Answer:
Step-by-step explanation:
a) The sampling distribution of Y, the number of residents infected by the T-Virus, follows a binomial distribution. We can denote Y ~ B(n, p), where n is the number of residents selected (in this case, n = 10) and p is the probability of an individual being infected by the T-Virus (p = 0.65).
(b) To calculate the probability that half of the selected residents have been infected by the T-Virus, we can use the binomial probability formula. The probability mass function (PMF) of a binomial distribution is given by:
P(Y = k) = C(n, k) * p^k * (1 - p)^(n - k)
where C(n, k) is the number of combinations of n items taken k at a time.
In this case, we want to find P(Y = 5), which represents the probability that exactly 5 out of 10 residents are infected. We can calculate it as:
P(Y = 5) = C(10, 5) * (0.65)^5 * (1 - 0.65)^(10 - 5)
(c) To calculate the probability that at most 7 of the selected residents have been infected by the T-Virus, we need to sum the probabilities of Y being 0, 1, 2, ..., 7. We can calculate it as:
P(Y ≤ 7) = P(Y = 0) + P(Y = 1) + P(Y = 2) + ... + P(Y = 7)
(d) The mean (expected value) and standard deviation of a binomial distribution can be calculated using the formulas:
Mean (μ) = n * p
Standard Deviation (σ) = sqrt(n * p * (1 - p))
In this case, we can calculate the mean and standard deviation as:
Mean = 10 * 0.65
Standard Deviation = sqrt(10 * 0.65 * (1 - 0.65))
By plugging in the values, we can obtain the numerical results.