15E Preview

Preparing for the next classIn the next in-class activity, you will need to identify when the chi-square test for independence should not be used and be able to combine categories of a contingency table. You will also need to use the chi-square test for independence after combining categories in a contingency table and identify when to use Fisher’s Exact Test.In the previous activities, you learned about hypothesis testing for categorical data. These tests range from the 𝜒2 goodness of fit test to the test of homogeneity to the test for independence. The one thing all of these tests have in common is that the variables of interest are categorical.
Questions 1–4: An independent researcher wants to determine a relationship between the color of a motorcyclist’s helmet and whether an injury was sustained in a crash. They randomly obtain a sample of data and organize that data into the following contingency table.
Black helmet White helmet Red helmet Yellow/orange helmet
No injury 8 4 3 2
Injured or killed 20 2 1 1

Question 1

1) Assume the variables are independent and calculate the expected counts for the cells in the contingency table and write them below.Use the DCMP Chi-square Test tool at https://dcmathpathways.shinyapps.io/ChiSquaredTest/to calculate the expected values. Round the expected values to one decimal place.

Black helmet White helmet Red helmet Yellow/orange helmet
No injury
Injured or killed

Question 2

2) Would it be appropriate to perform a chi-square test of independence in this case? Explain. Hint: Recall the requirements for the test of independence.

Question 3

3) A 2004 study titled “Motorcycle rider conspicuity and crash related injury: case-control study” looked at a similar context and, based on the conclusions of that study,[1] the researcher decides to combine cells from the two-way table as follows. Note that as a result, white, red, and yellow/orange are all combined into one category. In the following table, fill in the missing observed values.

Black helmet Other helmet color
No injury 8
Injured or killed 20

Hint: Add together the observed values for white, red, and yellow/orange.

Question 4

4) Assuming the variables are independent, calculate the expected countsfor this new table, rounding to one decimal place.Use the DCMP data analysis tool. Fill in the table with the expected counts.

Black helmet Other helmet color
No injury
Injured or killed

Question 5

5) Using the table in Question 4, verify thatall ofthe conditions for using the chi-square test for independence have been met:
Independence/Randomness Condition-Is the sample an independent random sample, or is it an independent sample that can be considered representative of the population? Is this condition met? Explain.
Large Sample Size Condition-The sample size must be large enough so that the expected count in each cell is at least five. Is thiscondition met? Explain.

Question 6

6) The researcher wants to determine at the 5% significance level if the color of a motorcyclist’s helmet (black or some other colors) and whether an injury was sustained in a crash are independent. Using the table in Question 3, answer the following questions.
Part A: What is the null hypothesis?
Part B: What is the alternative hypothesis?
Part C: Since the conditions for the chi-square test for independence were already verified, use the DCMP Chi-square Testtool at https://dcmathpathways.shinyapps.io/ChiSquaredTest/to obtain the test statistic and the P-value for this test.
Part D: Will the null hypothesis be rejected? Explain.
Part E: Write your conclusion in a sentence.

Question 7

7) Suppose that the researcher wants to know if the color of a motorcyclist’s helmet (red or some other colors) and whether an injury was sustained in a crash are independent. Fill in the following table with the counts. (The original data are copiedas well, for convenience.)

Black helmet White helmet Red helmet Yellow/orange helmet
No injury 8 4 3 2
Injured or killed 20 2 1 1
Red helmet Other helmet color
No injury 3
Injured or killed 1

Hint: Add together the observed values for white, black, and yellow/orange.

Question 8

8) Assuming that the colorof a helmetand injury are independent, calculate the expected counts for this new table, rounding to one decimal place.

Red helmet Other helmet
No injury
Injured or killed

Hint: Use the DCMP data analysis tool.Note that in thistwo-way table, one of the expected values is still less thanfive, preventing us from using the test for independence. However, now that the data have been collapsed into a 2×2table, we can use Fisher’s Exact Test. Fisher’s Exact Test is used for data in a 2×2contingency table where one or more of the expected frequencies are less than fiveand certain conditions (detailed later) are met. We can also use this test when the sample size is small. This test will provide us with an exact P-value and does not require any approximations. This test will be covered during the nextin-class activity.

Looking ahead
In the next class, you will be analyzing data from the International Union for Conservation of Nature’s Red List (IUCN Red List). Since 1964, the IUCN has been collecting data on the extinction risks of animal, fungus, and plant species. On the IUCN Red List, animals, fungi, and plants are categorized by their extinction levels. (See the following table for a summary of a few of these levels.)[2] For more information on the IUCN’s purpose, data collection, and background, visit https://www.iucnredlist.org/about/background-history. A species, family, or class is…Least Concern (LC) Widespread and doesn’t qualify for another category. Near Threatened (NT) Close to qualifying for a threatened category in the near future.Vulnerable (VU) Considered to be facing a high risk of extinction in the wild and meets a set of criteria.Endangered (EN) Considered to be facing a very high risk of extinction in the wild and meets a set of criteria.Critically Endangered (CR)Considered to be facing an extremely high risk of extinction in the wild and meets a set of criteria.
  1. Wells, S., Mullin, B.,Norton, R., Langley, J., Connor, J., Lay-Yee, R., & Jackson, R. (2004, April 10). Motorcycle rider conspicuity and crash related injury: case-control study. BMJ (Clinical research ed.), 328(7444), 857. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC387473/ ↵
  2. International Union for Conservation of Nature. (n.d.). 2001 IUCN Red list categories and criteria (version 3.1) -IUCN -SSC cetacean specialist group. https://iucn-csg.org/red-list-categories/ ↵