A LOOK AT USED COROLLAS:a limited statistical exploration of PRICE in CHICAGO and MIAMI

for model years 1999-2009

Friday, May 9, 2014

PURPOSE:

individual: each used, for-sale Corollas (1999-2009) in either CHICAGO or MIAMI

population: all the used, for-sale Corollas (1999-2009) in either CHICAGO or MIAMI

variables: explanatory variable-----The MILEAGE of a Corolla in Miami (quantitative) response variable------------ The PRICE of a Corolla in Miami (quantitative)

compare the prices of used cars in Chicago and Miamiexplore the relationship between price and mileageestimate the average price of Corollas in MIAMI with a margin of error and a degree of confidence.

OBSERVATIONAL STUDY:

Friday, May 9, 2014

DATA COLLECTION METHODS:

CHICAGO: population 276, (n=40)

autotrader.com collected 10/21/13

by Sean Bradley

MIAMI: population 245, (n=38)

autotrader.comcollected 03/22/14

by myself

We did our best to eliminate bias and collect a representative sample from each city by choosing a high-volume, national website, autotrader.com. We created an SRS (goal of n=40) for each city using random.org as our randomizer.

statcrunch: calculations, graphs, & numerical values

Friday, May 9, 2014

Of 245 listings we created an SRS of n=40, noting:

1) random listing number2) year of car3) mileage4) price

Listing # Year Mileage Price18 2008 83,344 $ 8,97719 2009 42,000 $ 8,45020 2009 36,191 $ 11,00029 2009 86,604 $ 8,98830 2009 86,604 UNLISTED37 2006 123,731 $ 5,99939 2002 124,000 $ 4,30045 2004 89,505 $ 6,99851 2006 91,981 $ 8,75963 2009 75,366 $ 9,98871 2006 157,827 $ 5,99273 2009 59,658 $ 10,17578 2003 98,282 $ 4,50082 1999 58,097 $ 5,90097 2009 26,529 $ 10,990

105 2008 80,526 $ 9,988106 2003 119,907 $ 6,988126 2009 89,684 $ 9,995133 2005 73,000 $ 6,600141 2003 70,710 $ 5,450148 2008 84,789 $ 7,995153 2006 106,142 $ 6,750158 2007 15,519 $ 12,771161 1999 104,819 $ 3,888167 2007 89,749 $ 8,391169 2000 51,534 $ 3,950178 2009 71,588 $ 9,335194 2006 UNLISTED UNLISTED202 2009 55,745 $ 11,498210 2007 58,208 $ 6,500214 2006 122,357 $ 6,900218 2003 114,001 $ 7,490221 2002 188,587 $ 3,995226 2001 32,115 $ 6,995228 1999 127,000 $ 3,900229 2006 78,901 $ 7,981236 2002 110,000 $ 3,500239 2008 66,000 $ 8,100242 2002 151,958 $ 3,990245 2006 159,052 $ 6,900

RAW DATA FOR MIAMI:

Friday, May 9, 2014

DOTPLOT OF PRICE: a subjective visual comparison of distributions

guess at: shape, span, center, mean, median, outliers~useful in the big picture~

*Two even, symmetrical distributions without obvious outliers, wider distribution in CHICAGO, more tightly packed in MIAMI,

i.e., does not appear to have skewness.

Friday, May 9, 2014

BOXPLOT OF PRICE:a subjective visual comparison of distributions helps see: span (whiskers), center (median),

IQR (in blue), outliers (none)

* The midrange 50% (IQR) in MIAMI are cheaper than in CHICAGO. *The median of CHICAGO (~$8K) > MIAMI (~$7K).

*No outliers, tighter span and IQR in MIAMI.

Friday, May 9, 2014

SUMMARY STATISTICS:an objective numerical comparison of samples

find actual: center values (mean and median), outliers, span, IQR~useful in the big picture~

*no outliers: no values < Q1-(IQR x 1.5) or > Q3+(IQR x 1.5).*median is quite close to the mean, reflecting evidence that our datasets are reasonably symmetrical (or, not biased by skewness). *comparison of centers shows us cars are between

$800-1,000 cheaper in MIAMI than in CHICAGO!

n Mean Std. dev. Median Range Min Max Q1 Q3 IQR

CHI PRICE 38 8,186 2,958 7,995 11,381 2,290 13,671 6,200 9,850 3,650

MIAMI PRICE 38 7,391 2,433 6,997 9,271 3,500 12,771 5,900 8,988 3,088

Friday, May 9, 2014

CONFIDENCE INTERVALS:

We are 95% confident the average price for all Corollas in Miami are between $6,590 and $8,192 (~P<0.0002, Statcrunch).

Variable Sample Mean Std. Err. DF T-Stat P-value

MIAMI PRICE 7391 394.66121 37 18.727988 <0.0001

The average price for a Corolla (1999-2009) in Miami is $7,391 (ME ± $801.16, 95% confidence)(statcrunch).

Friday, May 9, 2014

SCATTERPLOT:The correlation coefficient is -0.60108812; the R2 is 0.3612. Thus, the percentage of PRICE attributed to MILEAGE can be said to be 36% (~P<0.0002, statcrunch randomized test for correlation).

Friday, May 9, 2014

HYPOTHESES:H0 µM = µC: The average price of cars in Miami is equal to

the average price of cars in Chicago.HA µM ≠ µC: The average price of cars in Miami is different than the average price of cars in Chicago.

OR

µM - µC = 0 In this case, there’d be 0 difference by subtraction in the average prices. µM - µC ≠ 0 In this one, there’d be a difference.

Friday, May 9, 2014

TESTING HYPOTHESES:

1) Concerning the average prices in either city, the difference between what we expected ($0 difference) and what we observed ($795 difference) was not statistically significant (P=0.2049, statcrunch).

2) We have little to no evidence against the the null hypothesis, so we decline to reject the null hypothesis.

3) There is a strong reason to believe the cost disparity between MIAMI and CHICAGO can be attributed to sampling variability.

*A further resampling with randomization test for two means in 10,000 trials confirms this with an estimated P of 0.1904 (statcrunch).

Difference Sample Diff.

Std. Err. DF T-Stat P-value

µ1 - µ2 794.89474 621.23771 71.346986 1.2795339 0.2049

Friday, May 9, 2014

OVERALL CONCLUSION:A comparison of the prices of used cars in Chicago and Miami at first reveals less expensive cars (between $800-1000) in Miami, but hypothesis testing, t-testing, and resampling reveals this was likely due to sampling variability. There is little to no evidence (high P-value) to believe any Corolla in the whole population really would be $800-1000 less in Miami. As most purchasers of cars already know, there are known and lurking variables within our data set (year, trim level, condition, private/dealer seller, mileage). These common-sense variables are at least worth $800-1000 of PRICE in most purchasable cars.

An exploration of the relationship between price and mileage in MIAMI revealed a strong association between mileage and price to the tune of a factor of 36% (P-value <0.002, statcrunch). We have reason to believe our sample was reflective of the population of MIAMI due to our resampling and low P value.

An estimation of price in MIAMI concluded, with 95% confidence, the average price for all Corollas in Miami were between $6,590 and $8,192 (~P<0.0002, Statcrunch); OR, the average price for a Corolla (1999-2009) in Miami is $7,391 (ME ± $801.16, 95% confidence).

Friday, May 9, 2014