Probability and Statistical Inference

Lesson 8

Math

Unit 8

11th Grade

Lesson 8 of 13

Objective


Derive and calculate population percentages based on a normal distribution of data.

Common Core Standards


Core Standards

  • S.IC.A.2 — Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
  • S.IC.B.4 — Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Foundational Standards

  • S.ID.A.4

Criteria for Success


  1. Calculate the proportion of data that falls within one standard deviation, two standard deviations, and three standard deviations from the mean. 
  2. Recognize that the proportions of data one and two standard deviations from the mean are not exact when the distribution is not exactly normal but still give a useful approximation. 
  3. Identify population proportions based on a normal curve and assign probabilities given the standard deviation and the mean.

Tips for Teachers


  • Students studied standard deviation and some population proportions based on a normal distribution in Algebra 1 but will need a refresher. This (and the previous) lesson serves as that refresher. 
  • Anchor Problem #1 starts the work of having students identify the population proportions associated with standard deviations from the mean. Students should be given (or record) the normal curve with standard deviations broken into at least whole standard deviations, if not half standard deviations. 
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Anchor Problems

25-30 minutes


Problem 1

The mean of this distribution is approximately 8, and the standard deviation is approximately 2. The number across the top describes how many balls landed in each numbered bin. There were 5,498 balls that were put in the Plinko game. 

What percent of the balls fall between one standard deviation less and one standard deviation more than the mean?

What percent of the balls fall between two standard deviations less and two standard deviations more than the mean?

Guiding Questions

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References

PhET Resource: Plinko Probability

PhET Interactive Simulations
University of Colorado Boulder
https://phet.colorado.edu

Problem 2

Below is a normal distribution that shows the height of 8-year-old boys in inches. The relative frequency of each height is shown along the $${y-}$$axis, and the height, in inches, is shown along the $${x-}$$axis. 

The mean is 50 and the standard deviation is 2.

  • Draw vertical lines to show one and two standard deviations from the mean. 
  • If you are taking kids on a field trip and a ride won’t let students on unless they are 48 inches or taller, what percent of the 8-year-old boys would you expect to be able to ride the ride?

Guiding Questions

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Target Task

5-10 minutes


The histogram below shows the distribution of heights (to the nearest inch) of 1,000 young women.

  1. What does the width of each bar represent? What does the height of each bar represent?
  2. The mean of the distribution of women's heights is 64.6 in., and the standard deviation is 2.75 in. Interpret the mean and standard deviation in this context.

References

EngageNY Mathematics Algebra II > Module 4 > Topic B > Lesson 9Exit Ticket, Question #1 and #2

Algebra II > Module 4 > Topic B > Lesson 9 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Consider making histograms using the simulators to generate new problems
  • Include non-examples that are not normally distributed

Next

Use $${z-}$$scores to identify population percentiles.

Lesson 9
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Probability

Topic B: The Normal Distribution

Topic C: Statistical Inferences and Conclusions

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