Question

a positive test actually has a food allergy? use the drop-down menus to explain. actually has allergy? yes 8%, no 92%. test shows allergy? for yes (actually has), yes 90%, no 10%. for no (actually has), yes 15%, no 85%. click the arrows to choose an answer from each menu. the probability that a child will receive a positive test is choose... . for a child with a positive test, the probability that the child actually has a food allergy is approximately choose... . the new test choose... likely to be a reliable indicator that the child has a food allergy.

Explanation:

Step1: Calculate P(Positive Test)

To find the probability of a positive test, we use the law of total probability. The probability of having an allergy (Yes) is 8% or 0.08, and the probability of testing positive given allergy is 90% or 0.9. The probability of not having an allergy (No) is 92% or 0.92, and the probability of testing positive given no allergy is 15% or 0.15.

So, \( P(\text{Positive}) = P(\text{Allergy}) \times P(\text{Positive}|\text{Allergy}) + P(\text{No Allergy}) \times P(\text{Positive}|\text{No Allergy}) \)

Substituting the values: \( 0.08 \times 0.9 + 0.92 \times 0.15 \)

First, calculate \( 0.08 \times 0.9 = 0.072 \) and \( 0.92 \times 0.15 = 0.138 \)

Then, add them: \( 0.072 + 0.138 = 0.21 \) or 21%

Step2: Calculate P(Allergy|Positive) using Bayes' Theorem

Bayes' Theorem states that \( P(\text{Allergy}|\text{Positive}) = \frac{P(\text{Positive}|\text{Allergy}) \times P(\text{Allergy})}{P(\text{Positive})} \)

We know \( P(\text{Positive}|\text{Allergy}) = 0.9 \), \( P(\text{Allergy}) = 0.08 \), and \( P(\text{Positive}) = 0.21 \)

Substituting: \( \frac{0.9 \times 0.08}{0.21} \)

Calculate numerator: \( 0.9 \times 0.08 = 0.072 \)

Then, divide by 0.21: \( \frac{0.072}{0.21} \approx 0.3429 \) or approximately 34.3%

Step3: Analyze Reliability

Since the probability of actually having an allergy given a positive test is around 34.3%, which is less than 50%, the test is not very reliable (or "not very" likely to be a reliable indicator).

Answer:

1. The probability that a child will receive a positive test is 21% (or 0.21).
2. For a child with a positive test, the probability that the child actually has a food allergy is approximately 34.3% (or \(\frac{24}{70} \approx 0.3429\)).
3. The new test is not very likely to be a reliable indicator that the child has a food allergy.