Final answer:
Calculations for the binomial distribution with n=24 and p=0.50 involve understanding the normal approximation conditions (np and nq must be greater than 5) and using z-scores for probabilities related to the number of children in a household.
Step-by-step explanation:
Given the binomial distribution parameters n = 24 and p = 0.50, we can calculate various probabilities for the number of children in a household. To answer these questions, it is crucial to employ the appropriate binomial probability formulas or functions in a statistical software or calculator that supports binomial calculations. However, since specific probabilities are not provided in the student's details and the fact that normal approximation can be applied only if np and nq are both greater than 5, which they are in this case (np = 24*0.5 = 12 and nq = 24*(1-0.5) = 12), we will use normal approximation to the binomial distribution whenever necessary.
- (a) The probability of having exactly 15 or 22 children is the sum of the two individual probabilities P(X=15) + P(X=22).
- (b) The probability of having 20 or fewer children is P(X ≤ 20).
- (c) The probability of having 19 or more children is P(X ≥ 19).
- (d) The probability of having fewer than 22 children is P(X < 22), which can also be written as P(X ≤ 21) because the number of children is a discrete variable.
- (e) The probability of having more than 20 children is P(X > 20), which is also known as the complement of P(X ≤ 20).
The attempted solutions involve correctly identifying these probabilities using statistical methods or a calculator. For the normal approximation, one would calculate the z-scores using the formula z = (X - μ) / σ, where μ = np and σ = √npq, and then find the corresponding probability from the standard normal distribution table or software.