Welcome to Chapter 9 Part 2!
Hello everyone! In this section, we're diving deeper into the world of sampling and sampling distributions. We'll be focusing on the distribution of sample proportions and exploring different sampling methods. Get ready to expand your statistical toolkit!
Sampling Distribution of the Sample Proportion
Let's start by understanding the sampling distribution of the sample proportion. Imagine you're repeatedly taking samples from a population and calculating the proportion of a certain characteristic in each sample. The distribution of these sample proportions has some important properties:
- It's approximately a normal distribution, provided the population is infinite and the sample size is sufficiently large.
- The mean of the sample proportions, denoted as $\mu_{\hat{p}}$, is equal to the population proportion, $p$. Mathematically, this is expressed as: $$\mu_{\hat{p}} = E(\hat{p}) = p$$
- The standard deviation of the sample proportions, denoted as $\sigma_{\hat{p}}$, is calculated as: $$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$$, where $n$ is the sample size. This tells us how much the sample proportions are likely to vary around the true population proportion.
Example: Suppose a sample of 400 people is used to perform a taste test. If the true fraction in the population that prefers Pepsi is really 0.5, what is the probability that less than 0.44 of the persons in the sample will prefer Pepsi?
Given: $p = 0.5$, $n = 400$. We want to find $P(\hat{p} < 0.44)$.
First, calculate the standard deviation: $$\sigma_{\hat{p}} = \sqrt{\frac{0.5(1-0.5)}{400}} = 0.025$$
Next, calculate the z-score: $$z = \frac{x - \mu}{\sigma} = \frac{0.44 - 0.50}{0.025} = -2.4$$
The probability $P(z < -2.4)$ is approximately 0.0082. Therefore, there is approximately a 0.82% chance that less than 44% of the sample will prefer Pepsi.
Other Forms of Sampling
Besides simple random sampling, there are several other useful sampling techniques:
- Judgment Sample: Observations are selected by an expert in the field.
- Convenience Sample: Observations are easily obtained and not random.
- Systematic Sample: Choose a starting point and then select every $k^{th}$ member of the population.
- Cluster Sampling: Divide the population into clusters and randomly select clusters. This is used when "natural" groupings are evident.
- Stratified Sampling: Divide the population into strata (sub-populations) based on identifiable characteristics.
Example: A social researcher in Florida wants to determine the average number of children per family in the state. Consider different sampling methods (Simple Random, Cluster, Stratified) and their steps. What is the most cost-effective method? Justify your answer!
Keep Practicing!
Understanding sampling distributions and different sampling methods is crucial for making accurate inferences about populations. Keep practicing with different examples, and don't hesitate to ask questions. You've got this!