115 Distribution of Sample Proportions (4 of 6)

The simulations on the previous page reinforce what we have observed about patterns in random sampling.

• Proportions from random samples approximate the population proportion, p, 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 p. In other words, a sampling distribution for large samples has less variability.
• The distribution of sample proportions appears normal (at least for the examples we have investigated).

We can describe the sampling distribution with a mathematical model that has these same features.

The Sampling Distribution of Sample Proportions

For a categorical variable, imagine a population with a proportion p of successes. (For example, for the variable gender, imagine a population of part-time college students with p = 0.60 female. Note that a success is the category of interest. It is what we are counting. Here a success is a female.) We create a mathematical model that describes the sample proportions from all possible random samples of size n from this population. The model has the following center, spread, and shape.

Center: Mean of the sample proportions is p, the population proportion.

Spread: Standard deviation of the sample proportions is [latex]\sqrt{\frac{p(1-p)}{n}}[/latex]. The standard deviation of the sampling distribution is also called the standard error.

Shape: A normal model is a good fit if the expected number of successes and failures is at least 10. We can translate these conditions into formulas: [latex]np≥10\text{}\mathrm{and}\text{}n(1-p)≥10.[/latex]

Comment

The distribution of sample proportions for ALL samples of the same size is called the sampling distribution of sample proportions.

In a simulation, we collect thousands of random samples to examine the distribution of sample proportions. But when we model this distribution, our model describes the sampling distribution that comes from ALL possible random samples of the same size.