A simple random sample of size n=75 is obtained from a...

A simple random sample of size $n=75$ is obtained from a population whose size is $N=10,000$ and whose population proportion with a specified characteristic is $p=0.8$. (a) Describe the sampling distribution of $\hat{p}$ (b) What is the probability of obtaining $x=63$ or more individuals with the characteristic? That is, what is $P(\hat{p} \geq 0.84) ?$ (c) What is the probability of obtaining $x=51$ or few individuals with the characteristic? That is, what is $P(\hat{p} \leq 0.68) ?$

A simple random sample of size $n=75$ is obtained from a population whose size is $N=10,000$ and whose population proportion with a specified characteristic is $p=0.8$.
(a) Describe the sampling distribution of $\hat{p}$
(b) What is the probability of obtaining $x=63$ or more individuals with the characteristic? That is, what is $P(\hat{p} \geq 0.84) ?$
(c) What is the probability of obtaining $x=51$ or few individuals with the characteristic? That is, what is $P(\hat{p} \leq 0.68) ?$

Key Concepts

Sampling Distribution of the Sample Proportion

This concept describes how the sample proportion, which is the proportion of individuals in a sample having a specific characteristic, is distributed across many samples drawn from the same population. Its mean is equal to the population proportion (p) and its variance is determined by the sample size and the inherent variability in the population. When the sample is drawn randomly, the distribution reflects the randomness of the sampling process.

Central Limit Theorem for Proportions

The Central Limit Theorem (CLT) for proportions states that if the sample size is sufficiently large, the sampling distribution of the sample proportion approximates a normal distribution regardless of the population's original distribution shape. This approximation facilitates the use of z-scores and standard normal probability tables to calculate probabilities corresponding to different values of the sample proportion.

Finite Population Correction Factor

When sampling is conducted without replacement from a finite population, the observations are not entirely independent. The finite population correction (FPC) factor adjusts the standard error of the sample proportion to reflect the reduced variability due to the population's limited size. Its effect becomes significant especially when the sample size is not very small relative to the population size.

Transcript

Please give Ace some feedback