# MA.912.DP.1.4

Estimate a population total, mean or percentage using data from a sample survey; develop a margin of error through the use of simulation.

### Examples

Algebra 1 Example: Based on a survey of 100 households in Twin Lakes, the newspaper reports that the average number of televisions per household is 3.5 with a margin of error of ±0.6. The actual population mean can be estimated to be between 2.9 and 4.1 television per household. Since there are 5,500 households in Twin Lakes the estimated number of televisions is between 15,950 and 22,550.

### Clarifications

Clarification 1: Within the Algebra 1 course, the margin of error will be given.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Data Analysis and Probability
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Random Sampling
• Simulation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students solved real-world problems involving percentages and they calculated means of numerical data sets. In grades 7 and 8, students explored the relationship between theoretical probabilities and experimental probabilities. In Algebra I, students use means and percentages from data obtained from statistical experiments to make predictions about actual means and percentages in populations and they relate the experimental data to actual values through margins of error. In later courses, students will work more formally with margins of error and use the normal distribution to estimate population percentages.
• In Algebra I, students are not expected to master the skill of estimating a margin of error using simulation. But instruction may include such an activity, to allow students to experience the need for a margin of error whenever a mean or population percentage is estimated using data.
• Instruction includes introducing examples from the media of reports of population means and percentages that include margins of error.
• Instruction includes the understanding that an actual population mean or percentage is not necessarily within the margin of error of the estimated mean or percentage. Even if the data has been carefully collected, these estimated quantities are only within the margin of error with a “high degree of confidence.” If the data has not been carefully collected, or if the situation has changed significantly since the data were collected (as is often the case with election polls), the margin of error may not be very meaningful.

### Common Misconceptions or Errors

• When asked for population totals, students may forget to complete the final calculation with the margin of error and only report the mean or percentage as their final answer.
• Students may confuse three percentages: the estimated population percentage, the margin of error expressed as a percentage and the actual population percentage. Students can use a number line to visualize how the first two quantities determine an interval that is likely to contain the third quantity.

### Strategies to Support Tiered Instruction

• Teacher provides a list of definitions to students to eliminate misunderstandings that may be caused by the key terms. Because it is important that students understand the meanings of the terms when interpreting sample surveys, this should be an entry in a math journal.
• Instruction includes providing multiple scenarios and asking for identification of the key terms associated with the percentages given in the context.
• Teacher models the use of a number line to visualize how the estimated population percentage and the margin of error determine an interval that is likely to contain the actual population percentage.

• A packaging company prints 100,000 boxes per day. They determine that their printing machines are making a mistake on an average of 0.11% of boxes per day with a margin of error of ±0.01%. How many boxes will need to be recycled due to a printing error in one week?

### Instructional Items

Instructional Item 1
• Based on a survey of 150 students at Long Lake High School, the average number of hours spent on social media per week is 30.5 with a margin of error of 3.25.
• Part A. Give a range of values, based on this data, for the actual mean numbers of hours spent on social media.
• Part B. If there are a total of 723 students at the high school, give a range of values for the total number of weekly hours spent by all students.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1210300: Probability and Statistics Honors (Specifically in versions: 2014 - 2015, 2015 - 2019, 2019 - 2022, 2022 and beyond (current))
7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.DP.1.AP.4: Given the mean or percentage and the margin of error from a sample survey, identify a population total.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Lesson Plans

Standard Deviation and the Normal Curve in Kahoot!:

In this three-day lesson, students learn about standard deviation, the normal curve, and how they are applied. Your students will be engaged and learning when they collect and analyze data using a free Kahoot! quiz.

Type: Lesson Plan

How do we measure success?:

Students will use the normal distribution to estimate population percentages and calculate the values that fall within one, two, and three standard deviations of the mean. Students use statistics and a normal distribution to determine how well a participant performed in a math competition.

Type: Lesson Plan

The Cereal Prize Estimation:

How many boxes of cereal would you have to purchase to win all six prizes?

This lesson uses class data collected through simulations to allow students to answer this question. Students simulate purchasing cereal boxes and create a t-confidence interval with their data to determine how many boxes they can expect to buy.

Type: Lesson Plan

## Perspectives Video: Experts

Statistical Sampling Results in setting Legal Catch Rate:

Fish Ecologist, Dean Grubbs, discusses how using statistical sampling can help determine legal catch rates for fish that may be endangered.

Type: Perspectives Video: Expert

Mathematically Modeling Hurricanes:

<p>Entrepreneur and meteorologist Mark Powell discusses the need for statistics in his mathematical modeling program to help better understand hurricanes.</p>

Type: Perspectives Video: Expert

Statistical Inferences and Confidence Intervals :

<p>Florida State University Counseling Psychologist&nbsp;discusses how he uses confidence intervals to make inferences on college students' experiences on campus based on a sample of students.</p>

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Nestle Waters & Statistical Analysis:

<p>Hydrogeologist&nbsp;from Nestle Waters discusses the importance&nbsp;of statistical tests in monitoring&nbsp;sustainability and in maintaining consistent&nbsp;water quality in bottled water.</p>

Type: Perspectives Video: Professional/Enthusiast

Fishery Independent vs Dependent Sampling Methods for Fishery Management:

<p>NOAA&nbsp;Scientist Doug Devries discusses the differences between fishery independent surveys and fishery independent surveys. &nbsp;Discussion&nbsp;includes trap sampling as well as camera sampling. Using&nbsp;graphs to show changes in population of red snapper.</p>

Type: Perspectives Video: Professional/Enthusiast

Camera versus Trap Sampling: Improving how NOAA Samples Fish :

<p>Underwater sampling with cameras has made fishery management more accurate for NOAA&nbsp;scientists.</p>

Type: Perspectives Video: Professional/Enthusiast

Sampling Strategies for Ecology Research in the Intertidal Zone:

<p>Will Ryan describes methods for collecting multiple random samples of anemones in coastal marine environments.</p>

Type: Perspectives Video: Professional/Enthusiast

## Perspectives Video: Teaching Ideas

Pitfall Trap Classroom Activity:

<p>Patrick&nbsp;Milligan&nbsp;shares a teaching idea for collecting insect samples.</p>

Type: Perspectives Video: Teaching Idea

Ecological Sampling Methods and Population Density:

Dr. David McNutt explains how a simple do-it-yourself quadrat and a transect can be used for ecological sampling to estimate population density in a given area.