๐ What is the Normality Condition for Sample Proportions?
The sampling distribution of the sample proportion is approximately normal under certain conditions. This allows us to use normal-based methods for inference (like confidence intervals and hypothesis tests) about a population proportion.
๐ History and Background
The concept of a sampling distribution and its normality is rooted in the Central Limit Theorem (CLT). While the CLT applies to sample means, similar principles govern sample proportions. The idea is that, with a large enough sample size, the distribution of sample proportions will approach a normal distribution, regardless of the shape of the population distribution.
๐ Key Principles
- ๐ Randomization Condition: The data must come from a random sample or a randomized experiment. This ensures that the sample is representative of the population.
- ๐ข Independence Condition: Sampled values must be independent of each other. This is often satisfied if the sampling is done with replacement or if the population is much larger than the sample. A common rule of thumb is the 10% condition.
- ๐ฏ 10% Condition: When sampling without replacement, the sample size ($n$) should be no more than 10% of the population size ($N$). Mathematically, $n \leq 0.10N$. This ensures that the observations are approximately independent.
- โ๏ธ Success/Failure Condition: The sample size must be large enough so that both $np$ and $n(1-p)$ are at least 10, where $n$ is the sample size and $p$ is the population proportion. This ensures that the sampling distribution of the sample proportion is approximately normal.
- โ
$np \geq 10$ (Expected number of successes)
- โ $n(1-p) \geq 10$ (Expected number of failures)
๐ Real-World Examples
Let's look at some examples to understand these conditions better.
Example 1: Polling
Suppose a polling agency wants to estimate the proportion of adults who support a particular political candidate. They randomly sample 500 adults from a population of 10,000. The sample showed 60% support the candidate.
- ๐ฒ Randomization: The sample was randomly selected.
- ๐งโ๐คโ๐ง Independence (10% Condition): 500 is less than 10% of 10,000 ($500 < 0.10 * 10,000 = 1,000$), so the independence condition is met.
- โ๏ธ Success/Failure:
- $np = 500 * 0.60 = 300 \geq 10$
- $n(1-p) = 500 * 0.40 = 200 \geq 10$
Both conditions are met, so the sampling distribution of the sample proportion is approximately normal.
Example 2: Manufacturing
A manufacturer produces light bulbs and wants to estimate the proportion of defective bulbs. They take a random sample of 100 bulbs from a large production run and find that 5 are defective.
- ๐ฒ Randomization: The sample was randomly selected.
- ๐งโ๐คโ๐ง Independence (10% Condition): Assume the production run is very large (e.g., 10,000+ bulbs). Then 100 is less than 10% of the population size.
- โ๏ธ Success/Failure:
- $np = 100 * (5/100) = 5 \nless 10$
- $n(1-p) = 100 * (95/100) = 95 \geq 10$
Since $np$ is less than 10, the success/failure condition is not met. The sampling distribution of the sample proportion may not be approximately normal. Therefore, using normal-based methods may not be appropriate.
๐ Conclusion
Ensuring the randomization, independence, and success/failure conditions are met is crucial for the validity of statistical inference involving sample proportions. Always check these conditions before applying normal-based methods!