Chapter 6, Problem 6.26E

To determine

The two-way table for the given situation may vary. One of the possible answers is given below:

To give: The two-way for the given situation.

To show: The obese smokers and obese non-smokers who are more likely to die earlier than people who are not obese.

To check: Whether the Simpson’s paradox holds good for the given situation.

Answer to Problem 6.26E

The two-way table for smokers is given below:

	Obese
Death	Yes	No
Yes	8	57
No	192	1,443
Total	200	1,500

The two-way table for non-smokers is given below:

	Obese
Death	Yes	No
Yes	8	6
No	592	594
Total	600	600

The result tells that 4% of obese smokers had died, 1.3% of obese non-smokers had died. However, only 3.8% of not obese smokers had died and 1% of not obese non-smokers had died.

The Simpson’s paradox holds good for the given situation because the two individual tables tell that obese people are more prone to death but the combined table tells that non-obese people who are smoking are more prone to death.

Explanation of Solution

Given info:

Two-way table can be constructed using obese (yes or no) and death (yes or no) for smokers and non-smokers. Also, combine the two-way table of smokers and non-smokers with respect to obese and death.

Calculation:

The two-way table for smokers is constructed by taking death as a row variable and obese as a column variable.

The two-way table for smokers is given below:

Table 1

	Obese
Death	Yes	No
Yes	8	57
No	192	1,443
Total	200	1,500

The two-way table for non-smokers is given below:

Table 2

	Obese
Death	Yes	No
Yes	8	6
No	592	594
Total	600	600

Percentage of obese smokers who died is given below:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of}\\ \text{obese smokers}\\ \text{who were died}\end{array}\right)=\frac{\text{Number of people who are obese given that died early}}{\text{Total number of people who are obese}}\\ =\frac{8}{200}\times 100\\ =0.04\times 100\\ =4\end{array}$

Thus, 4% of obese smokers had died.

Percentage of obese nonsmokers who died is given below:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of}\\ \text{obese nonsmokers}\\ \text{who were died}\end{array}\right)=\frac{\text{Number of people who are obese given that died early}}{\text{Total number of people who are obese}}\\ =\frac{8}{600}\times 100\\ =0.013\times 100\\ =1.3\end{array}$

Thus, 1.3% of obese non-smokers had died.

Percentage of not obese smokers who died is given below:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of}\\ \text{nonobese smokers}\\ \text{who were died}\end{array}\right)=\frac{\text{Number of people who are not obese given that died early}}{\text{Total number of people who are not obese}}\\ =\frac{57}{1,500}\times 100\\ =0.038\times 100\\ =3.8\end{array}$

Thus, 3.8% of not obese smokers had died.

Percentage of non-smokers who are not obese died is given below:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of}\\ \text{not obese nonsmokers}\\ \text{who were died}\end{array}\right)=\frac{\text{Number of people who are not obese given that died early}}{\text{Total number of people who are not obese}}\\ =\frac{6}{600}\times 100\\ =0.01\times 100\\ =1\end{array}$

Thus, 1% of not obese non-smokers had died.

Combined table:

The combined table is obtained by combining the corresponding values from Table 1 and Table 2.

Table 3

	Obese
Death	Yes	No
Yes	16	63
No	784	2,037
Total	800	2,100

Percentage of non-obese smokers who died is given below:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of non-obese smokers}\\ \text{who have died}\end{array}\right)=\frac{\text{Number of people who are not obese but died}}{\text{Total number of people who are not obese}}\\ =\frac{63}{2,100}\times 100\\ =0.03\times 100\\ =3\end{array}$

Thus, 3% of non-obese smokers had died.

Percentage of obese smokers who died is given below:

$\begin{array}{c}\left(\begin{array}{l}\text{Percentage of obese smokers}\\ \text{who have died}\end{array}\right)=\frac{\text{Number of people who are obese also died}}{\text{Total number of people who are obese}}\\ =\frac{16}{800}\times 100\\ =0.02\times 100\\ =2\end{array}$

Thus, 2% of obese smokers had died.

The percentage of deaths for smokers is given below:

	Obese
Death	Yes	No
Yes	4%	3.8%

The percentage of deaths for non-smokers is given below:

	Obese
Death	Yes	No
Yes	1.3%	1%

The percentage of deaths under combined table is given below:

	Obese
Death	Yes	No
Yes	2%	3%

Justification:

Simpson paradox:

Conclusion drawn from aggregated table might go wrong by drawing conclusions from the individual tables.

Thus, the scenario of Simpson’s paradox is stated. The individual tables tell that obese people are more prone to death, but the combined table tells that non-obese people who are smoking are more prone to death.