Final answer:
The probability that the sample mean lifespan of male mosquitoes exceeds 10 days is approximately 0.02, or 2.28%.
The correct answer is option (a).
Step-by-step explanation:
The question asks for the probability that the sample mean lifespan of male mosquitoes exceeds 10 days, given a mean of 8 days and a standard deviation of 6 days, with a sample size of 36. To calculate this, we utilize the Central Limit Theorem, which implies that the sampling distribution of the sample mean will be approximately normally distributed because the sample size is large (n≥36).
First, let's compute the standard error (SE) using the formula:
SE = σ / √n
Where σ is the population standard deviation and n is the sample size. Here, SE = 6 / √36 = 6 / 6 = 1 day.
We then find the z-score for a sample mean of 10 days using the formula:
z = (X - μ) / SE
Where X is the sample mean, and μ is the population mean. Substituting the given values, we find z = (10 - 8) / 1 = 2.
Using a z-table or normal distribution calculator, we find the probability that z exceeds 2. The corresponding probability is approximately 0.0228 or 2.28%. This means that the probability the sample mean lifespan (x) exceeds 10 days P(x>10) is approximately 0.02, and hence the correct answer is option a.
This probability helps biologists understand the lifespan distribution among a population of mosquitoes and can be crucial in vector control strategies.