- Use technology to create a sampling distribution of a sample proportion given [latex]n[/latex] and [latex]p[/latex].
- Calculate the mean and standard deviation for a sampling distribution of a sample proportion.
- Recognize the difference between the standard deviation and the standard error of a sample proportion.
The Mean and Standard Deviation of a Sampling Distribution of a Sample Proportion
When dealing with a population, it is often not practical or feasible to examine every single individual in the population to gather data. Instead, we typically work with a smaller subset of the population known as a sample. However, if we were to take every possible sample from the population, we could create what is called an exact sampling distribution.
An exact sampling distribution would show all the possible sample proportions that could be obtained from every possible sample of a given size. This distribution would give us a comprehensive understanding of how sample proportions vary and provide precise information about their behavior.
Unfortunately, in most cases, sampling every possible sample is simply not feasible due to time, cost, and logistical constraints, especially when dealing with large populations. This is where mathematical theory comes to the rescue!
The theoretical expressions for the mean and standard deviation of the sampling distribution are derived based on the characteristics of the population and the sample size.
The mean and standard deviation of the sampling distribution of a sample proportion
When taking many random samples of size [latex]n[/latex] from a population distribution with population proportion [latex]p[/latex]:
- The mean of the distribution of sample proportions is [latex]\mu_\hat{p}=p[/latex].
- The standard deviation of the distribution of sample proportions is [latex]\sigma_\hat{p}=\sqrt{\frac{p(1-p)}{n}}[/latex].