Objective
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.
Common Core Standards
Core Standards
The core standards covered in this lesson
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
7.SP.A.1
Statistics and Probability
7.SP.A.1— Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
Criteria for Success
The essential concepts students need to demonstrate or understand to achieve the lesson objective
- Describe the potential bias of one sample and mitigate bias by using multiple random samples.
- Distinguish between a population characteristic and a sample statistic, defining the characteristic as what is true about a population and a sample statistic as an estimate of that characteristic based on a smaller, representative group.
- Calculate sample statistics such as the sample mean and sample variability between random samples.
- Describe the variability of multiple samples and use this information to assess the accuracy in the samples to predict a population characteristic.
Tips for Teachers
Suggestions for teachers to help them teach this lesson
- For Anchor Problem #1, the discussion and synthesis will have to be fairly teacher directed. The idea that you could take multiple random samples and compare the means will a big step for many students. Projecting the applet and completing the suggestions that students have for how the sampling size and number of samples drawn in real time in front of the class will be key to students intuitively understanding this concept. An alternative is to give students access to technology and have them do this in pairs or on their own.
- The post, "Seeing Stars with Random Sampling" on MathCoachBlogis a great resource for understanding and finding some good activities/understanding on these topics.
- Sample Size Calculator by Creative Research Systems isa good calculator for sample size that you can use to gauge student understanding of sample size.
- This lesson borders on the “central limit theorem” through Anchor Problem #2. You do not need to give the students this language.
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Anchor Problems
Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Problem 1
Sam is a penny collector and has 1,000 pennies of various ages dating back more than 50 years.
Because Sam has kept careful records, he knows the age of each of the pennies and has a set of descriptive statistics on the entire collection. Sam has numbered each of his pennies with an identifying code. He entertains himself one day by randomly selecting 10 ID numbers from his collection and comparing the sample to the population. Below is a histogram of the population data and a dot plot of the sample data.
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Sam is baffled. He KNOWS that he actually randomly sampled the population of pennies. He also knows that random sampling is supposed to produce a sample that represents the population.
How can you explain to Sam why his sample does not match the population?
Guiding Questions
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References
Rossman/Chance Resource:One Variable with Sampling Applet—Used to create graphs
One Variable with Sampling Applet made available by Rossman/Chance.© 2012-2016 Beth and Frank Chance. Accessed June 5, 2018, 3:52 p.m..
Problem 2
Sam has decided on two methods, described below.
Option A: A larger sample; draw the sample only once
Option B: A smaller sample; draw the sample multiple times
Which of these do you think gives Sam more accurate data? Explain your reasoning.
Guiding Questions
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References
Rossman/Chance Resource:One Variable with Sampling Applet—Used to create graphs
One Variable with Sampling Applet made available by Rossman/Chance.© 2012-2016 Beth and Frank Chance. Accessed June 5, 2018, 3:52 p.m..
Target Task
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
Below are three dot plots of the proportion of tails in 20, 60, or 120 simulated slips of a coin. The mean and standard deviation of the sample proportions are also shown for each of the three dot plots. Match each dot plot with the appropriate number of flips. Clearly explain how you matched the plots with the number of simulated flips.
References
EngageNY Mathematics Algebra II > Module 4 > Topic C > Lesson 15—Exit Ticket
Algebra II > Module 4 > Topic C > Lesson 15 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 USlicense.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.
- Two applets are suggested below for students to describe and practice this idea.
- Simulation-Based and Exact One Proportion Inference(Coin Tosses) by Rossman/Chance
- Reeses Piecesby Rossman/Chance
- Illustrative Mathematics Guess the Probability
- EngageNY Mathematics Algebra II > Module 4 > Topic C > Lesson 15—Problem Set and Example #1 (This is excellent for students assessing the validity of a claim using one of the applets)
- Mathematics Vision Project: Secondary Mathematics Three Module 8: Statistics (Teacher Notes)—Lesson 8.7
Lesson 10
Lesson 12
