Explanation:
(a)
P(X ≤ 2) can be calculated as follows:
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
Using the binomial probability formula: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
P(X = 0) = (25 choose 0) * 0.05^0 * (1 - 0.05)^(25 - 0)
P(X = 1) = (25 choose 1) * 0.05^1 * (1 - 0.05)^(25 - 1)
P(X = 2) = (25 choose 2) * 0.05^2 * (1 - 0.05)^(25 - 2)
Using a statistical software or calculator, these probabilities can be calculated. For example, using the SALT program:
P(X ≤ 2) = 0.594
P(X < 2) can be calculated as follows:
P(X < 2) = P(X = 0) + P(X = 1)
Using the binomial probability formula:
P(X < 2) = (25 choose 0) * 0.05^0 * (1 - 0.05)^(25 - 0) + (25 choose 1) * 0.05^1 * (1 - 0.05)^(25 - 1)
Using a statistical software or calculator:
P(X < 2) = 0.531
(b)
P(X ≥ 3) can be calculated as follows:
P(X ≥ 3) = 1 - P(X < 3)
Using the binomial probability formula:
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
Using a statistical software or calculator:
P(X ≥ 3) = 1 - 0.531 = 0.469
(c)
P(1 ≤ X ≤ 2) can be calculated as follows:
P(1 ≤ X ≤ 2) = P(X = 1) + P(X = 2)
Using the binomial probability formula:
P(X = 1) = (25 choose 1) * 0.05^1 * (1 - 0.05)^(25 - 1)
P(X = 2) = (25 choose 2) * 0.05^2 * (1 - 0.05)^(25 - 2)
Using a statistical software or calculator:
P(1 ≤ X ≤ 2) = P(X = 1) + P(X = 2)
(d)
E(X) represents the expected value of X, which can be calculated using the formula:
E(X) = n * p
Using the given values:
E(X) = 25 * 0.05
Using a calculator:
E(X) = 1.25
The value of X is the number of children in the sample who have a food allergy, which can vary depending on the sample. It can take on any integer value from 0 to 25, inclusive.
(e)
If we consider a sample of 30 children, the probability that none of them has a food allergy can be calculated using the binomial probability formula:
P(X = 0) = (30 choose 0) * 0.05^0 * (1 - 0.05)^(30 - 0)
Using a statistical software or calculator:
P(X = 0) = 0.443