Question

Equine infectious anemia (EIA) is considered the main infectious disease in Brazilian equine farming, for which there is no effective vaccine or treatment. It is caused by a retrovirus of the genus Lentivirus, which affects horses, donkeys and mules and is transmitted in nature mainly by hematophagous insects of the genus Tabanidae. Researchers analyzed the records of 9,439 equids from Acre, submitted to the agar gel immunodiffusion test (AGID) for equine infectious anemia (EIA), between 1986 and 1996. Of these, 6199 tested positive for equine infectious anemia (EIA) . Knowing that the age of AIE-positive horses follows a Normal distribution with a mean of 5 years and a standard deviation of 1.5 years, determine the expected number of AIE-positive horses in the Acre sample that will be aged less than or equal to 3 years. ATTENTION: Provide the answer to exactly FOUR decimal places.

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To determine the expected number of AIE-positive horses in the Acre sample that will be aged less than or equal to 3 years, we need to calculate the probability of a horse being less than or equal to 3 years old given that it tested positive for equine infectious anemia (EIA).
Let's denote X as the age of a positive horse. We know that X follows a normal distribution with a mean (μ) of 5 years and a standard deviation (σ) of 1.5 years.
To find the probability of a positive horse being less than or equal to 3 years old, we need to calculate the z-score corresponding to the age of 3 years using the formula:
z = (X - μ) / σ
z = (3 - 5) / 1.5
z = -2 / 1.5
z = -1.3333
Now, we can use a standard normal distribution table or a calculator to find the probability corresponding to the z-score of -1.3333.
Using a standard normal distribution table or a calculator, we find that the probability is approximately 0.0918.
To calculate the expected number of AIE-positive horses aged less than or equal to 3 years, we multiply the probability by the total number of AIE-positive horses:
Expected number = Probability * Total number
Expected number = 0.0918 * 6199
Expected number ≈ 569.0682
Therefore, the expected number of AIE-positive horses in the Acre sample that will be aged less than or equal to 3 years is approximately 568.7242. Rounded to four decimal places, the answer is 569.0682

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