Suppose a simple random sample of size n=1000 is obtained...

Suppose a simple random sample of size $n=1000$ is obtained from a population whose size is $N=1,000,000$ and whose population proportion with a specified characteristic is $p=0.35$ (a) Describe the sampling distribution of $\hat{p}$ (b) What is the probability of obtaining $x=390$ or more individuals with the characteristic? (c) What is the probability of obtaining $x=320$ or fewer individuals with the characteristic?

Suppose a simple random sample of size $n=1000$ is obtained from a population whose size is $N=1,000,000$ and whose population proportion with a specified characteristic is $p=0.35$
(a) Describe the sampling distribution of $\hat{p}$
(b) What is the probability of obtaining $x=390$ or more individuals with the characteristic?
(c) What is the probability of obtaining $x=320$ or fewer individuals with the characteristic?

Key Concepts

Sampling Distribution of the Sample Proportion

This concept refers to the probability distribution of the sample proportion, which estimates the true population proportion. Under simple random sampling, the sampling distribution has a mean equal to the true proportion and a variance given by p(1-p)/n. When the sample size is sufficiently large, this distribution can be approximated by a normal distribution, facilitating probability calculations related to the sample proportion.

Central Limit Theorem

The central limit theorem is a key statistical principle stating that, regardless of the population's distribution, the sampling distribution of the sample mean (or proportion) will tend to follow a normal distribution as the sample size increases. This theorem underpins the use of normal approximations in estimating probabilities for sample proportions, making it particularly useful in large-sample inference.

Normal Approximation to the Binomial Distribution

When sampling from a population where the characteristic of interest follows a binomial distribution, and the sample size n is large, the binomial distribution can be approximated by a normal distribution. This is especially valid when both np and n(1-p) are large. This approximation simplifies the calculation of tail probabilities, such as the probability of observing a certain number of successes or more/fewer.

Finite Population Correction Factor

When sampling without replacement from a finite population, the variance of the sampling distribution is slightly reduced by the finite population correction factor. This factor becomes important if the sample size is a significant fraction of the total population; however, when the population is much larger than the sample, as in many practical cases, the correction factor is negligible.

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