Wiki Page, Math 13 - Perez Sandbox : Module 17 - Wrap Up "Distribution of Sample Proportions"

Wiki Page
2026-01-17T21:15:23-08:00

Let’s Summarize

Inference is based on probability.

A parameter is a number that describes a population. A statistic is a number that describes a sample. In inference, we use a statistic to draw a conclusion about a parameter. These conclusions include a probability statement that describes the strength of the evidence or our certainty.

For a categorical variable, the parameter and the statistics are proportions. For a quantitative variable, the parameter and statistics are means.

For a given situation, we assume that the parameter is fixed. It does not change. However, statistics always vary. When we take random samples, the fluctuation in statistics is due to chance.

Larger samples have less variability.

For a categorical variable, we assume that the population has a proportion p of successes. When we select random samples from this population, the sample proportions have a pattern in the long run. We can describe this pattern with a mathematical model of the sampling distribution. 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

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:

When a normal model is a good fit for the sampling distribution, we can calculate a z-score. It allows us to use the standard normal model to find probabilities associated with the sampling distribution.

We can also write this as one formula:

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