Lesson 5: 7.5 The Central Limit Theorem
Duration of Days: 3
Lesson Objective
Students will be able to state the Central Limit Theorem and understand that for a large enough sample size, the sampling distribution of the sample mean is approximately Normal, regardless of the shape of the population distribution.
If the population of household incomes is heavily right-skewed (most people earn a little, a few earn a lot), why does the distribution of average incomes look like a symmetric bell curve?
What is the "magic number" for sample size that allows us to safely assume Normality for means?
How does the CLT allow us to calculate probabilities for populations whose shapes we don't even know?
Central Limit Theorem (CLT)
Normal Condition for Means
Large Sample Condition
HSS-ID.A.4: Use the mean and standard deviation of a data set to fit it to a normal distribution.
HSS-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process.
The SAT often assesses the validity of conclusions. The CLT is the justification for why a sample mean from a large enough sample is a reliable estimate. While you won't name the "CLT" on the SAT, you will use its logic to defend why a large sample provides a more predictable result than a small one.
DOK 2: Determine if the Central Limit Theorem applies to a given scenario based on the sample size and population shape.
DOK 3: Explain the difference between the "Large Counts Condition" (for proportions) and the "Central Limit Theorem" (for means).
The Problem: The time it takes for a certain chemical reaction to occur is highly skewed to the right, with population mean= 40 minutes and population standard deviation= 15 minutes.
Task 1: Can you calculate the probability that a single random reaction takes less than 35 minutes?
Task 2: You monitor a random sample of n = 45 reactions. Calculate the probability that the average time of these reactions is less than 35 minutes.
"The Population Becomes Normal": This is the most common error. Students think that if you take a big sample, the histogram of the data becomes Normal. False. The population stays skewed; only the sampling distribution (the map of all possible averages) becomes Normal.
The "N" Myth: Students think that if the population is already Normal, you still need the sample to be greater than or equal to 30. If the population is Normal, the sampling distribution is Normal for any sample size.
Support: "The Blur Effect." Use an image of a pixelated photo. One pixel (a small sample) tells you nothing about the shape. But as you add more pixels, the "average" view of the photo becomes clear and smooth.
Extension: Use a "Rice Paddy" or "Penny" applet online. Have students draw samples from a "Custom" distribution (where they draw a weird shape) and watch the sampling distribution turn into a bell curve in real-time.
Graphic Organizer: Create a "Conditions Table" for students to keep in their notebooks that separates the rules for Proportions and Means.
Teacher assigns examples from textbook and other resources.
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