For what percent of those tested for a certain infection was the test accurate; that is, positive for those who...

Understanding the Question

The question asks: For what percent of those tested was the test accurate?

Let's break this down. A test is "accurate" when it gives the correct result:

• Positive result for someone who has the infection (true positive)
• Negative result for someone who doesn't have the infection (true negative)

So we need to determine: What percentage of all tested people got the correct result?

What We Need to Determine

To find accuracy, we need to know:

• How many people were tested in total
• How many got correct results (true positives + true negatives)
• How many got incorrect results (false positives + false negatives)

Key Insight

Here's the crucial point: A test can be wrong in two ways:

1. False positives: Test says you have the infection, but you don't
2. False negatives: Test says you don't have the infection, but you do

To calculate overall accuracy, we must know BOTH types of errors, not just one.

Analyzing Statement 1

Statement 1 tells us: Of those who tested positive for the infection, \(\frac{1}{8}\) did not have the infection.

This means among people with positive test results, \(\frac{1}{8}\) are false positives and \(\frac{7}{8}\) are true positives.

What We Still Don't Know

While we know the false positive rate among positive results, we know nothing about false negatives. We have no information about how many infected people tested negative.

Testing with Concrete Numbers

Let's see why this matters:

Example: Suppose 100 people tested positive

• False positives: 12.5 people (\(\frac{1}{8}\) of 100)
• True positives: 87.5 people (\(\frac{7}{8}\) of 100)
• Critical unknown: How many infected people tested negative?

If 50 infected people tested negative, the accuracy would be very different than if 500 infected people tested negative. Without this information, we cannot calculate overall test accuracy.

Conclusion

Statement 1 alone is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: Of those who tested for the infection, 90 percent tested negative.

This means:

• 90% got negative results
• 10% got positive results

What We Still Don't Know

We know the distribution of test results, but we don't know:

• How many people actually have the infection
• Whether those negative results are accurate (true negatives) or inaccurate (false negatives)
• Whether those positive results are accurate (true positives) or inaccurate (false positives)

Why This Matters

Let's explore what could be happening:

Possibility 1: Only 5% of people actually have the infection

• With 10% testing positive → many false positives
• Most of the 90% negative results would be true negatives (accurate)
• High overall accuracy

Possibility 2: 20% of people actually have the infection

• With only 10% testing positive → many false negatives
• Many of the 90% negative results would be false negatives (inaccurate)
• Low overall accuracy

These scenarios would yield completely different accuracy percentages!

Conclusion

Statement 2 alone is NOT sufficient.

This eliminates choice B.

Combining Both Statements

Let's see what we know when we combine BOTH statements:

Combined Information

• From Statement 2: 90% tested negative, 10% tested positive
• From Statement 1: Among the 10% who tested positive, \(\frac{1}{8}\) are false positives

This means:

• If 1000 people were tested:
  • 100 tested positive
  • 900 tested negative
• Of the 100 positive tests:
  • 12.5 are false positives (\(\frac{1}{8} \times 100\))
  • 87.5 are true positives (\(\frac{7}{8} \times 100\))

The Critical Missing Piece

We still don't know: How many of the 900 people who tested negative actually have the infection?

Without knowing the actual infection rate in the population, we cannot determine the false negative rate. And without the false negative rate, we cannot calculate overall accuracy.

Demonstrating the Problem

Let me show you why this matters with two different scenarios:

Scenario 1: Total infected = 87.5 (only those who tested positive and have it)

• False negatives = 0
• True negatives = 900 - 0 = 900
• Accuracy = \(\frac{87.5 + 900}{1000} = 97.5\%\)

Scenario 2: Total infected = 187.5

• False negatives = 187.5 - 87.5 = 100
• True negatives = 900 - 100 = 800
• Accuracy = \(\frac{87.5 + 800}{1000} = 88.75\%\)

Different infection rates lead to completely different accuracy percentages!

Conclusion

Even combining both statements, we still cannot determine test accuracy because we don't know the actual infection rate, which means we can't calculate the false negative rate.

The statements together are NOT sufficient.

[STOP - Not Sufficient!]

The Answer: E

The statements together are not sufficient because:

1. We can determine the false positive rate from the given information
2. But we cannot determine the false negative rate without knowing the actual infection rate
3. Both error rates are needed to calculate overall test accuracy

Answer Choice E: "The statements together are not sufficient."