collected by The Lloyd real estate agency. The Lloyd real estate agency is involved in retail selling of houses at Adelaide. Information regarding the number of houses sold by the agency in difference suburbs of Adelaide were collected from the organization. The information pertains to the number of houses sold and their average prices for a particular suburb. Information for the year 2017 and 2018 were collected from the agency.

For the analysis of the prices of the houses initially the descriptive statistics of the prices is undertaken. We extend the descriptive statistics to investigate the distribution of the prices. Further we test whether the number of houses sold in every suburb of Adelaide. Finally, we test the prices of the houses.

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Table 1: Descriptive Statistics for the prices of Houses

Houses Prices 2017 Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper
Mean	565875.1
Standard Error	12990.59
Median	513750
Mode	370000
Standard Deviation	248524.6
Sample Variance	6.18E+10
Kurtosis	3.964733
Skewness	1.572198
Range	1635000
Minimum	165000
Maximum	1800000
Sum	2.07E+08
Count	366
1st Quartile	393125
3rd Quartile	687875
IQR	294750

The above table presents the descriptive statistics for the prices of the houses sold in 2017. From the analysis it is found that the average prices of the houses sold is 565875.1 with a standard deviation of 248524.6. The median prices of the houses sold was found to be 513750. Since the mean prices of the houses sold is higher than the median prices hence it can be inferred that the prices of the houses are skewed to the right. Since, the median selling price is 513750 hence it can be said that 50% of the houses were sold below 513750.

Moreover, the analysis showed that maximum number of houses were sold at a price of 370000.  Further, it is found that selling prices of the houses range from a minimum of 165000 to a maximum of 1800000. Thus, it is found that the range of prices is 1635000.

The 1st and 3rd quartile of the selling prices of the houses were 393125 and 687875 respectively.  Thus, it can be interpreted that 25% of the houses were sold below 393125. Similarly, it can be said that the selling price of 25% of the houses are above 687875. Thus, it is found that the IQR of the prices of the houses was 294750. Thus, 50% of the sold were within a range of 294750.

Table 2: Distribution of the selling prices of the houses

Price Range	Frequency
> 300000	26
300000 – 500000	147
500000-700000	106
700000-900000	56
900000-1100000	18
1100000-1300000	7
1300000-1500000	3
1500000-1700000	2
< 1700000	1

Figure 1: Distribution of Selling Price of houses

The mean prices of all houses sold in 2017 at Adelaide is 565875. The standard deviation of the prices of houses sold is 248525. Let’s assume that the selling prices of houses is normally distributed.

Recently Lloyd sold a house for 594966.

The z-score of 594966 would express how much the price of house sold is away from the mean price of all houses sold in 2017.

The z-score for 594966 is given through

The z-score of the house sold informs us that the selling price is 0.117 times the standard deviations is from the mean.  

Chi-square test is used to investigate if the number of houses sold in every suburb was equal.

Distribution of selling prices of houses

Thus, the number of houses sold in each and every suburb was aggregated. Thus, the number of houses sold in each suburb was observed. The expected number of houses sold in each suburb is 22.875.

Null hypothesis: The average number of houses sold is independent of the suburb

Alternate hypothesis: The average number of houses sold are equal in each suburb

The Chi-square test is used to test the hypothesis.

Level of Significance: 0.05 level of significance is used to test the hypothesis.

Decision Rule: The degrees of freedom = 15. At 0.05 level of significance and 15 degrees of freedom χ² crit value is 24.996. Thus, if the calculated χ² value is more than χ² -crit values then we reject Null Hypothesis else accept Alternate Hypothesis.

The χ² value is calculated as

Table 3: Observed and Expected number of houses sold

Description	Observed	Expected	Calculation
ADELAIDE HILLS	15	22.875	2.711
BURNSIDE	25	22.875	0.197
CHARLES STURT	39	22.875	11.367
GAWLER	10	22.875	7.247
HOLDFAST BAY	11	22.875	6.165
MARION	26	22.875	0.427
MITCHAM	28	22.875	1.148
NORWOOD PAYNEHAM & ST PETERS	18	22.875	1.039
ONKAPARINGA	39	22.875	11.367
PLAYFORD	22	22.875	0.033
PORT ADELAIDE ENFIELD	47	22.875	25.443
PROSPECT	5	22.875	13.968
SALISBURY	23	22.875	0.001
TEA TREE GULLY	23	22.875	0.001
UNLEY	15	22.875	2.711
WEST TORRENS	20	22.875	0.361
Grand Total	366	χ2 =	84.186

Table 4: chi-Square test Calculations

Statistics	Value
a	0.05
df	15
χ2	84.186
p-value	0.000
χ2 crit	24.996

From the analysis it is found that χ² value = 84.186. Since χ² value is more than χ² crit, hence we reject Null Hypothesis. Thus it is found that the average number of houses sold in each suburb is equal to 22.875.

We assume that the selling prices of houses sold in 2018 by Lloyd is normally distributed. Further we test the following probabilities.

The probability that the selling price of a house is 390000.

The probability that the selling price of a house is above 690000

The probability that the selling price of a house is between 390000 and 690000.

From an analysis of houses sold in 2018 it is found that the average price of houses sold in Adelaide is 594966. The standard deviation of selling price is 306579.

Thus the probability that the selling price of a house is 390000  

From z-table it is found that

Hence, it can be inferred that the probability that the selling price of a house is 390000 = 0.2514

From an analysis of houses sold in 2018 it is found that the average price of houses sold in Adelaide is 594966. The standard deviation of selling price is 306579.

Thus the probability that the selling price of a house is more than 690000   we first calculate for

From z-table it is found that

Thus, the probability that

Hence, it can be inferred that the probability that the selling price of a house is more than 690000 = 0.3783

The average price of all houses sold in 2018 is 594966

The standard deviation of the prices of houses sold in 2018 is 306579

Thus, the probability that a house is sold for more than 390000

Thus, the probability that a house is sold for more than 69000

Thus, the probability that the selling price of a house would be between 390000 and 690000 is 0.3703

We further investigated whether the mean prices of the houses sold was equal to 600000.

Null Hypothesis: The mean prices of the selling prices of the houses in 2017 is equal to 600000.

Alternate Hypothesis: The mean prices of the selling prices of the houses in 2017 is not equal to 600000.

Z-Scores

Level of Significance: 0.05 level of significance is used to test the hypothesis.

Decision Rule: Degrees of freedom = 365. At 0.05 level of significance and 365 degrees of freedom t-crit values for two-tailed t-test are -0.0627, 0.0627. Thus, if the calculated t-stat is more extreme than t-crit values then we reject Null Hypothesis else accept Alternate Hypothesis.

Calculation: The t-stat is calculated through:

Table 5: Hypothesis test for

Hypotheses
Null Hypothesis	µ	=	600000
Alternative Hypothesis	µ	<>	600000
Test Type			Two
Level of significance
		α	0.95
Critical Region
Degrees of Freedom	365
Lower Critical Value	-0.0627
Upper Critical Value	0.0627
Sample Data
Sample Standard Deviation	248525
Sample Mean	5,65,875
Sample Size	366
Standard Error of the Mean	12990.5872
t Sample Statistic	-2.6269
p-value	0.0090
Decision
Reject Null Hypothesis

Decision: The value of the t-statistics is -2.6269. Since the value of t-stat is higher than 0.0627, hence we reject the Null Hypothesis. Hence, it can be said that the mean selling price of the houses in 2017 is not equal to 600000. Thus it is inferred that the mean selling price of the houses in 2017 is less than 600000.

Further, we tested if the prices of houses sold in 2017 is equal to 2018

Null Hypothesis: The mean prices of the selling prices of the houses in 2017 and 2018 are equal

Alternate Hypothesis: The mean prices of the selling prices of the houses in 2017 and 2018 are not equal

Type of test: Two tailed is test is used to

Level of Significance: 0.05 level of significance is used to test the hypothesis.

Decision Rule: Degrees of freedom = 700. At 0.05 level of significance and 700 degrees of freedom t-crit values for two-tailed t-test are -1.963, 1.963. Thus, if the calculated t-stat is more extreme than t-crit values then we reject Null Hypothesis else accept Alternate Hypothesis.

Table 6: t-Test: Two-Sample Assuming Unequal Variances

	Houses Prices 2017	House Prices 2018
Mean	565875	594966
Variance	61764459939	93990501276
Observations	366	366
Hypothesized Mean Difference	0
df	700
t Stat	-1.410
P(T<=t) one-tail	0.079
t Critical one-tail	1.647
P(T<=t) two-tail	0.159
t Critical two-tail	1.963

Decision: The value of the t-statistics is -1.410. Since the value of t-stat is lower than -1.963 hence we do not reject the Null Hypothesis. Thus, it is found that the selling prices of the houses in 2017 and 2018 are equal.

Conclusion 

For the present report we have considered the information from Lloyd real estate agency. The agency is involved in real estate in the state of Adelaide. The organization has branches in different suburbs of Adelaide. The organization was generous enough to provide us with information regarding the number of houses sold in different suburbs in 2017 and 2018. They also provided us with the information of the prices of houses sold in different suburbs.

From the study of descriptive statistics for the prices of houses sold in 2017, the mean prices of houses were found to be higher than the median prices of the houses. Thus it is found that the prices of house are skewed to the right. Further, the histogram was used to visualize the distribution of the houses. The histogram also proves that the prices of the houses are skewed to the right. The spread of houses is investigated thorough the use of minimum and maximum price of a house. In addition, the quartile values are also used to explore the spread of the prices of the house.  

z-score was used to study the assumption of how far the price of a house from the mean.

Moreover, the normal distribution was used to study the probability of the price of a house.

The chi-square test is used to check for the independence of the number of houses sold in different suburbs. From the chi-square test it is found that the number of houses sold in a suburb is not independent of the number of houses sold. Thus, we find that the average number of houses in each suburb of Adelaide are equal.  

In addition, two hypothesis test is done. In the first hypothesis test a one sample t-test is used. The one-sample t-test is used to investigate if the price of a house is equivalent to a given price. The one-sample t-test proves that the mean prices of houses sold in 2017 is less than 600000. In the second hypothesis test independent sample t-test is used. The independent sample t-test is used to investigate is the mean price of a house sold in 2017 is different than 2018. In addition, we find that the mean prices of houses sold in 2017 is equal to the mean prices sold in 2018.