Probability and Statistical Inference

Lesson 12

Math

Unit 8

11th Grade

Lesson 12 of 13

Objective


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

Common Core Standards


Core Standards

  • 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. Describe margin of error, or confidence interval, as an interval that has a 95% probability of including the population characteristic.
  2. Identify that the margin of error does not take into account sampling error, poorly worded questions, lack of participation, or other examples of poor study design.
  3. Describe that there are three main methods for finding the margin of error. The coefficient of 2 is the $${{z-}}$$score for two standard deviations. For other percentages of data included, use a different $${{z-}}$$score. 
    1. Doubling the standard deviation of the sample means. 
    2. Doubling the standard deviation of the sample means when you are only looking at one (or have taken only one sample. The formula for this is $${2{s\over\sqrt{n}}}$$, where s is the standard deviation of that one sample and n is the sample size. 
    3. Using the population proportion of a desired statistic by doubling the square root of the quotient of the product of the success/not success by the size of the sample, or $${2\sqrt{{\hat{p}}(1-{\hat{p}})\over n}}$$, where $${\hat{p}}$$ is the proportion of a sample of an event occurring, $$1-{\hat{p}}$$  is the proportion of a sample of an event not occurring, and n is the sample size. 
  4. Describe that the margin of error indicates the two standard deviations above or below the sample mean, thus encompassing 95% of the data. 
  5. Describe that values near the mean are more likely than other values within the margin of error, but any value within the margin of error should be expected.
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Anchor Problems


Problem 1

Flares are used to warn cars that there is an obstacle on the road. Flares are often part of cars’ emergency kits and burn for a long time.

Fred’s Flare Company used to make flares that burned for 100 minutes on average. Fred changed the chemical formula in his flares, and he thinks that his new formula will burn for slightly over 120 minutes. Fred wants to make the claim on his packaging that his flares “Now burn for over 2 hours on average!!” 

To make this claim, Fred needs to back it up with data. He decides that performing some random sampling is the best way to do this. Since he runs a 24-hour production operation, he decides that taking a sample each hour is the easiest way to do this. 

Fred has tried two different ways of sampling, as shown below.

Sample size: 20 flares

24 total samples (one per hour of production)

Sample size: 100 flares

24 total samples (one per hour of production)

If we further extrapolate that Fred continues to take samples for 30 days in the same manner as above, he charts his results in histograms shown below. 

The margin of error is the largest expected difference between the estimate of a population parameter and the actual population parameter. 

What would you guess is the mean and the margin of error of burn time of Fred’s flares (with the new and improved formula)?

Guiding Questions

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References

Illustrative Mathematics Fred's Flare Formula

Fred's Flare Formula, accessed on June 5, 2018, 4:13 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem 2

The margin of error can also be calculated in by $${ME=2\sqrt{{\hat{p}}(1-{\hat{p}})\over n}}$$, where $${\hat{p}}$$ is the proportion of a sample of an event occurring, $$1-{\hat{p}}$$ is the proportion of a sample of an event not occurring, $$n$$ is the sample size, and $${\sigma}$$ is the sample standard deviation. 

Suppose you draw a random sample of 36 chips from a mystery bag and find 20 red chips. Find: 

  • The sample proportion of red chips
  • The standard error (standard deviation from the mean)
  • The margin of error 

Guiding Questions

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References

EngageNY Mathematics Algebra II > Module 4 > Topic C > Lesson 17Exercise #8

Algebra II > Module 4 > Topic C > Lesson 17 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..

Modified by Fishtank Learning, Inc.

Target Task


  1. Find the estimated margin of error when estimating the proportion of red chips in a mystery bag if 18 red chips were drawn from the bag in a random sample of 50 chips. 
  2. Explain what your answer to Problem 1 tells you about the number of red chips in the mystery bag. 
  3. How could you decrease your margin of error? Explain why this works. 
  4. The students doing a project collected 50 random samples of 10 students each and calculated the sample means. The standard deviation of their distribution of the 50 sample means was 0.6. Based on this standard deviation, what is the margin of error for their sample mean estimate? Why can you just double the standard deviation to find the margin of error with 95% confidence?

References

EngageNY Mathematics Algebra II > Module 4 > Topic C > Lesson 20Exit Ticket

Algebra II > Module 4 > Topic C > Lesson 20 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..

Modified by Fishtank Learning, Inc.
EngageNY Mathematics Algebra II > Module 4 > Topic C > Lesson 17Exit Ticket

Algebra II > Module 4 > Topic C > Lesson 17 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..

Modified by Fishtank Learning, Inc.

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.

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Lesson 11

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Lesson 13

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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