To find the probability that a randomly chosen plant is detected incorrectly, we need to consider both the false positives and false negatives in the context of the given population.
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False Negatives: This occurs when the test fails to detect the protein in plants that actually have it. Since the test is 98% accurate for identifying plants with the protein, the false negative rate is:
- False Negative Rate = 1 - True Positive Rate = 1 - 0.98 = 0.02 (or 2%)
Given that 3.5% of corn plants actually have the protein, the probability of false negatives in the population is:
- Probability of False Negative = 0.035 (percentage of plants with the protein) * 0.02 (false negative rate) = 0.0007 (or 0.07%)
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False Positives: This occurs when the test incorrectly detects the protein in plants that do not have it. Since the test is 97% accurate for identifying plants without the protein, the false positive rate is:
- False Positive Rate = 1 - True Negative Rate = 1 - 0.97 = 0.03 (or 3%)
Given that 96.5% of corn plants do not have the protein, the probability of false positives in the population is:
- Probability of False Positive = 0.965 (percentage of plants without the protein) * 0.03 (false positive rate) = 0.02895 (or 2.895%)
Now, we combine these two probabilities to find the total probability that a randomly chosen plant is detected incorrectly:
- Total Probability of Incorrect Detection = Probability of False Negative + Probability of False Positive = 0.0007 + 0.02895 = 0.02965
Thus, the probability that a randomly chosen plant is detected incorrectly is 0.02965.
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