Black and white – is there really a difference?

An observational study of vehicle wheel sizes.

Abstract

This report seeks to answer the following question: Do black vehicles have a greater mean wheel diameter than white vehicles? The underlying, safe assumption in this regard is that there is no significant difference in mean wheel diameter between black vehicles and white vehicles (null-hypothesis). This paper reports on an observational, statistical investigation that was performed based on two independent, random samples for vehicles (one sample for black vehicles and the other for white vehicles). The samples were drawn from a very specific population of vehicles and great care was taken to ensure that the samples were as close as possible to a simple random samples (SRS). The samples’ distributions facilitated in utilising the t-test.  The statistical investigation suggests a moderate relationship between the reported vehicle colours and mean wheel sizes.  So, can the choice of vehicle colour lead the consumer to favour a different wheel size? This report suggests that it might not be the case.

Introduction

Picking a vehicle colour is never an easy choice and buyers are usually trapped between their emotions and their practicality with many of them wondering, “…should I go for the bad-ass black or the more sensible and safe white?” . According to news.com.au around 15% of all vehicles on Australian roads are black and 21% white . On the surface it seems to be a close split between the adventurous buyers and the practical buyers. Black is usually associated with high performance and/or luxurious vehicles as opposed to white vehicles being associated with average performance and/or standard vehicles. Apart from vehicle performance, there is also a safety aspect to consider – white is a more visible colour. Then, there are some practical aspects to consider – black vehicles tend to show dirt easier (or so it is perceived by the general populous*) and black vehicles have excessively warmer interiors since black doesn’t reflect any sunlight. The choice between a black or white coloured vehicle is a subtle social difference that may or may not lead to the average buyer’s choice of wheels fitted to the vehicle. This then, is also the main motivation for this investigation. From this background information and the general assumption that black vehicles have a greater average wheel diameter than white vehicles, the following hypotheses are drawn:

Null hypothesis: H0 : µblack = µwhite

Alternative hypothesis: HA : µblack > µwhite

µblack = mean rim diameter for black vehicles.

µwhite = mean rim diameter for white vehicles.

*The only vehicle which hides dirt the best is most likely a dirt coloured vehicle…? [Author’s note]

This document reports on a statistical method that evaluates the above stated hypotheses and, in the process, seeks to better understand the correlation between black and white vehicles and those two groups’ wheel sizes (diameter of the rim). Two random data samples (one for black vehicles and another for white vehicles) were collected by ways of observation and several measures were put in place to minimize bias and sampling errors.

Data

The following table lists the details of the two independent samples captured during the study.

Table 1 – Data Details

The wheel rim diameter (response variable - b in Table 1) is read directly from the sampled vehicle’s tyre code. E.g. from the wheel code: 185/65R15, the R15 implies that the rim diameter is 15 inches. This then, is the number recorded for the sample. The following procedures were put in place to reduce bias. Firstly, a very specific geographical boundary was chosen for the population - Westfields mall parking lot at Knox City centre. Secondly, a narrow timeslot was set for sampling. It was impossible to ensure that the population did not change over the course of the data collection time slot (10am -11am). The only way to ensure that, would have been to lock down the parking lot for an hour. That would have caused panic, mayhem and widespread riots and probably landed the data capturers in custody. So, a small margin of population turnover, i.e. an error of approximately 3%, must be taken into consideration for vehicles entering and existing during the data collection time slot. The exact number of parking spaces at Westfields mall, as sourced from the centre’s management, is 6000. A random number generator was then utilized to generate 20 random numbers from a uniform distribution - one set for black vehicles and another set for white vehicles. The random numbers correlate to physical locations in the parking lots. If neither a black nor white vehicle was parked in the randomly chosen position, then the next random number is selected and so forth. For a parking lot with 6000 spaces it is impractical to try and capture more than 20 vehicles in 1 hour, hence the sample size of 20 was chosen. In this regard the experimental setup had to sacrifice specificity to enhance the SRS aspect of the samples.

Methods

The R-Studio analytical software package, version 1.1.453, was utilized to analyse the data. The captured results were entered into R-data-frames and initial EDA was done utilising the following R commands: c, table, qqnorm, hist, boxplot and summary. Initial EDA suggested that both the samples’ distributions were normal. Since two independent samples, which both conforms to normal distributions is used in this analysis, the t-test was chosen to determine whether there is a significant statistical difference between the two samples .

To reject the null-hypothesis and to avoid committing a type I error, a significance level of 5% is chosen (α = 0.05). By utilising the following R commands: mean, sd and var, both the samples’ means, standard deviations and variances were calculated. From these calculations it was noticed that the variances for the two samples differed and therefore the t.test function’s parameter: var.equal=false. The setup of the null-hypothesis resulted in setting the t.test function’s parameter: alternative=”greater”. From this statistical test method and by utilising the listed R commands a conclusive statistical result was determined.

Results and Discussion

From Table 1, which lists the basic statistical properties of the two samples, it can be observed that the sample for the black vehicles has a higher mean than the sample for the white vehicles and that the two samples have different variances. The first part of the latter observation adds support towards the alternative hypothesis.

Table 2 – Black and White Sample statistics

From Figure 1, it can be observed that the 1st and 3rd quartiles of both samples are very much the same and that there are no outliers for any of the two samples.

Figure 1 – Black and White boxplots

From Figure 2, it can be observed that both samples have relatively normal distributions. From that observation it can be deduced that the larger population of motor vehicles from which the samples where collected, is also normally distributed .

Figure 2 – Black and White Quartile plots

From the results of the t-test, as listed in Table 3, it can be observed that a p-value of 0.2599 was calculated with 37.961 degrees of freedom (df = 37.961). This p-value is much larger than the initially set confidence level of α = 0.05. This is a strong indication to not reject the null-hypothesis. Therefore, there is no statistically significant difference between the two samples captured for this study.

Table 3 - T.Test Results

Conclusions

Even though the black vehicle sample had a higher mean value than the white vehicle sample, the t-test results conclusively showed that there is no real significant statistical difference between the two samples (p-value > α). Given the small sample sizes (20), it would be interesting to see what results a larger sample size would generate. Even though it seems that the general populous seems to think that black cars have larger wheel diameters than white cars, this statistical analysis suggests otherwise.

Appendix A

Samples