During public health crises, governments often aim to distribute resources and aid to citizens equitably. However, in practice, this may not always be the case.We will explore some mock data about vaccine distribution in New York during a pandemic.Imagine a vaccine for a pandemic is produced, but there is a limited supply. Although the government promises that the vaccine will be distributed equitably, New Yorkers wonder whether the government is preferencing distributing to wealthy individuals.
Question 1
1) The State of New York claims that each county receives a number of vaccines that is proportional to its population size. An independent inspector is granted access to obtain a random sample of vaccine box shipping labels in a warehouse shipping to only three New York counties: the Bronx, Queens, and Westchester. The median income is highest in Westchester County, followed by Queens and finally the Bronx. Below are the populations of all three counties (as of 2019). Find the percentages of vaccine doses in the warehouse that should go to each county:
| County | Population Size |
| Queens | 2,287,000 |
| Bronx | 1,435,000 |
| Westchester | 968,000 |
The independent inspector collects a random sample of 500 shipping labels for vaccine boxes being distributed by the warehouse (all boxes have an equal number of vaccine doses). She would like to test if there’s convincing evidence against the claim that vaccine shipments are proportionalto each county’s population sizeusinga chi-squaretest for goodness of fit.
Question 2
2) Write the null and alternative hypotheses for this test.Now, we have to check conditions for this test to make sure it’s valid. Here are the conditions for the chi-square test for goodness of fit:
•Random:Observed counts must come from a random sample (to ensure our conclusions are free from sampling bias).
•10%:The sample size must be less than a tenth of the population size (to satisfy independence assumptions).
•Large Sample:The sample is large enough such that theexpected counts are allfive or greater (to ensure our sampling distribution resembles a chi-squaredistribution).
Question 3
3) For the sample collected by the independent inspector, check each of the conditions described previously.Show your work in checking the large sample condition.
Question 4
4) Below are the observed counts of vaccine shipments designated to each county.
| County | Queens | Bronx | Westchester |
| Observed Count | 204 | 132 | 164 |
Goto theDCMPChi-Square Testtool athttps://dcmathpathways.shinyapps.io/ChiSquaredTest/. Select the Goodness of Fittabat the top of the data analysis tool. Under “Enter Data,” choose “Contingency Table.” Change each of the counts to the observed counts from the previous table. Change each of the “props” to the expected proportions. Then press“Submit.”Report thedegrees of freedom, chi-squaretest statistic value, and the P-value.
Question 5
5) Was your chi-squarevalue large or small? Explain.What does its value indicate about the claim that vaccine shipments are made proportionally to each county’s population size?
Question 6
6) Interpret the meaning of the P-value in the context of this problem.CountyQueensBronxWestchesterObserved Count204132164
Question 7
7) Draw a conclusion aboutthe hypotheses based on your test results.Use a significance level of 𝛼=0.05.In addition, comment on whether your test results are consistent with the idea that vaccine shipments flow disproportionately to wealthier areas.
