Question

Critical Thinking: Interpreting results from a test for smoking

It is estimated that roughly half of smokers lie when asked about their smoking involvement. Pulse Co-Oximeters may be a way to get information about smoking without relying on patients’ statements. Pulse CO-oximeters use light that shines through a fingernail, and it measures carbon monoxide in blood. These devices are used by firemen and emergency departments to detect carbon monoxide poisoning, but they can also be used to identify smokers. The accompanying table lists results from people aged 18–44 when the pulse CO-oximeters is set to detect a 6% or higher level of carboxyhemoglobin (based on data from “Carbon Monoxide Test Can Be Used to Identify Smoker,” by Patrice Wendling, Internal Medicine News, Vol. 40., No. 1, and Centers for Disease Control and Prevention).

CO-Oximetry Test for Smoking

	Positive Test Results	Negative Test Results
Smoker	49	57
Non-smoker	24	370

Negative Predictive Value Find the negative predictive value of the test by finding the probability that the subject does not smoke, given that the test yields a negative result..

Short answer

The negative predictive value of the test is 0.866.

Step-by-step solution

Given information

The results from people aged 18-44 to detect the level of carboxyhemoglobin are based on the data from “Carbon Monoxide Test” used to identify smokers.

Describe Conditional probability

A conditional probability of event is a probability obtained with the additional information that some other event has already occurred.

Notation for conditional probability isdenotes the conditional probability of event B occurring, given that event A has already occurred.

Formula for conditional probability is as,

$P\left(B|A\right)=\frac{P\left(A \text{and} B\right)}{P\left(A\right)}$

Where,

$P\left(A \text{and} B\right)= \text{Probability} \text{of} \text{A} \text{and} \text{B} \text{both} \text{occurring}$

$P\left(A\right)=\text{Probability} \text{of} \text{occurring} \text{event} A$

Summarize the given information

The test for smoking involves 500 people. Out of the 500 people, 106 are smokers and 394 are non-smokers. And, 73 people got the positive test results and 427 got the negative test results.

The above information is summarizedin the following table,

	Positive Test Results	Negative Test Results	Total
Smoker	49	57	106
Non-smoker	24	370	394
Total	73	427	500

Define the events

The probability that a subject does not smoke, given that the test yields a negative result. Here,define the two events as:A be the event that the test yields a negative result, and B be the event that the subject does not smoke.

Find the probability that a subject smoke, given that the test yields a positive result.

The probabilitythat a subject does not smoke, given that the test yields a negative result,$P\left(B|A\right)$.

By using conditional probability,

$P\left(B|A\right)=\frac{P\left(A \text{and} B\right)}{P\left(A\right)} ...\left(1\right)$

Find the probability that the test yields a negative result

The probability that the test yields a negative result is,

$$\begin{gathered} P\left(A\right)=\frac{\text{Number}\text{of}\text{subjects with negative} \text{test} \text{reults}}{\text{Total subjects surveyed}} \\ =\frac{427}{500} \end{gathered}$$

Find the probability that a subject does not smoke and had negative test result

The probability that a subject does not smoke and had negative test result is,

$$\begin{gathered} P\left(A \text{and} B\right)=\frac{\text{Number of subjects who does not}\text{smoke and yield negative test result}}{\text{Total}} \\ =\frac{370}{500} \end{gathered}$$

Find the negative predictive value 

The negative predictive value is the probability that a subject does not smoke, given that the test yields a negative result.

Substitute $P\left(\text{subject does not smoke and yield neagtive test result}\right)$and$P\left(\text{Negative result of test}\right)$ in equation (1),

$$\begin{gathered} P\left(\text{B|A}\right)=\frac{P\left(A \text{and} B\right)}{P\left(A\right)} \\ =\frac{\frac{370}{500}}{\frac{427}{500}} \\ =\frac{370}{427} \\ =0.8665 \end{gathered}$$

Thus, the probability that a subject smoke, given that the test yields a negative result is 0.866.