Analyzing Ushabti Auction Prices Using Statistical Methods

This assignment provides a statistical analysis of Ushabti auction prices, including exploratory data analysis, distribution plots, box plots, scatter plots, and t-tests. The exploratory analysis identifies the most expensive Ushabtis and potential outliers. Distribution plots and histograms are used to assess the distribution of hammer prices and the suitability of logarithmic transformation for OLS regression. Box plots compare hammer prices for different materials, while scatter plots examine the relationship between hammer price and lowdate. T-tests are conducted to compare mean prices between auction houses and locations. Bivariate and multivariate OLS regression analyses explore the correlation between various characteristics and the hammer price, revealing the impact of material, lowdate, size, year, and publication status on Ushabti values. The analysis concludes that Sotheby's generally sells more expensive Ushabtis than Christies, and London has higher priced Ushabtis compared to New York. Stone, faience and metal materials command a price premium compared to other materials.

Statistics
Name:
Institution:
20th March 2019

Q1: Exploratory analysis
The first analysis performed was aimed at identifying the most expensive ushabtis in the
dataset. The three most expensive ushabtis are given in the table below;
Table 1: Most expensive ushabtis
Lot
number
material Auction
house
Year Mont
h
Location
27 limestone Sothebys 2010 12 New York
49 wood Sothebys 2008 6 New York
40 serpentine Christies 2015 4 London
As can be seen, the most expensive ushabtis had limestone material and was auctioned in
Sothebys house in December 2010 in New York City. The second most expensive ushabtis
was also auctioned in Sothebys house in New York City in June 2008. Lastly, the third most
expensive ushabtis was made from serpentine and auctioned in Christies house in April 2015
in London.
The three most expensive values presented above seems to be outliers and most likely they
were incorrectly recorded. The suitable remedy to deal with the entries would be to remove
them in case the actual values cannot be obtained.
Q2: Distribution plots (histogram)
Next, we present the distribution of “hammerpriceinpounds” and its (natural) logarithmic
transformation in histograms. The plots are given in figure 1 and 2 below;
0 .1 .2 .3 .4
Density
4 6 8 10 12 14
loghammerpriceinpound
Histogram of log hammer price in pounds
Figure 2: Histogram of log hammer price
0 5.0e-06 1.0e-05 1.5e-05 2.0e-05 2.5e-05
Density
0 200000 400000 600000 800000
hammerpriceinpounds
Histogram of hammer price in pounds
Figure 1: histogram of hammer price

From the above graphs, I would use the logarithmic transformation for OLS regression
analysis since the raw data is excessively skewed (right skewed). However, looking at the
histogram for the log hammer we can see that it is close to normal distribution (David ,
2009). One of the assumptions for conducting OLS regression is that the data needs to follow
normal distribution which the logarithmic transformation is close to.
Q3: Box plots
Box plots are presented below to try and compare lnhammerprice for wooden, stone, and
faience ushabtis.
6 8 1 0 1 2 1 4
lnh am m e rprice
Faience Stone Wood
Box plot of Inhammer price
Figure 3: Box plot of Inhammer price by material
As can be seen from the plot, the stone material seems to have on average larger Inhammer
price as compared to the wooden and faince (Hubert & Vandervieren , 2008).
Q4: Scatter plot
The next plot is a scatter plot of “lnhammerprice” and “lowdate. This plots tries to check
whether there is relationship between the two variables (Emerson, Green, Schoerke, &
Crowley, 2013).

4 6 8 1 0 1 2 1 4
lnh am m e rprice
-2000 -1500 -1000 -500 0
lowdate
A scatter plot of Inhammer price vs lowdate
Figure 4: A scatter plot of Inhammer price vs lowdate
The results of the scatter plot between Inhammerprice and lowdate given above shows that
there is no clear relationship between the two variables.
Q5: T-test
In this section, we sought to find out whether the claim that Christies sells more expensive
ushabtis than Sotheby’s on average was statistically valid. An independent samples t-test was
performed and results presented below;
Table 2: T-test of mean price for Christies and Sotheby
Pr(T < t) = 0.0015 Pr(|T| > |t|) = 0.0031 Pr(T > t) = 0.9985
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
Ho: diff = 0 degrees of freedom = 310
diff = mean(Christie) - mean(Sothebys) t = -2.9820
diff -20661.37 6928.714 -34294.62 -7028.11
combined 312 15800.46 3063.577 54113.56 9772.499 21828.42
Sothebys 80 31164.04 11165.28 99865.27 8940.111 53387.96
Christie 232 10502.67 1348.31 20536.85 7846.115 13159.23
Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
Two-sample t test with equal variances
. ttest pricehighestimatedollars, by(auctionhouses)
An independent samples t-test was done to compare the mean price of Ushabtis for the
Christies and Sotheby (Fay & Proschan, 2010). Results showed that the Sotheby (M =
31164.04, SD = 99865.27, N = 80) was significantly more expensive than Christies (M =

10502.67, SD = 20536.85, N = 232), t (310) = -2.98, p < .05, two-tailed. The difference of
20661.37 showed a very significant difference. Essentially results showed that the claim that
Christies sells more expensive ushabtis than Sotheby’s on average was not statistically valid
instead Sotheby was found to sell more expensive ushabtis than Christies.
Next we sought to find out whether the claim that London had higher priced ushabtis are
traded was statistically valid. An independent samples t-test was performed and results
presented below;
Table 3: T-test of mean price for London and New York
Pr(T < t) = 0.0002 Pr(|T| > |t|) = 0.0003 Pr(T > t) = 0.9998
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
Ho: diff = 0 degrees of freedom = 424
diff = mean(London) - mean(NewYork) t = -3.6136
diff -18939.22 5241.159 -29241.11 -8637.329
combined 426 12288.81 2359.51 48699.74 7651.05 16926.57
NewYork 115 26115.33 7870.509 84401.81 10523.91 41706.75
London 311 7176.113 1311.171 23122.76 4596.192 9756.033
Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
Two-sample t test with equal variances
. ttest hammerpriceinpounds, by( location)
An independent samples t-test was performed to compare the mean price of Ushabtis for
London and New York cities (Sawilowsky, 2005). Results showed that the London (M =
7176.11, SD = 23122.76, N = 311) had significantly more expensive than Ushabtis than New
York (M = 26115.33, SD = 84401.81, N = 115), t (424) = -3.61, p < .05, two-tailed. The
difference of 18939.22 showed a very significant difference. Essentially results showed that
the claim that London had higher priced ushabtis are traded was statistically valid.
Q6: Bivariate OLS regression
A bivariate OLS regression analysis was performed to explore the correlation between the
characteristics such as lowdate, sizeincm, Year and whether an object has been “published”
(i.e. is considered of scholarly importance) with the lnhammerprice paid. Results are
presented in table 4 below;

Table 4: Bivariate OLS regression results
Model 1 Model 2 Model 3 Model 4
Lowdate -.0009*
Sizeincm .1186*
Year .0548*
Published .6893*
Constant 7.285* 6.193* -102.065* 7.974*
R-Squared 0.0575 0.1673 0.0335 0.0424
* Significant at 5% level of significance
As can be seen in table 4 above, there was significant correlation between Inhammerprice and
all the four characteristics presented (p < 0.05). Three of the four characteristics (sizeincm,
year and published) had positive relationship with the Inhammerprice while lowdate had a
negative correlation (negative relationship) with the Inhammerprice (Tofallis, 2009).
The coefficient of lowdate is -.0009; this means that a unit increase in lowdate would result to
a decrease in the Inhammerprice by 0.0009. Similarly, a unit decrease in lowdate would result
to an increase in the Inhammerprice by 0.0009.
The coefficient of sizeincm is .1186; this means that a unit increase in sizeincm would result
to an increase in the Inhammerprice by 0.1186. Similarly, a unit decrease in sizeincm would
result to a decrease in the Inhammerprice by 0.1186.
The coefficient of year is .0548; this means that a unit increase in year would result to an
increase in the Inhammerprice by 0.0548. Similarly, a unit decrease in year would result to a
decrease in the Inhammerprice by 0.0548.
The coefficient of published is .6893; this means that published objects are more likely to
fetch more Inhammer price by 0.6893 as compared to non-published objects.
Looking at the values of R-Squared, we can see that sizeincm explains the highest proportion
of variation (R2 = 0.1673) in the Inhammer price as compared to all the four characteristics.
Year explained the least proportion of variation in the Inhammer price (R2 = 0.0335).

Q7: Multivariate OLS regression analysis
Lastly, a multivariate OLS regression analysis was performed to explore the effect of material
on the hammer price. In the model, we used “lowdate” and “Size”, and “Year” as control
variables and tested whether wooden, metal, stone, and faience objects command a price
premium (compared to composition / terracotta / other). Results are given in table 5 below.
Table 5: Multiple OLS regression analysis
Coefficient Standard
Error
t-value p-value [95% Conf. Interval]
Wood .0806 .2302 .35 0.726 -.3719 .5331
Stone .9042* .1831 4.94 0.000 .5444 1.2641
Faience .8761* .1387 6.32 0.000 .6036 1.1487
Metal 1.3111* .3215 4.08 0.000 .6791 1.9431
Lowdate -.0006* .0002 -2.95 0.003 -.0009 -.0002
Sizeincm .1104* .0132 8.35 0.000 .0844 .1364
Year .0596* .0120 4.97 0.000 .0360 .0831
Constant -114.2697* 24.0575 -4.75 0.000 -161.5574 -66.9819
R-Square (R2) 0.3511
* Significant at 5% level of significance
As can be seen in table 5 above, there was significant relationship between Inhammerprice
and metal, stone, and faience objects while controlling for lowdate, Size, and Year (p < 0.05).
However, wood material did not have significant command on the hammer price (p > 0.05).
The coefficient of stone was found to be .9042; this means objects made from stone materials
are likely to fetch more Inhammer price by 0.9042 as compared to composition / terracotta /
other.
The coefficient of faience was found to be .8761; this means objects made from faience
materials are likely to fetch more Inhammer price by 0.8761 as compared to composition /
terracotta / other.
The coefficient of metal was found to be 1.3111; this means objects made from metal
materials are likely to fetch more Inhammer price by 1.3111 as compared to composition /
terracotta / other.

Looking at the value of R-Squared, we can see that the value is 0.3511; this implies that
35.11% of the variation in the hammer price is explained by the four variables (while
controlling for lowdate, Size, and Year) in the model.

References
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