Question

Simpson’s paradoxAccident victims are sometimes taken by helicopter from the accident scene to a hospital. Helicopters save time. Do they also save lives? The two-way table summarizes data from a sample of patients who were transported to the hospital by helicopter or by ambulance.

(a) What percent of patients died with each method of transport? Here are the same data broken down by severity of accident:

(b) Calculate the percent of patients who died with each method of transport for the serious accidents. Then calculate the percent of patients who died with each method of transport for the less serious accidents. What do you notice?

(c) See if you can explain how the result in part (a) is possible given the result in part (b).

Note: This is an example of Simpson’s paradox, which states that an association between two variables that holds for each value of a third variable can be changed or even reversed when the data for all values of the third variable are combined.

Short answer

Part (a) Helicopter 32% and Ambulance 23.64%

Part (b) Serious accidents: Helicopter 48% and Ambulance 60%

Less Serious Accidents: Helicopter 16% and Ambulance 20%

Part (c) More persons are taken by ambulance than by helicopter, although more serious accidents are more likely to be transported by helicopter than less catastrophic accidents.

Step-by-step solution

Part (a) Step 1. Given information.

The given table is:

Part (a) step 2. Find the percent of patients who died with each method of transport.

Total number of persons transported through helicopter: 200

Persons died in Helicopter: 64

Therefore, the required percent is:

$\frac{64}{200}=32\%$

Total number of persons transported through Ambulance: 1100

Persons died in Ambulance: 260

Therefore, the required percent is:

$\frac{260}{1100}=23.64\%$

Part (b) step 1. Find the percentage of patients who died with each method of transport for serious accidents.

The Percent of patients who died with each method of transport in case of serious accidents:

Helicopter:

$\frac{48}{100}=48\%$

Ambulance:

role="math" localid="1653568964878" $\frac{60}{100}=60\%$

The percent of patients who died with each method of transport for the less serious accidents:

Helicopter:

$\frac{16}{100}=16\%$

Ambulance:

$\frac{200}{1000}=20\%$

Patients transported with a helicopter were more likely to die overall (as a result of the previous exercise) than patients transported with an ambulance, but patients in serious accidents who were transported with a helicopter were less likely to die than patients in less serious accidents who were transported with an ambulance, and patients in less serious accidents who were transported with a helicopter were less likely to die than patients in less serious accidents who were transported with an ambulance.

Part (c) Step 1. Explanation.

From Part (b) result: We note that patients transported by helicopter were more likely to die overall (as a result of the previous exercise) than patients transported by ambulance, while patients in serious accidents who were transported by helicopter were less likely to die than patients in less serious accidents who were transported with a helicopter, and patients in less serious accidents who were transported with a helicopter were less likely to die.

This is owing to the fact that there were far more individuals taken by ambulance than by helicopter, although more serious accidents tend to employ a helicopter as a mode of transportation more frequently than less serious accidents.