Unit 11: Testing Claims about Means

The Necessity of the t-Distribution: Why we must use a "t-score" instead of a "z-score" when the true variation of the entire population is unknown and we only have the variation from our sample.

Degrees of Freedom: The understanding that the shape of the t-distribution changes based on the sample size, becoming more like a Normal curve as the sample size increases.

Independent vs. Dependent Groups: The conceptual difference between comparing two entirely separate groups (like students in two different cities) versus comparing measurements from the same group twice.

The Null Hypothesis of "No Difference": The logic that we start by assuming there is no actual difference between two group averages and only change our minds if the sample data is highly unlikely.

Robustness: The idea that t-procedures are reliable even if the data isn't perfectly bell-shaped, as long as there are no extreme outliers or very strong skewness.

Verify the Three Essential Conditions: For every test, students will check for Randomness, the 10% Rule (for independence), and the Normal/Large Sample condition (using the Central Limit Theorem or by graphing the data).

Navigate the t-Table or Technology: Use degrees of freedom to find the appropriate probability values for one-sided (less than/greater than) or two-sided (different from) tests.

Execute the 4-Step Process: Utilize the State-Plan-Do-Conclude framework to organize their statistical evidence and reasoning.

Graph Data for Small Samples: Create and analyze dotplots, boxplots, or histograms for samples smaller than 30 to justify that the data is "safe" to test.

Students will demonstrate their mastery by writing formal contextual conclusions that link the calculated probability value back to the original claim, such as stating they have convincing evidence that a new method results in a higher average than an old one. In a free-response format, they will show they can correctly choose between a one-sample test and a two-sample test based specifically on the study's design.

They will also perform an error analysis by describing the real-world consequences of a False Positive—claiming a difference exists when it doesn't—versus a False Negative—failing to find a difference that actually exists—for a specific scenario.

Finally, they will demonstrate a deep understanding of the relationship between different inference methods by showing how a significance test at a specific level relates to a corresponding confidence interval for the same set of data.