Analysis of Prices

We use the dataset of 60 observations given to us for 3 districts to answer a range of questions on prices across places and other features like ocean view, and type of dwelling-unit or house. The 3 districts covered are Sydney, Wollongong and Newcastle. 2 other categorical variables are provided for each data point- the type of dwelling can be unit or a house.    We are also told about the absence or presence of ocean view with the dwelling. The focus of the report is on PRICES of dwellings and how these vary across regions, dwelling type and presence of an ocean view.

We use Microsoft Excel to answer a range of queries pertaining to this data. We use concepts like measures of central tendency, dispersion, correlation, confidence intervals, and hypothesis testing. We use t distribution to deal with the hypothesis testing. Visual charts are included – pie chart, bar chart, and histogram to aid in our analysis.

Analysis:

This section is divided into sub sections, where each subsection deals with a separate query. We note that we have 4 variables in all, out of which only 1 variable is quantitative. This is prices of dwellings. All other variables are categorical in nature.

We begin with an analysis of prices irrespective of location, dwelling type and ocean view.  A snapshot of prices in the following histogram is given. We have used 6 classes here with width of $150 each. 

This chart is based on the following data. We can see that prices are relatively normally distributed. This is all seen from the descriptive statistics given below. 

PRICE
Mean	543.0481
Standard Error	24.64311
Median	528.7699
Standard Deviation	190.8847
Sample Variance	36436.97
Kurtosis	0.844758
Skewness	0.770584

< 300	6
30-450	14
450-600	21
600-750	11
750-900	4
> 900	4

The mean price is $543, whereas the median is $528. So we have 50% dwellings with a price that exceeds $528. As mean exceeds median we know that the distribution is positively skewed, but not by a large degree. The skewness value is only 0.77.

Next we disaggregate the data by location. Each location has 20 data points, which are analysed in table below. As can see that mean price is highest for Sydney.

Variance in prices is also highest in Sydney, showing the highest dispersion in prices.

The lowest average price is for Newcastle, which also has lowest dispersion value.

To compare average against dispersion we use the CV- coefficient of variation value. It is given as the ratio of standard deviation to mean value. It is a relative measure of the dispersion. As shown the CV is highest for Newcastle, whereas it is lowest for Wollongong. This data is not in line with variance / standard  deviation. The latter is a an absolute measure of dispersion, whereas CV is an absolute measure devoid of units. CV is therefore better measure to compare dispersion of different series.        

	SYDNEY	WOLLONGONG	NEWCASTLE
Mean	717.2859	532.6064044	379.252
Standard Error	38.79888	24.32388522	23.33847
Median	668.4485	515.1707706	364.8505
Standard Deviation	173.5139	108.7797217	104.3728
Sample Variance	30107.06	11833.02784	10893.69
Kurtosis	0.500424	-0.4987024	-0.52924
Skewness	0.930083	0.208633606	0.507136
CV	0.241903	0.204240356	0.275207

A visual comparison is shown below. The mean, standard error, median  and standard deviation are all highest for Sydney followed by Wollongong and then lowest for Newcastle.  

 

While the above look at absolute values of prices across regions, we now check if these differences are statistically different. We use an ANOVA test to test for differences in average prices across locations.

Variation of Prices by Location

Ho:               µ1 = µ2 = µ3

H1:                 µ1 ≠ µ2 ≠ µ3

We produce the ANOVA results below.

Summary
Groups	Count	Sum	Average	Variance
Sydney	20	14345.71726	717.2859	30107.06
Wollongong	20	10652.12809	532.6064	11833.03
Newcastle	20	7585.040681	379.252	10893.69
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	1145940	2	572969.8	32.53429	3.75E-10	3.158843
Within Groups	1003842	57	17611.26
Total	2149781	59

 As we can see the F test value is 32.53, while its p value is zero. This shows that at all confidence levels, we do not accept the null hypothesis. There is statistical evidence that prices differ across locations. The alternate hypothesis is supported.

We now move to investigate if the prices are different across dwelling type.

	House	Unit
Mean	626.12	459.98
Standard Error	38.41	22.79
Median	585.53	466.33
Standard Deviation	210.41	124.83
Sample Variance	44270.41	15582.18
Kurtosis	0.11	-0.78
CV	0.33	0.27

As shown the prices are higher for HOUSES. The average price for a house ($626)  is higher than for a unit type ($460). Both sets have different skewness. While houses have positively skewed prices, unit type dwellings have negative skewness.    This is also seen in the median price for houses being lower than average price, while the median for unit type is higher than average price of unit type dwellings.

Despite the large difference in average prices we can test for this difference in a statistical way. Using a t test with unequal variances, we find that the t test value is 3.719We use a 1 tail test here as we investigate if house prices exceed unit prices.

Ho: µH = µU

H1: µH >  µU

t-Test: Two-Sample Assuming Unequal Variances
	HOUSE	UNIT
Mean	626.1199759	459.9762249
Variance	44270.40993	15582.18227
Observations	30	30
Hypothesized Mean Difference	0
df	47
t Stat	3.719659246
P(T<=t) one-tail	0.000265725
t Critical one-tail	1.677926722
P(T<=t) two-tail	0.00053145
t Critical two-tail	2.01174048

Using a p value approach we can see that p value = 0.0002. As this p value is less than 0.01 we can conclude that at 99% level we do not accept the null hypothesis. There is statistical evidence that houses are higher priced than unit dwellings. Even if we use a 90% or 95% level we still reach the same conclusion.

We now investigate if prices are systematically higher for dwellings with an ocean view. We look at this difference separately for units and houses.

We sort data twice- first in terms of type of dwellings, and then each category in terms of ocean view.

Let us consider UNIT type first. We have 15 data points for each segment- unit dwellings with ocean view and those unit dwellings without ocean view. The data below shows that average price of a unit with a ocean view is $624 while it is higher for those without the view by $3 only. Both these have similar standard deviation, but the data is spread differently. Both are positively skewed, but the degree is much higher for those units with a view. ( 0.791 > 0.094).   

	view	no view
Mean	624.917	627.323
Standard Error	54.294	56.263
Median	587.332	567.314
Standard Deviation	210.280	217.904
Sample Variance	44217.575	47482.314
Kurtosis	2.051	-1.032
Skewness	0.791	0.094

Using a 1 tail t- test we check if the prices of units with ocean view are higher than for units without the view.

Ho: µV = µNov

H1: µV > µNov

The t test value is -0.03, which is less than the critical value of 0.97. so we have NO evidence that unit prices with ocean view are higher than unit prices without the ocean view. 

t-Test: Two-Sample Assuming Unequal Variances
	With view	Without view
Mean	624.9166524	627.3232994
Variance	44217.57504	47482.31412
Observations	15	15
Hypothesized Mean Difference	0
df	28
t Stat	-0.03078035
P(T<=t) one-tail	0.487831533
t Critical one-tail	1.701130908
P(T<=t) two-tail	0.975663066
t Critical two-tail	2.048407115

Next we look at Houses type of dwellings. We again have 15 observations in each category.

The data below shows that average price of a house with a ocean view is $455 while it is higher for those without the view by $10 approximately. Both these have similar standard deviation, but the data is spread differently. Both are negatively skewed, but the degree is higher for those houses without a view in an absolute sense. ( 0.022 < 0.082).

	With view	Without view
Mean	455.386	464.567
Standard Error	33.705	31.824
Median	465.708	466.951
Standard Deviation	130.539	123.255
Sample Variance	17040.522	15191.702
Kurtosis	-0.866	-0.459
Skewness	-0.022	-0.082

Comparison of Prices Across Dwelling Types

Using a 1 tail t- test we check if the prices of houses with ocean view are higher than for houses without the view.

Ho: µV = µNov

H1: µV > µNov

The t test value is – 0.198, which is less than the critical value of 0.84. so we have NO evidence that house prices with ocean view are higher than house prices without the ocean view.

t-Test: Two-Sample Assuming Unequal Variances
	yes	no
Mean	455.3859	464.5665793
Variance	17040.52	15191.70228
Observations	15	15
Hypothesized Mean Difference	0
df	28
t Stat	-0.19805
P(T<=t) one-tail	0.422218
t Critical one-tail	1.701131
P(T<=t) two-tail	0.844436
t Critical two-tail	2.048407

We now look at Wollongong exclusively and unit dwellings in it. We have 10 such observations with 4 having an ocean view and 6 without the view. The average price is higher for those with an ocean view ($474) while the price averages $436 without the view. Both dataets are negatively skewed though the data without the view is more skewed in absolute sense.

view		no view
Mean	474.0248	436.156
Standard Error	34.83078	24.19601
Median	477.4105	451.7202
Mode	#N/A	#N/A
Standard Deviation	69.66157	59.26789
Sample Variance	4852.734	3512.683
Kurtosis	0.994989	2.800395
Skewness	-0.27972	-1.57714

We now look into systematic differences in prices , beyond a simple numerical comparison. Using a 1 tail t test we have ( V= view and NoV = no view)

Ho: µV = µNov

H1: µV > µNov

t-Test: Two-Sample Assuming Unequal Variances
	view	no view
Mean	474.0248493	436.156024
Variance	4852.733889	3512.68253
Observations	4	6
Hypothesized Mean Difference	0
df	6
t Stat	0.89291651
P(T<=t) one-tail	0.203143754
t Critical one-tail	1.943180274
P(T<=t) two-tail	0.406287508
t Critical two-tail	2.446911846

The t test value is 0.89, and the critical value is 1.94. as test value < critical value we ACCEPT the null hypothesis. There is no evidence that Wollongong units with an ocean view are higher priced than those without the view. A numerical comparison using mean shows  a difference but it is not supported statistically. 

Conclusion

The data given for 3 locations is fairly evenly distributed for prices of dwellings. Also we have equal number of data points for each qualitative attribute.  We use the data given to use in different ways to check for significant differences in prices that can be attributed to type of dwelling, ocean view and location. We find that Sydney is most expensive and there are systematic differences across locations. There is no evidence that dwellings – units or houses with an ocean view are more expensive than those without the view. This is confirmed if we look at Wollongong units only. Here units with or without view have no difference in average prices units with a view are not higher priced than those without the view. On an average house are higher priced than units on average basis.

All these results are based on data given. Their applicability must be seen in terms of the sampling procedure used and the population from which the sample data is derived.

Reference

Anon., n.d. choosing the number of bins. [Online] Available athttps://statweb.stanford.edu/~susan/courses/s60/split/node43.html  [Accessed 9 Oct 2017].

Anon., n.d. How to choose no of bins. [Online] [Accessed 11 Oct 2017].

Anon., n.d. Hypothesis Testing. [Online] Available at:   

https://onlinecourses.science.psu.edu/statprogram/node/138  [Accessed 14 Oct 2017].

Anon., n.d. Hypothess testing. [Online] Available athttps://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/  [Accessed 21 Oct 2017].

Anon., n.d. Mean, median, mode. [Online] Available at:     

https://www.bbc.co.uk/schools/gcsebitesize/maths/statistics/measuresofaveragerev6.shtml  [Accessed 12 Oct 2017].

Cfcc.edu, n.d. Tests of hypothesis. [Online] Available at:     

https://cfcc.edu/faculty/cmoore/0801-HypothesisTests.pdf  [Accessed 15 Oct 2017].

Cyclismo.org, n.d. calculating confidence intervals. [Online] Available at

https://www.cyclismo.org/tutorial/R/confidence.html  [Accessed 15 Oct 2017].

Insee.fr, 2016. Coefficeient of Varaiation/CV. [Online] Available athttps://www.insee.fr/en/metadonnees/definition/c1366  [Accessed 11 Oct 2017].

Kean.edu, n.d. Confidence Inteval for Mean. [Online] Available athttps://www.kean.edu/~fosborne/bstat/06amean.html  [Accessed 16 Oct 2017].

Learn,bu.edu, n.d. The 5 steps in Hypothesis testing. [Online] Available athttps://learn.bu.edu/bbcswebdav/pid-826908-dt-content-rid-2073693_1/courses/13sprgmetcj702_ol/week04/metcj702_W04S01T05_fivesteps.html  [Accessed 14 Oct 2017].

LEarn.bu.edu, n.d. The fice steps for hypothesis testing. [Online] Available at:   https://learn.bu.edu/bbcswebdav/pid-826908-dt-content-rid-2073693_1/courses/13sprgmetcj702_ol/week04/metcj702_W04S01T05_fivesteps.html  [Accessed 13 Oct 2017].

Online courses.science.psu.edu, n.d. Interval estimate of population mean. [Online] Available at:  

https://onlinecourses.science.psu.edu/stat505/node/61  [Accessed 17 Oct 2017].

Rgs.org, n.d. Sampling techniques. [Online] Available at:   

https://www.rgs.org/OurWork/Schools/Fieldwork+and+local+learning/Fieldwork+techniques/Sampling+techniques.htm  [Accessed 18 Oct 2017].

Simon.cs.vt.edu, n.d. Measuresof dispersion. [Online] Available athttps://simon.cs.vt.edu/SoSci/converted/Dispersion_I/  [Accessed 17 Oct 2017].

stat.yale.edu, n.d. Sampliing in Statistical Inference. [Online] Available at:   https://www.stat.yale.edu/Courses/1997-98/101/sampinf.htm  [Accessed 17 Oct 2017].

Statistics. laerd.com, n.d. Measures of Spread. [Online] Available at:   https://statistics.laerd.com/statistical-guides/measures-of-spread-range-quartiles.php  [Accessed 17 Oct 2017].