Lesson 8 | Probability and Statistical Inference | 11th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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?

High School — Statistics and Probability

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.

High School — Statistics and Probability

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

The foundational standards covered in this lesson

S.ID.A.4

High School — Statistics and Probability

S.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. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Calculate the proportion of data that falls within one standard deviation, two standard deviations, and three standard deviations from the mean.
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.
Identify population proportions based on a normal curve and assign probabilities given the standard deviation and the mean.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

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

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

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

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

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

What does the width of each bar represent? What does the height of each bar represent?
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 9 — Exit 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

Mathematics Vision Project: Secondary Mathematics Three Module 8: Statistics (Teacher Notes) — Lesson 8.2
Teaching Statistics #Made4Math - Normal Distribution Cards — (Use these for students to calculate the population percentages)

Lesson 7

Lesson 9

Lesson Map

Topic A: Probability

Determine probabilities of mutually exclusive events.

S.CP.A.1

Determine probabilities of events that are not mutually exclusive.

S.CP.A.1 S.CP.B.6 S.CP.B.7

Calculate conditional probabilities.

S.CP.A.3

Determine when events are independent and describe independent events using everday language.

S.CP.A.2 S.CP.A.3 S.CP.A.5

Calculate relative frequencies in two-way tables to analyze data and determine independence.

S.CP.A.4

Use conditional probability to make decisions about medical testing.

S.CP.A.2 S.CP.A.3

Topic B: The Normal Distribution

Describe the center, shape, and spread of distributions by reasoning visually about the mean, standard deviation, and shape of a histogram.

S.IC.A.1

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

S.IC.A.2 S.IC.B.4

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

S.IC.B.4

Topic C: Statistical Inferences and Conclusions

Describe and compare statistical study methods.

S.IC.B.3 S.IC.B.6

Use multiple random samples to estimate a population mean or proportion and verify the validity of the sampling method by analyzing the means and standard errors of samples.

S.IC.B.4

Calculate and describe the margin of error in context and use larger sample sizes to minimize the margin of error.

S.IC.B.4

Compare two treatments in experimental data and determine if the difference between the two treatments is significant.

S.IC.B.5

Probability and Statistical Inference

Objective

Common Core Standards

Core Standards

High School — Statistics and Probability

High School — Statistics and Probability

Foundational Standards

High School — Statistics and Probability

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Target Task

References

Additional Practice

Lesson Map

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