- Check the conditions for normal approximation of a sampling distribution of a sample proportion.
- Use the normal distribution to calculate probabilities and percentiles from a sampling distribution.
- Find the sample size needed for a sampling distribution to have a desired standard deviation.
The Normal Condition for a Sampling Distribution of Sample Proportions
Central Limit Theorem
The Central Limit Theorem states that, as the sample size gets larger, the distribution of the sample proportion will become closer to a normal distribution.
In order to estimate probabilities from our sampling distribution, we need the distribution to be approximately normal.
The Normal Condition for Proportions
Given a population proportion, [latex]p[/latex] and a sample size [latex]n[/latex], if [latex]np\ge10[/latex] and [latex]n(1-p)\ge10[/latex] then we assume that the sampling distribution of the sample proportion is approximately normal.
- Proportions from random samples approximate the population proportion, [latex]p[/latex], so sample proportions average out to the population proportion.
- Larger random samples better approximate the population proportion, so large samples have sample proportions closer to [latex]p[/latex]. In other words, a sampling distribution for large samples has less variability.
- The distribution of sample proportions appears normal when the sample size is sufficiently large