Question

A researcher working on the shopping habits of those living in a particular city
He made. During the meeting with 400 families, 225 families came from the countryside to the city.
He determined that he had settled and 175 families were always settled in the city. From these families
54 and 52 respectively stated that they shop in big chain markets.
a) Depending on this example, these two groups differ depending on where they shop.
Decide with a hypothesis test at 5% significance level.
b) Those who come from the countryside and settle in the city are buying less than people who always settled in the city. Can we said that in 5% significance level? Search with the hypothesis test.

Answer 1

• Here, we are to test if there is difference between the proportion of people who shop in big chain markets.
  
  It is given that, 54 out of 225 families who came from the countryside to the city shop in big chain markets.
  It is also given that 52 out of 175 families who were always settled in the city shop in big chain markets.
  
  Hence, we are to test:
  
  where, is the proportion of the families who came from the countryside to the city shop in big chain markets.
  And, is the proportion of the families who were always settled in the city shop in big chain markets.
  
  The test statistic is given by:
  
  Now,
  
  
  
  
  Hence,
  
  
  Under Null,
  
  Hence, we reject at 5% level of significance iff
  Now,
  
  
  Now, it is observed that,
  
  
  Hence, we fail to reject at 5% level of significance, and conclude that there is no difference between the proportion of people who shop in big chain markets.
  
  
• Hence, we are to test:
  
  The test statistic is given by:
  
  Now,
  
  
  
  
  Hence,
  
  Under Null,
  
  Hence, we reject at 5% level of significance iff
  Now,
  
  
  Now, it is observed that,
  
  Hence, we fail to reject at 5% level of significance, and conclude that the proportion of the families who came from the countryside to the city shop in big chain markets is not less than the proportion of the families who were always settled in the city shop in big chain markets.

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