MATH1324 Assignment 2

Supermarket Price Wars

Group/Individual Details

• Shishen Chen (s3442140)
• Shipren Jayadev (s3744421)

Executive Statement

Is there a statistical significant difference in prices between Coles and Woolworths?

That is the aim of this investigation. To begin we have chosen to randomly sample products from both supermarkets primarily focusing on the meats and vegetables section. The reason we chose this product range is that they are easily comparable in terms of price and will have the least influence by branding other than the branding from Coles and Woolworths.

We have elected to choose a sample size of 40 products from each supermarket. The method we obtained the prices was through both supermarkets online store obtaining their prices per unit kilogram.

After conducting the hyphothesis test below, we can conclude that there is no statistical difference in the prices between Woolworths and Coles given the variables and samples taken for this investigation.

Load Packages and Data

library(tidyr)
library(dplyr)
library(magrittr)
library(car)

Summary Statistics

Use R to summarise the data from the investigation. Include an appropriate plot to help visualise the data. Describe the trend.

#Summary Stats
price <- read.csv("Price.csv")
price_g <- price %>% gather('Coles','Woolworths',key = "supermarket",value = "Price")
price_g <- price_g %>% mutate(supermarket = as.factor(supermarket))
price_g %>% group_by(supermarket) %>% summarise(Min = min(Price,na.rm = TRUE),
                                                Q1 = quantile(Price,probs = 0.25,
                                                              na.rm = TRUE),
                                                Median = median(Price,na.rm = TRUE),
                                                Q3 = quantile(Price,probs = 0.75,
                                                              na.rm = TRUE),
                                                Max = max(Price,na.rm = TRUE),
                                                Mean = mean(Price, na.rm = TRUE),
                                                SD = sd(Price,na.rm = TRUE),
                                                n = n(),
                                                Missing = sum(is.na(Price)))

price_g %>% boxplot(Price~supermarket, data = ., ylab = "Price($)")

Hypothesis Test

Use R to perform an appropriate hypothesis test to determine which supermarket is the cheapest. You need to explain your choice of hypothesis test, any assumptions and the significance level.

1.Comparing two independent sample means with unknown population variance.

2.Large sample used (n>30 for both groups) so normality can be assumed.

3.Population homogeneity of variance.

Mc = Mean of Coles Price

Mw = Mean of Woolworths Price

Variance1^2 = Variance of Coles Price

Variance2^2 = Variance of Woolworths Price

H0: Mc = Mw (No difference in prices between Supermarkets)

HA: Mc != Mw (Prices between Supermarkets are different)

HA: Mc > Mw (Coles is more expensive than Woolworths)

HA: Mc < Mw (Woolworths is more expensive than Coles)

# H0 : Variance1^2 = Variance2^2
# HA : Variance1^2 != Variance2^2
leveneTest(Price~supermarket, data = price_g)

Levene's Test for Homogeneity of Variance (center = median)
      Df F value Pr(>F)
group  1  0.0129   0.91
      78               

#Two Sample t-test
t.test(Price~supermarket, data = price_g, var.equal = TRUE, alternative = "two.sided")

    Two Sample t-test

data:  Price by supermarket
t = 0.13941, df = 78, p-value = 0.8895
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -3.130926  3.602426
sample estimates:
     mean in group Coles mean in group Woolworths 
                 8.81150                  8.57575 

Interpretation

After doing the Levene Test, since p > 0.05, we can assume the variances are homogeneous. Samples can also be assumed to be approximately normal since they are both more than 30 in size.

The estimated diffrence in means between the two supermarkets is Mc - Mw = 8.81150 - 8.57575 = 0.23575 The 95% CI is [-3.130926, 3.602426]

p-value = 0.8895.

Since the p-value is more than 0.05, we will not be rejecting the null hyphothesis.

There is no statistical difference in the prices between Coles and Woolworths.

Discussion

Our findings confirmed our preliminary assumption wihch was there is no significant difference in the two supermarkets. The strength of our investigation comes from the product range we sampled from as meats and vegetables tend to be less influenced by brand names and tend to have a similar supplier cost so it limits price influences other than the ones imposed on the products by Coles and Woolworths.

One weakness would be that the sample size is small and a better approach in the future would be to select even larger sample sizes or to even consider stratified sampling.