Question

9.42. T test and the cost of levies’ jeans and H&M dresses in Halifax: Numbeo is a crowd-sourced web site that gathers data on cities and countries around the world. Visitors to the site can add information on variables such as cost of living, traffic and crime. The data is searchable by city and country. For example, when we looked up Halifax, Canada, we discovered that a pair of jeans go for an average of $62.50. The range was $50 to $ 75. Add a summer dress from somewhere like Zara or H&M goes for an average of $48, with a range of $40 to $60. Numbeo also tells us that their Halifax data is based on contributions from 80 different people. a.) Let’s say the Levi Stratus & Co. agreed to tell you the mean price (in Canadian dollars) of Levi’s 501 jeans around the world. If you wanted to test whether Levi’s 501 jeans in Halifax coast a different amount from the world price, what hypothesis test would you use? Explaine your answer. b.) What additional information would you need to conduct the hypothesis test that you identified in part a? c.) Thinking back to chapter 5, what kind of sample is this? What concerns might you have about this kind of sample? d.) Numbeo doesn’t tell us how many of the 80 contributors answered each of these questions. Why might the sample be even smaller than 80 for the average cost of summer dresses?

Answer 1

a)

Supposing that Levi Stratus & Co. agreed to tell me the mean price in Canadian dollars of Levi's 501 jeans.

Let us assume the price of the Levi's jeans approximately follows a normal distribution with mean and variance

That is we are assuming that the population which we are considering is the Levi's 501 jeans all around the world and the price of jeans comes from this normally distributed population.

Here we are interested in testing if mean price of Levi's 501 jeans costs different amount from the rest of the world

That is we are actually interested in performing a test of mean based on a normal population.

Hence, we can set up the following hypothesis:

  

versus

Where is the price of Levi's 501 jeans as stated by Levi Stratus and Co. , that is is a known quantity.

b) The additional information that we need are two.

First of all the price in rest of the world is not in Canadian Dollars.Hence whatever price we get from other part of the world has to be converted into the Canadian Dollar at first and then based on the random sample we can talk about testing the hypothesis of interest.

The next thing we want to know that the variance of the Normal population which we are considering.

If Levi Stratus and Co. gives us the variance of the price of Levi's 501 jeans in different parts of the world then we don't have to estimate it and we can directly use it to test the hypothesis and we can construct the hypothesis based on a statistic which follows normal distribution.

But, if we don't know the population variance then we have to estimate the variance using the sample variance statistic.

Let,

is the random sample. where each of

Hence

and

where,

But is unknown here and we have to estimate it using

and we know

and hence as S^{2 and are independent hence,}

^{Under H}₀ ,

Hence the test statistic, T under null hypothesis is

Here n is 80, the sample size.

c) The sample is a random sample assumed to be collected from a normal population. The sample should be a good representative of the population, otherwise it may give some absurd values resulting in false conclusion of the testing of hypothesis of interest.