MATH1324 Assignment 3

Supermarket Price Wars

Group/Individual Details

• Justin Munoz (s3581499)

Executive Statement

The aim of this assignment is to determine if there is a statistically significant difference between prices of matched products from major supermarkets, Coles and Woolworths. This investigation will require a collection of adequate data consisting of randomly sample matched products from each store.

There are many variables to consider for this investigation, first and foremost ensuring that all products used in the sample are matched between stores. To control this, products sold solely at these supermarkets (Coles Brand, Homebrand, etc) will not be considered in the study. Furthermore, the prices of these products should not be skewed by weekly specials or promotions. Collecting data from one instance may influence bias and lead to a poor representative of an average selling price. Hence, this investigation will focus on a particular line of products claimed to be fixed “everyday” prices. Coles has Everyday Low Prices and Woolworths has Low Price Always. This approach will aim to control the bias of temporary prices among both supermarkets. A sample size of 30 was used where 15 products were randomly selected from the Everyday Low Prices line (Coles) along with 15 products from the Low Price Always line (Woolworths). A convenience sampling method was used to randomise selection of products.

The data was imported into R where a hypothesis test was performed accordingly. Woolworths appeared to be cheaper with a mean difference of 0.13, however the results of the hypothesis test failed to find a statistically significant difference. The outcome is quite underwhelming as the results of this investigation is expected when conducting research in such a competitive market where both supermarkets are rival competitors.

Load Packages and Data

library(dplyr)
library(car)
library(granova)

PriceWars <- read.csv("/Users/Justin/Desktop/PriceWars.csv")
PriceWars$d <- PriceWars$CPrice - PriceWars$WPrice

Summary Statistics

PriceWars %>% summarise(
  `Mean Coles Price` = mean(CPrice, na.rm = TRUE),
  `SD Coles Price` = sd(CPrice, na.rm = TRUE),
  `Mean Woolworths Price` = mean(WPrice, na.rm = TRUE),
  `SD Woolworths Price` = sd(WPrice, na.rm = TRUE),
  `Mean Difference` = `Mean Coles Price` - `Mean Woolworths Price`,
  `SD Difference` = sd(CPrice - WPrice, na.rm = TRUE),
  n = n()
)

granova.ds(
  data.frame(PriceWars$CPrice, PriceWars$WPrice),
  xlab = "Coles Price ($)",
  ylab = "Woolworths Price ($)"
)

            Summary Stats
n                  30.000
mean(x)             4.882
mean(y)             4.754
mean(D=x-y)         0.128
SD(D)               0.880
ES(D)               0.146
r(x,y)              0.943
r(x+y,d)            0.029
LL 95%CI           -0.200
UL 95%CI            0.457
t(D-bar)            0.799
df.t               29.000
pval.t              0.431

The data shows an amplitude of similarity in terms of price differences with a majority of the scatter plot sitting amongst the confidence interval. This trend corresponds with the homogeneous qualities of the mean and standard deviation of the prices of both supermarkets as can be seen in the initial summary data frame.

The mean difference suggests that Woolworths is slightly cheaper than Coles, however a hypothesis test will ensure if this is a significant difference. Despite the relatively low mean difference, some outliers are evident in the plot reaching differences of over 2.0 but overall don’t appear to influence the results all too much.

Hypothesis Test

Hypotheses for the paired (dependent) samples t-test:

\[H_0: \mu_\Delta = 0\] \[H_A: \mu_\Delta \neq 0\]

Assumptions:

• Comparing the average price difference, μΔ, between two matched measurements of product prices, \[d_i=x_{i2}−x_{i1}\]

• Δ are normally distributed or large sample used (n>30).

A one sample t-test of the price differences, d, was used as the hypothesis test, where the hypothesized mean difference = 0 and the significance level = 0.05.

t.test(PriceWars$d, 
       mu = 0, 
       alternative = "two.sided")

    One Sample t-test

data:  PriceWars$d
t = 0.7992, df = 29, p-value = 0.4307
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.2000827  0.4567494
sample estimates:
mean of x 
0.1283333 

Interpretation

A paired-samples t-test was used to test for a significant mean difference between prices from two different supermarkets, Coles and Woolworths. A sample size of 30 was used. The mean price difference was found to be 0.13 (SD = 0.88).

As the Null hypothesised value (0) is captured within the CI and the p-value is greater than the significance level, the results of the Hypothesis test are not statistically significant, t(df=29)=0.8t, p>.431, 95% [-0.2 0.46].

Discussion

Prices were found to have no statistically significant difference between matched products from Coles and Woolworths. This investigation consisted of a few limitations such as controlling a standardised price for products of both supermarkets and finding a way to randomise selection. The strengths include the overwhelming selection of potential data available and the feasibility of collecting this data online.

In regards to improvements, devising a way to use stratified or cluster sampling to compile the data would result in a sample that is more likely to be representative of the prices at both supermarkets. Another approach to attain a better representative would be to gather a large random sample size over a period of time. Using this method would produce an average sell price for each product in the sample. This allows the investigation to be open to more matched products regardless of temporary specials.