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

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?

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

Continuity Correction

Continuity correction is an adjustment made when a discrete distribution (such as the binomial) is approximated by a continuous distribution (such as the normal). This correction involves adding or subtracting 0.5 to the discrete variable's value to improve the approximation. It is especially important when calculating cumulative probabilities for specific counts to achieve more accurate results.

Normal Approximation

Normal approximation is the process of using the normal distribution to estimate probabilities for a binomial random variable or sample proportions when the sample size is large. This method simplifies calculations by approximating the discrete binomial distribution with a continuous normal distribution, which is particularly useful when calculating probabilities related to the number of successes in a sample.

Sampling Distribution of the Sample Proportion

This concept refers to the probability distribution of the sample proportion (p?) that is obtained from repeated sampling from a population. It describes how p? varies from one sample to another and is characterized by a mean equal to the true population proportion (p) and a standard error that quantifies its variability. This distribution is essential for making inferences about population parameters based on sample data.

Central Limit Theorem

The Central Limit Theorem (CLT) states that, given a sufficiently large sample size, the sampling distribution of the sample proportion will be approximately normally distributed, regardless of the shape of the population distribution. This theorem is crucial for justifying the use of normal approximation techniques when calculating probabilities and creating confidence intervals, particularly when both np and n(1-p) are large.

Standard Error of the Proportion

The standard error of the sample proportion is a measure of the variability of p? from sample to sample. It is calculated using the formula ?(p(1-p)/n), where p is the true population proportion and n is the sample size. This metric is a key component in constructing confidence intervals and performing hypothesis tests about a population proportion.

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