7.1 Introducing Statistics: Why Should I Worry About Error?

Part of Inference for Quantitative Data: Means in AP® Statistics. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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The Challenge of Unknown Population Variance

In Unit 6, we performed z-tests for proportions. The standard error used the parameter ${p_0}$ (known under ${H_0}$ ).

For means, we need:

\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}

Problem: In practice, we rarely know the population standard deviation ${\sigma}$ .

Solution: Replace

{\sigma}

with the sample standard deviation

{s}

SE_{\bar{X}} = \frac{s}{\sqrt{n}}

But using ${s}$ (a random variable) instead of ${\sigma}$ (a fixed constant) introduces extra uncertainty. This extra uncertainty means the standardized statistic no longer follows the standard normal distribution ${N(0,1)}$ .

Instead, it follows the t-distribution — a family of symmetric distributions with heavier tails than normal.

In this unit

9 lessons

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