Final answer:
The normal distribution is used for concluding if the proportion of all pet owners who celebrate their pet's birthday is less than 0.25, based on the central limit theorem and given that np and nq are greater than 5.
Step-by-step explanation:
The appropriate distribution to use when concluding whether the proportion of pet owners who celebrate their pet's birthday is less than 0.25 is the normal distribution. This conclusion stems from the central limit theorem for proportions which states that if the sample size is large enough, and np and nq are both greater than 5, the sampling distribution of the sample proportion p' will approximate a normal distribution with mean μ = p and standard deviation σ = √(pq/n), where q = 1 - p.
In the context of the question, knowing that P' = 0.2 and n = 1,000 suggests that np (200) and nq (800) are both well above 5, confirming the normal distribution assumption is reasonable. This is because a large sample size and the central limit theorem allow the use of the normal approximation to binomial distribution for hypothesis testing regarding proportions.