Some researchers claim there is an association between getting a tattoo and contracting hepatitis C. They took a random sample of people living in Los Angeles, California, and recorded whether they had a tattoo and whether they were infected with hepatitis C. The results are shown in the following two-way relative frequency table.
| | Has Hepatitis C | Does Not Have Hepatitis C | Row Totals |
|---|---|---|---|
| Does not have a tattoo | 0.01 | | 0.12 |
| Has one tattoo | | 0.48 | 0.50 |
| Has more than one tattoo | 0.03 | 0.35 | |
| Column Totals | 0.06 | 0.94 | 1 |
What additional data could be placed on the table?
A. Has one tattoo and has Hepatitis $C =0.02$
B. Has one tattoo and has Hepatitis $C =0.11$
C. Does not have a tattoo and does not have Hepatitis $C =0.02$
D. Does not have a tattoo and does not have Hepatitis $C=0.38
Answer (2)
Calculate 'Does not have a tattoo and Does not have Hepatitis C': 0.12 − 0.01 = 0.11 .
Calculate 'Has one tattoo and Has Hepatitis C': 0.50 − 0.48 = 0.02 .
Calculate 'Has more than one tattoo' row total: 0.03 + 0.35 = 0.38 .
The additional data are 0.11 , 0.02 , and 0.38 . 0.11 , 0.02 , 0.38
Explanation
Understand the problem and provided data We are given a two-way relative frequency table with some missing values. Our goal is to find these missing values using the information provided in the table. The table shows the relationship between having a tattoo and contracting hepatitis C.
Calculate the missing value in the first row To find the missing value 'Does not have a tattoo and Does not have Hepatitis C', we subtract the known value (0.01) from the row total (0.12).
Perform the subtraction 0.12 − 0.01 = 0.11
State the result So, the missing value 'Does not have a tattoo and Does not have Hepatitis C' is 0.11.
Calculate the missing value in the second row To find the missing value 'Has one tattoo and Has Hepatitis C', we subtract the known value (0.48) from the row total (0.50).
Perform the subtraction 0.50 − 0.48 = 0.02
State the result So, the missing value 'Has one tattoo and Has Hepatitis C' is 0.02.
Calculate the missing row total To find the missing row total for 'Has more than one tattoo', we sum the known values in that row (0.03 and 0.35).
Perform the addition 0.03 + 0.35 = 0.38
State the result So, the missing row total for 'Has more than one tattoo' is 0.38.
Final Answer In summary, the additional data that can be placed on the table are:
Does not have a tattoo and Does not have Hepatitis C = 0.11
Has one tattoo and Has Hepatitis C = 0.02
Has more than one tattoo (Row Total) = 0.38
Examples
Two-way relative frequency tables are commonly used in epidemiology to study the association between risk factors and diseases. For example, researchers might use such a table to investigate the relationship between smoking and lung cancer, or between diet and heart disease. The relative frequencies help to understand the proportion of the population affected and to assess the strength of the association between the variables.
The calculations of the missing values yield that 'Does not have a tattoo and does not have Hepatitis C' is 0.11, 'Has one tattoo and has Hepatitis C' is 0.02, and the total for 'Has more than one tattoo' is 0.38. The correct answer from the multiple-choice options is A, which states 'Has one tattoo and has Hepatitis C = 0.02'.
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