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

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?

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 distribution obtained by calculating the proportion of successes in many repeated simple random samples from a population. Its mean is equal to the population proportion, and its variability or spread is measured by the standard error, which is determined by both the population proportion and the sample size.

Central Limit Theorem

The Central Limit Theorem states that, given a sufficiently large sample size, the sampling distribution of the sample proportion will approximate a normal distribution regardless of the original population distribution. This is crucial for making probability statements using the normal model.

Standard Error of a Proportion

The standard error quantifies the expected variation of the sample proportion from the true population proportion. It is calculated as the square root of [p(1-p)/n], where p is the population proportion and n is the sample size. A smaller standard error indicates that sample estimates are more consistently close to the true proportion.

Normal Approximation to the Binomial Distribution

When dealing with a large number of trials, the binomial distribution, which describes the number of successes in a fixed number of independent trials, can be approximated by a normal distribution. This simplifies probability calculations since the normal distribution is continuous and its properties are well-understood.

Continuity Correction

The continuity correction is used when a discrete distribution, like the binomial, is approximated by a continuous distribution, such as the normal. In order to more accurately calculate probabilities for discrete outcomes, an adjustment of 0.5 is added or subtracted from the discrete variable's value to better align with the continuous setting.

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