Lesson 5: 2.5 Normal Distributions: Finding Areas from Values
Duration of Days: 3
Lesson Objective
Students will be able to calculate the proportion of observations in a specified interval of a Normal distribution by standardizing (calculating z-scores) and using Table A or technology.
If Table A only gives us the area to the left, how can we use subtraction to find the area to the right or the area between two values?
Why is the probability of a single, exact value considered to be 0 in a continuous density curve?
How can we use a z-score to "translate" a real-world measurement into a common language (standard units)?
Standard Normal Distribution
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Standardized Proportion
HSS-ID.A.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
The SAT frequently asks students to interpret the "proportion" of a population that falls within a certain range. Understanding that "Area = Proportion = Probability" is a fundamental concept for the Problem Solving and Data Analysis section.
This section focuses on the forward direction of Normal calculations. Students learn the procedural flow: Observation --> z-score --> Area.
The Problem: The distribution of heights of young women aged 18 to 24 is approximately Normal with mean of 64.5 inches and standard deviation of 2.5 inches.
Task: What percent of young women are between 62 and 67 inches tall? Show the z-score calculations and a sketch of the shaded curve.
The "Between" Error: Students sometimes calculate two z-scores and then subtract the z-scores themselves rather than subtracting the areas associated with those z-scores.
Directionality: Forgetting to subtract from 1 when looking for the area to the right ("at least" or "more than").
Support: Use a physical "shading" activity. Give students several printed Normal curves and have them shade the region described by phrases like "at most," "more than," and "at least" before doing any math.
Extension: Ask students to calculate the area beyond 3 standard deviations to see why the Empirical Rule says "99.7%" rather than 100%.
Teacher assigns examples from the textbook and other resources.
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