Question

(a) You wish to estimate the mean percentage price change for all supermarkets in the Foodmart chain. From the sample of 150 stores, we have calculated the sample mean,

x = 4.092 and sample standard deviation, s = 0.795.
(i) Calculate the 99% confidence interval estimate of the mean percentage change

for all supermarkets in the Foodmart chain.

(ii) Provide a plain language interpretation of the interval you have just constructed.

(iii) Nationally, the Food component of the Australian CPI increased by 3.9% for the year to June 2008. From your earlier results, does it appear that food price changes at Foodmart are any different to those as a nation overall? Explain your reasoning.

Answer 1

i) Since the population standard deviation is unknown, hence we would use the t-statistics to calculate the estimate of the mean percentage change.

The sample mean = 4.092

Sample standard deviation = 0.795

Degree of freedom = 150- 1 = 149

Significance level = 0.01

t-critical = 2.609

99% confidence interval estimate of the mean percentage change

ii) Interpretation: If we take many samples of size 150 and calculate the percentage price change for each of the samples then in 99% of the cases, the mean percentage change in price will lie in the interval [3.923, 4.261].

iii) A national percentage price change of 3.9% lies outside of the 99% confidence interval that we just calculated. Hence, the food price changes at Foodmart are different from those as a nation overall.

Alternatively,

The null hypothesis is that the mean percentage price change is the same for food mart to the national value.

There is a probability of 0.99 that a price change percentage will lie in the calculated given interval. So, there is a probability of less than 0.01 that a price change percentage will lie outside that interval. Since 3.9% lies outside that interval hence, the p-value for a price change percentage of 3.9 will be less than 0.01. If the p-value if less than 0.01 then we cannot accept the null hypothesis. Hence, the food price changes at Foodmart are different from those as a nation overall.

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