Quantitative Methods

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ASSIGNMENT QUESTIONS

Background A clothing company is considering opening a new store in a middle sized city in the United States (US). They are considering two particular cities: Boise (Idaho) and Des Moines (Iowa). Past research on their customers in their existing US stores has revealed that most of their customers are wealthy females between the ages of 18 and 40. They thus aim to open new stores in cities where there are more wealthy females of this age.

To assist with their choice of which city to open their new store in, the clothing company obtained information on the individual annual income levels of females aged 18 to 40 in these two cities. Such information was originally collected by the US Census Bureau using random samples of individuals across the US. The clothing company obtained the resulting data from the Census Bureau website.

Individual income levels (the amount they received during the year) for samples of females aged 18 to 40 living in the two cities are provided on the sheet 1 in the Excel file entitled “Incomes_Assign3.xlsx” that is available from the LMS under the “Assignment 3” menu item. Sheet 2 provides the incomes in log form for females both cities that is required for parts (d) to (h).

Complete the following tasks

  1. Provide histograms and numerical descriptive statistics (measures of central location and dispersion) for female incomes in each of the two cities separately that allow the firm to compare income distributions (central location and dispersion). Use the income level data provided on sheet 1 in the file “Incomes_Assign3.xlsx”. Don’t forget to label your graphs! b. Provide a written comparison of the two sets of data based on the graphs and summary statistics from part (a). Describe the shape of each city’s income distribution for females. c. Can you construct a 95% confidence interval estimate for the difference in mean incomes of females aged 18 to 40 between the two cities using this data based on the methods we have learnt in this course? Explain why or why not? In your answer you should discuss the relevant CI estimator and its conditions.
  2. Economists often work with the income data in log form. One of the reasons is that it reduces skewness in the data. The second sheet in the “Incomes_Assign3.xlsx” file provides the income data for Boise and Des Moines in log form. (The excel command “=LN(cell)” has been used compute the natural log of the income data).

Provide numerical descriptive statistics (measures of central location and dispersion) and histograms of the income data in log form for both cities. Use the income data in log form provided in sheet 2. Briefly comment on the shape of the income distributions in log form and compare it with the shape of the histograms provided in part (a).

3

  1. In the past the clothing company has sometimes used the rule to open a store if the mean log income in the city was above 9.95 (around 21,000 Dollars). Conduct a hypothesis test at the 5% level to test whether the mean log incomes in the Boise and Des Moines are over this threshold (You need to conduct a separate test for each city). For each test, follow the 6 steps of the hypothesis test from the lectures and tutorials and show your workings. Make sure to provide evidence to support any assumptions that you make when constructing your test. f. Use Excel to compute the p-value for the two tests you conducted in part (e).  Interpret your results. How strong is our statistical evidence in support of the alternative hypothesis in Boise and Des Moines? Where should the company open the store? g. Now, suppose that the company has just obtained access to additional information from the previous Census which tells you that the population variances of log incomes are 0.35 for Boise and 0.25 for Des Moines.   Construct a 95% confidence interval estimate for the difference in mean log incomes of females aged 18 to 40 between the two cities using this data. Show all your workings. You should also provide evidence to support any assumptions that you make when constructing your estimate. Fully interpret your confidence interval estimate in words. h. Again, using the new Census information on the population variances, provide the company with more statistical evidence to make a decision regarding the location of the store and conduct a hypothesis test at the 1% level to test whether log income is higher in Des Moines than in Boise. Follow the 6 steps from the lectures and tutorials. Make sure to provide evidence to support any assumptions that you make when constructing your test.

Your answers should not exceed 8 pages.

(50 marks)

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