Suppose a simple random sample of size n=1460 is obtained...

Suppose a simple random sample of size $n=1460$ is obtained from a population whose size is $N=1,500,000$ and whose population proportion with a specified characteristic is $p=0.42$ (a) Describe the sampling distribution of $\hat{p}$ (b) What is the probability of obtaining $x=657$ or more individuals with the characteristic? (c) What is the probability of obtaining $x=584$ or fewer individuals with the characteristic?

Suppose a simple random sample of size $n=1460$ is obtained from a population whose size is $N=1,500,000$ and whose population proportion with a specified characteristic is $p=0.42$
(a) Describe the sampling distribution of $\hat{p}$
(b) What is the probability of obtaining $x=657$ or more individuals with the characteristic?
(c) What is the probability of obtaining $x=584$ or fewer individuals with the characteristic?

Key Concepts

Sampling Distribution of a Sample Proportion

This concept refers to the probability distribution of sample proportions obtained from repeated random samples drawn from a population. It shows how the estimator (the sample proportion) behaves in terms of its center, spread, and shape. With a known population proportion, the sampling distribution has a mean equal to the population proportion and a variance that depends on the size of the sample and the variability inherent in the binary outcome.

Central Limit Theorem

The Central Limit Theorem (CLT) underpins much of sampling theory by asserting that, for a sufficiently large sample size, the sampling distribution of the sample proportion will be approximately normal, regardless of the original distribution of the data. This is crucial because it allows for the use of normal probability models to calculate probabilities and confidence intervals for the sample proportion.

Standard Error of the Proportion

The standard error measures the variability or spread of the sampling distribution of a sample proportion. It is calculated using the formula sqrt[p(1-p)/n], where p is the population proportion and n is the sample size. This measure is essential for quantifying the uncertainty inherent in any single sample estimate of the population proportion.

Normal Approximation

Given the conditions provided (large sample size and a small sampling fraction relative to the population), the normal approximation is applied to the sampling distribution of the sample proportion. This approach simplifies the calculation of probabilities regarding sample outcomes by converting the sample proportion into a Z-score, which can then be referenced against the standard normal distribution.

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