Question

Question One)

Discuss what needs to be satisfied in order for the distribution of the sample proportion to be considered approximately normal. From what probability distribution does this requirement come from?

Question 2 ) Discuss the requirements that need to met in order to state that the distribution of the sample mean follows a normal distribution.

Answer 1

1) we have a good sense of what happens as we take random samples from a population. Our simulation suggests that our initial intution about the shape and centre of the sampling distribution is correct. If the population has a propprtion of p, then random samples of the same size drawn from the population will have sample proportions close to p. More specifically, the distribution of sample proportions will have a mean of p. We also observed that for this situation, the sample proportions are approximately normal. We will see later that this is not always the case. But if sample proportions are normally diatributed, then the distribution is centered at p. Now we want to use simulation to help us think more about the variability we expect to see in the sample proportions. From intutions we tell that larger samples will better approximate the population, so we might expect less variability in large samples. A sampling distribution is a probability distribution of a stastic obtained from a larger number of samples drawn from a specific population. The sampling distribution of a given population is the distribution of frequencies of a range of different outcomes that could possibly occur for a stastic of a population. In stastics, a population is the entire pool from which a stastic sample is is drawn. A population may refer to an entire group of people, objects, events, hospital visuts, or measurments. A population can thus be said be an aggregate onservation of subjects grouped together by common feature.

2) Distribution sampling:- A sampling distribution is a probibility distribution of a stastic (such as the mean) that results from selecting an infinte number of random samples of the same size from a population. The given sampling distribution of a given population is the distribution of frequencies of a range of different outcomes that could possibly occure for a stastic of a population. The central limit theorem states that if you have a population with mean and standard deviation and take sufficiently large random samples from population with replacement, them the distribution of the sample means will be approximatly normally distributed. This will hold true regardless of weather the source population is normal skewed, provided the sample size is suffiviently large(usually small and is greater than or equal to 30) . If the population is normal, then the theorem holds true even for samples smaller than 30.in fact, this also holds true even if population is boinomial, provided that min (np, N(1-p)) greater than or equal to 5,where an is the sample size and p is the probability of success in the population. This means that we can use the normal probability model to quantify uncertaintly when making interference about a population mean based on the sample mean.