Lesson 1: 11.1 Tests About a Population Mean
Duration of Days: 4
Lesson Objective
Students will be able to perform a one-sample $t$-test for a population mean, including stating hypotheses, checking conditions, calculating the test statistic, and determining the P-value.
1. Why do we use a t-statistic instead of a z-statistic for means?
2. How do we determine the "degrees of freedom" for a single sample?
3. What does "reject/fail to reject the null" mean in the context of a problem?
AP Stats CED: SRE-3.A (Hypotheses for population means), SRE-3.B (Conditions), SRE-3.C (Calculating t* and P-values).
Description
This section covers the One-Sample t-test. Students use the sample mean and sample standard deviation to calculate a t-score, which measures how many estimated standard errors the sample mean is from the hypothesized null value. They then find the P-value using a t-table or calculator and state a conclusion based on a designated significance level.
Purpose
To provide a rigorous way to test claims about averages (e.g., "The average commute time is 25 minutes") when the population spread is unknown. It is the most common test used in scientific research for single-group data.
DOK Level
Level 3 (Strategic Thinking): Students must examine sample data to verify the "Normal/Large Sample" condition and justify their decision to proceed with the test.
Relating the outcome of a test to the correct conclusion tends to be difficult for students. The Low p-value relating to STRONG evidence against the null seems, for some, counterintuitive. Students also feel inclined to say that a high p-value indicates that the null is true, when in fact it just indicates that there is not strong evidence that it is false.
Struggling Learners: Use a "Parameter vs. Statistic" anchor chart. Remind them that the Null Hypothesis always uses the population symbol. Use a visual "P-value Scale" to show that smaller numbers represent "Stronger Evidence."
Advanced Learners: Explore Power and Effect Size. Ask: "If we have a very large sample size (n=500), can a tiny, practically meaningless difference become 'statistically significant'?" This helps them distinguish between statistical significance and practical importance.
ELL Learners: Focus on the Directional Vocabulary in word problems. Create a "Word-to-Symbol" bank"; give a template for conclusions in context "Since the p-value is ______ (less than/ greater than) _____ (alpha), we ________ (reject/fail to reject) the null. There ______ (is/ is not) strong evidence that_________ (the alternative).
Real-world application to scenario involving Parkinsons Disease; Quiz