Name: Assignment 21: Goodness of Fit Tests and Simpson's Paradox 1. Three towns in Maine have boxes of ballots from the recent election (each box contains the same number of ballots). Here's the number of boxes for each town, broken down by ballot type: Springfield Rome Houlton Total in-person 10 12 30 mail-in 10 20 Total 10 20 20 50 Using this data, we will test the hypothesis that location (which city in which a ballot was cast) is independent of the type of ballot submitted. You should be able to do this problem without a calculator. (a) What are the hypotheses? (b) Find the expected frequencies. Springfield Rome Houlton | Total in-person mail-in Total (c) Find and simplify the test statistic. (d) Use a significance level of of = 0.05 to test the hypothesis. Draw a picture showing the test statistic, critical values(s), and shading the critical region. (e) What is the conclusion of our hypothesis test?2. We now return to the problem we began on A20 about TB death rates. If you did goodness of fit tests on both tables, you would find that for the TB death table, the test statistic is x- > 950 with a p-value that is far less than .000001, while on the Total Population table the test statistic is x > 540,000 with a p-value that is also far less than .000001. So, these data also don't fit the expected frequencies at all. It's worth investigating this further. For this problem, you should show all computations in both fraction form, and also as a decimal rounded to 5 decimal places. (a) Find the TB death rate for the combined total of both cities. (b) How do the rates of TB deaths for white people in NYC compare to the overall rate? (c) How do the rates of TB deaths for white people in Richmond compare to the overall rate? (d) How do the rates of TB deaths for non-white people in NYC compare to the overall rate? (e) How do the rates of TB deaths for non-white people in Richmond compare to the overall rate? 3. So it seems that the real difference in death rates is not between NYC and Richmond, but between white people and non-white people. And we can next show what makes the data so confusing (and a great example of Simpson's Paradox): these groups are not equally represented in each city. (a) Find the proportion of non-white residents in NYC. b) Find the proportion of non-white residents in Richmond. For more information on Simpson's Paradox, I highly recommend the Wikipedia article. There, I found the following definition: Simpson's paradox, which also goes by several other names, is a phenomenon in probability and statistics, in which a trend appears in several different groups of data but disappears or reverses when these groups are combined. This result is often encountered in social-science and medical-science statistics and is particularly problematic when frequency data is unduly given causal interpretations. The paradox can be resolved when causal relations are appropriately addressed in the statistical modeling. 31