Unit 1 Lesson 11 : The Normal Distribution – Applications and Unit Review

Handout 1: The Normal Distribution – Applications and Unit Review

Unit 1, Topic 1.10: The Normal Distribution – Applications and Unit Review

Overview

This lesson applies the normal distribution to real situations, interpreting probabilities like "chance of exceeding a value." It also reviews Unit 1 concepts (variables, tables, graphs, descriptions, summaries). Context, like who the data is from, matters because it makes probabilities meaningful. For example, P(height > 190 cm) ≈ 2.5% for adults, but context like "athletes" changes the claim.

The normal distribution (bell-shaped, μ center, σ spread) helps with z-scores and areas. We’ll interpret results, reverse questions from graphs, and review to fix misconceptions (e.g., always include all features in descriptions).

Assignment:

Part 1: Guided Practice Activity

Work on your own. Use the normal distribution for heights (μ = 170 cm, σ = 10 cm). Interpret probabilities and review key concepts.

Example: Shaded curve: Area right of 180 cm (z=1, top 16%).

Tasks:

1. Interpreting Probabilities:
   • Calculate z for 180 cm and find P(height > 180 cm) using empirical rule or table (≈ 16%).
   • Write 1-2 sentences interpreting in context (e.g., "16% of adults are taller than 180 cm, useful for basketball team selection.").
   • Extra Practice: For test scores (μ = 75, σ = 10), interpret P(score < 65) ≈ 16% with context.
2. Reviewing Unit Concepts:
   • Compare two distributions (e.g., heights vs. scores: shapes, centers, variabilities).
   • Describe a shape (e.g., normal as symmetric) and calculate a summary (e.g., mean for scores).
   • Write 1-2 sentences addressing a misconception (e.g., "Descriptions must include gaps, not just peaks.").
   • Extra Practice: Review a graph from Handout 4 (histogram); describe its features fully.
3. Reversing Interpretations (Individual):
   • For a shaded normal curve (e.g., middle 68% shaded), create 2 questions it answers (e.g., "What % are within 1σ?").
   • Write 2-3 sentences justifying with context (e.g., "This shows typical range for heights in a population.").

Part 2: Independent Practice

Review Unit 1 with this mixed task for commute times (μ = 30 min, σ = 10 min). Data points: 20, 30, 40, 50 min.

Tasks:

• Interpret P(commute > 40 min) ≈ 16%, justifying a claim in context.
• Compare to a skewed distribution (e.g., from Handout 6: skewed scores); note differences in shape/center.
• Write 2-3 sentences on a misconception (e.g., "Normal assumes symmetry, unlike skewed data where median > mean.").
• Extra Activity: Invent a normal scenario (e.g., weights); create a reversed question from a shaded curve and interpret.

Homework Assignment

• Complete the AP Classroom Personal Progress Check for Unit 1. Review misconceptions (e.g., full descriptions), and preview Unit 2 by noting how two-variable data (e.g., height vs. weight) differs from one-variable to share next class.