Suppose a simple random sample of size n=200 is obtained...

Suppose a simple random sample of size $n=200$ is obtained from a population whose size is $N=25,000$ and whose population proportion with a specified characteristic is $p=0.65$ (a) Describe the sampling distribution of $\hat{p}$ (b) What is the probability of obtaining $x=136$ or more individuals with the characteristic? That is, what is $P(\hat{p} \geq 0.68) ?$ (c) What is the probability of obtaining $x=118$ or fewer individuals with the characteristic? That is, what is $P(\hat{p} \leq 0.59) ?$

Suppose a simple random sample of size $n=200$ is obtained from a population whose size is $N=25,000$ and whose population proportion with a specified characteristic is $p=0.65$
(a) Describe the sampling distribution of $\hat{p}$
(b) What is the probability of obtaining $x=136$ or more individuals with the characteristic? That is, what is $P(\hat{p} \geq 0.68) ?$
(c) What is the probability of obtaining $x=118$ or fewer individuals with the characteristic? That is, what is $P(\hat{p} \leq 0.59) ?$

Key Concepts

Sampling Distribution of a Sample Proportion

This concept involves understanding that the sample proportion, denoted as p?, is a random variable whose distribution, when samples are repeatedly drawn from the population, has a specific mean and variability. Typically, the mean of the sampling distribution is equal to the true population proportion p, and its standard error (or standard deviation) reflects the spread of p? around p, calculated typically as sqrt[p(1-p)/n] for a large enough sample size.

Central Limit Theorem

The Central Limit Theorem is a fundamental principle that 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 allows practitioners to use normal probability models even when dealing with proportions derived from a binomial experiment.

Normal Approximation

Related to the Central Limit Theorem, the normal approximation is the practice of treating the distribution of the sample proportion as a normal distribution. This approximation is valid when the sample size is large enough such that both np and n(1-p) are greater than 10. It simplifies probability calculations significantly by using z-scores and standard normal tables.

Standard Error and Finite Population Correction

The standard error quantifies the variability of the sample proportion and is typically calculated as sqrt[p(1-p)/n]. When the sample is drawn from a finite population without replacement, a finite population correction factor, sqrt[(N - n)/(N - 1)], is applied to adjust the standard error. This adjustment accounts for the reduced variability expected when sampling without replacement from a finite set.

Z-Score Calculation

A z-score represents the number of standard errors a sample proportion is away from the true population proportion. By converting the sample proportion to a z-score using the formula (p? - p) divided by the standard error, one can use the standard normal distribution to compute probabilities associated with obtaining sample proportions above or below certain thresholds.

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