HSS.IC.B.4Common CoreMathStatistics and ProbabilityGrades 9-12
HSS.IC.B.4: Estimating a Population Mean or Proportion with a Margin of Error
In plain English: HSS.IC.B.4 is the Common Core statistics standard that asks students to use data from a random sample survey to estimate a population mean or proportion, and to find a margin of error by simulating many random samples. The key idea is that sample results vary, and the simulation shows how far an estimate is likely to be from the truth. It is usually taught in Algebra II or an introductory statistics course.
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.
Common Core State Standards for Mathematics · Domain: Making Inferences and Justifying Conclusions (IC) · Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies Also written as HSS-IC.B.4 or S-IC.4 · Official standard
Students use data from a random sample survey to estimate an unknown population value: a proportion (the fraction of voters who support a bond) or a mean (the average number of books students read). The sample proportion p̂ and the sample mean x̄ are the point estimates. The lesson then asks the question every poll must answer: how far from the truth could this estimate be?
Students answer it with simulation. They build a model population that matches the sample, draw hundreds of random samples of the same size from it, and look at how much the sample results vary. About 95% of the simulated results fall within two standard deviations of the center, and that distance is the margin of error. Students also see that larger samples give smaller margins of error, and they interpret an estimate written as "estimate ± margin of error" in context.
Learning Objectives
By the end of this lesson, students will be able to:
Use data from a random sample survey to compute a sample proportion or sample mean as an estimate of the population value
Build a simulation model for random sampling and use it to generate many samples of the same size
Find a margin of error from the spread of simulated sample results, using the middle 95% or about two standard deviations
Interpret an estimate with its margin of error as a range of plausible population values, in context
Explain how sample size affects the margin of error
Prior Knowledge Required
Students should already be comfortable with:
Using a random sample to draw inferences about a population 7.SP.A.2
Statistics as inference about a population from a random sample HSS.IC.A.1
Computing a mean from a list or frequency table and describing spread with standard deviation HSS.ID.A.2
Show one survey result and ask students to answer two questions on a sticky note:
Warm-Up Prompt
"A random sample of 50 of the 1,200 students at a school were asked whether they would buy a school hoodie for $30. Eighteen said yes. (1) About how many students in the whole school would buy one? (2) If another class took its own random sample of 50, would it also get exactly 18 yes answers?"
Collect answers. For (1), the sample proportion is 18/50 = 0.36, and 0.36 × 1,200 = 432 students. For (2), many students say "probably not", which is the point of the lesson: a different random sample gives a different result. Ask how far off 432 might be. Write the class's guesses on the board and return to them at the end.
Direct Instruction20-25 minutes
Part 1: Estimates from a sample survey. A parameter is a number that describes the population (p for a proportion, μ for a mean). It is usually unknown. A statistic is the matching number computed from the sample (p̂ or x̄), and we use it as the estimate. The estimate is only trustworthy if the sample was chosen at random, so that every member of the population had a chance to be picked.
Part 2: A margin of error from simulation. Show the steps below, then work the examples.
Build a model population that matches the sample. For a proportion, use a random number generator where each draw is "yes" with probability p̂. For a mean, put every sample value in a bag and draw from it with replacement.
Simulate one random sample of the same size n as the real survey, and record its proportion or mean.
Repeat at least 100 times (200 or more with technology) and plot the simulated results.
Measure the spread: find the interval that holds the middle 95% of the simulated results, or compute their standard deviation.
Report the margin of error as about half the width of the middle 95%, which is close to 2 × (standard deviation of the simulated results). Write the estimate as estimate ± margin of error.
Books read for pleasure in the last 3 months, random sample of 40 students
Books read
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Number of students
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1
Estimating a proportion
A random sample of 400 registered voters in a city: 228 support a library bond.
Equation: p̂ = 228/400 = 0.57, so an estimated 57% of the city's voters support the bond
Margin of error for a proportion, by simulation
200 simulated polls of 400 voters each, from a model where each voter says yes with probability 0.57 (Diagram 1). The simulated proportions have a standard deviation of about 0.023, and the middle 95% runs from about 0.53 to 0.62.
Equation: Margin of error ≈ 2 × 0.023 ≈ 0.05, so 0.57 ± 0.05: between 52% and 62% support
Estimating a mean
The table above: a random sample of 40 of the students at a school, asked how many books they read for pleasure in the last 3 months.
Put the 40 values in a bag, draw 40 with replacement, and record the mean. Repeat 200 times (Diagram 2). The simulated means have a standard deviation of about 0.25.
Equation: Margin of error ≈ 2 × 0.25 = 0.5, so 2.2 ± 0.5: between 1.7 and 2.7 books
Effect of sample size
Simulate polls from a model with p = 0.4, first with samples of 150, then with samples of 600.
Equation: Standard deviation ≈ 0.04 for n = 150 and ≈ 0.02 for n = 600: margins of error about 0.08 and 0.04. Four times the sample size gives half the margin of error
Stress three points. First, the margin of error measures only the chance variation from random sampling; it does not fix a biased sample. Second, the number of simulated samples is not the sample size: running more simulations makes the picture of the spread clearer, but only a larger survey makes the margin of error smaller. Third, "between 52% and 62%" is a range of plausible values for the population proportion, not a guarantee. For a quick check of a simulation, the spread of p̂ is close to √(p̂(1 - p̂)/n): √(0.57 × 0.43/400) ≈ 0.025, close to the simulated 0.023.
Guided Practice15-20 minutes
Pairs work two problems, then compare with another pair. (1) A random sample of 500 residents of a county: 160 say they recycle every week. Find p̂ (0.32). A simulation of 200 samples of 500 from a model with p = 0.32 gives simulated proportions with a standard deviation of 0.021. Find the margin of error (about 0.04) and the interval (0.28 to 0.36). (2) A random sample of 36 fans at a stadium spent a mean of $18.50 on food. Resampling the 36 values 200 times gives simulated means with a standard deviation of $1.10. Find the margin of error (about $2.20) and the interval ($16.30 to $20.70). For each, pairs write one sentence that interprets the interval in context. Listen for students who divide the standard deviation by 2 instead of doubling it, and for students who say the interval contains "95% of the fans".
Independent Practice15 minutes
Students work alone. (1) A random sample of 80 of the 900 seniors in a district: 48 plan to attend a four-year college. Estimate the proportion (0.60) and the number of seniors (540). (2) Using a spreadsheet, each student simulates 100 samples of 80 from a model with p = 0.60, with the formula =IF(RAND()<0.6,1,0) copied into 80 cells and averaged, and reports the standard deviation of the 100 simulated proportions (close to 0.055) and a margin of error (about 0.11). (3) Students compare their margin of error with a neighbor's and explain why they differ slightly but not much.
Closure5-10 minutes
Exit ticket: A news story says 41% of adults in a state plan to buy an electric car, with a margin of error of 5 percentage points, from a random sample. (1) Write the interval of plausible values. (Answer: 36% to 46%.) (2) Could the true percent be 50%? Explain. (Not plausible: 50% is outside the interval.) (3) Name one way to make the margin of error smaller. Return to the warm-up guesses about how far 432 hoodies might be off.
Differentiation Strategies
For Struggling Students
Give a partly completed simulation table with columns for sample number, number of yes answers and sample proportion, so students focus on the idea rather than the setup
Start with a small physical simulation (samples of 10 from the bead bag) before moving to technology
Provide a sentence frame: "We estimate that ___ of all ___ ___, give or take ___."
For Advanced Students
Ask students to simulate for n = 25, 100, 400 and 1,600 with the same p, plot margin of error against n, and find the pattern (it halves each time n is multiplied by 4)
Ask students to compare the simulated standard deviation with √(p(1 - p)/n) and explain why p = 0.5 gives the largest margin of error for a given n
Ask students to find the smallest sample size that gives a margin of error of about 0.03 when p is near 0.5, first by simulation and then with the formula
Assessment Guidance
What to Look For
Check that students name the statistic (p̂ or x̄) and the parameter it estimates, and that every estimate comes from a random sample. In simulations, look for a model that matches the real survey: the same sample size n and a success probability equal to p̂ (or a bag holding the sample values for a mean). Students should report the margin of error from the spread of the simulated results, not from the number of simulations, and should interpret "estimate ± margin of error" as a range of plausible values for the population, not as the range of individual responses.
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Classroom Activities
3 Activities
1
Bead Bag Polls
20 minGroups of 3
Each group gets a paper bag holding 300 beads: 120 red and 180 white, so the true proportion of red beads is 0.40. Students do not know this. They take random samples, estimate the proportion of red beads, and use the class's results to see how much sample proportions vary.
Procedure
Shake the bag, draw 25 beads without looking, record the number of red beads, return the beads and shake again
Each group repeats this 3 times and computes p̂ for each sample
Each group puts its 3 sample proportions as sticky notes on a class dot plot with a scale from 0 to 1
As a class, find the interval that holds the middle 95% of the dot plot, and use half its width as a margin of error
Reveal that p = 0.40. Count how many groups' intervals p̂ ± margin of error contain 0.40
Discussion Questions
Why did the groups get different values of p̂ from the same bag?
With samples of 25, the margin of error is close to 0.2. What would you change to make it smaller?
Would the results change if the bag held 3,000 beads in the same ratio? Why or why not?
Modification for Distance Learning
Replace the bag with a random number generator: a number from 1 to 300 counts as red if it is 120 or less. Students post their proportions in a shared spreadsheet that builds the class dot plot.
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Poll Simulator Sample Sizes
20 minPairs
Pairs use a spreadsheet or a free online simulation tool to simulate a poll whose sample proportion was p̂ = 0.45, and compare the margin of error for three sample sizes. The goal is to see why pollsters survey hundreds of people and not dozens.
Procedure
Set up a model where each simulated person answers yes with probability 0.45
Simulate 200 samples of size 50, record the 200 sample proportions, and find their standard deviation and the middle 95%
Repeat with samples of size 200 and then 800
Fill in a table: sample size, standard deviation of p̂, margin of error. Expected values are close to 0.07, 0.035 and 0.018 for the standard deviations, so margins of error of about 0.14, 0.07 and 0.035
Discussion Questions
What happens to the margin of error each time the sample size is multiplied by 4?
A pollster wants a margin of error of about 0.035. About how many people should be surveyed?
Does running 1,000 simulations instead of 200 change the margin of error much? Why not?
Challenge Variation
Pairs repeat the sample of 200 with p = 0.10 and p = 0.90 and compare with p = 0.45. They explain why the margin of error is largest when p is close to 0.5.
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Resampling Sleep Hours
25 minPairs
Pairs estimate the mean hours of sleep on a school night for the students at a school, from an invented random sample of 20 students, and develop a margin of error by resampling. This activity covers the population mean case of the standard.
Setup
Invented sample of 20 students (hours of sleep): 8, 7, 6.5, 9, 7.5, 8.5, 6, 7.5, 8, 7, 9.5, 6.5, 8, 7.5, 5.5, 8.5, 7, 8, 6.5, 7.5
Number the values 01 to 20. Each pair needs a random digit table or a random number generator
Procedure
Compute the sample mean (7.475, about 7.5 hours)
Draw 20 two-digit numbers from 01 to 20 with replacement (skip 00 and 21-99), write down the matching values, and find their mean. This is one simulated sample from the model population
Each pair makes 5 simulated means and adds them to a class dot plot; the class then has about 60 to 75 means
Find the standard deviation of the class's simulated means (expect about 0.2 to 0.25) and a margin of error (about 0.4 to 0.5 hours)
Write the estimate as 7.5 ± the class margin of error and interpret it in context
Discussion Questions
Why do we draw with replacement? What would happen without replacement?
The individual values range from 5.5 to 9.5 hours, but the simulated means stay much closer to 7.5. Why?
The estimate is about the whole school. What must be true about how the 20 students were chosen?
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Diagrams & Visual Aids
2 diagrams
Diagram 1: Simulated Sample Proportions and the Margin of Error
Histogram of 200 simulated sample proportions, each from a random sample of 400 drawn from a model in which 57% of voters support the bond. Bars are 0.01 wide and drawn to scale. The middle 95% of the results falls within about 0.05 of the center, so the margin of error for the survey estimate 0.57 is about 0.05.
Diagram 2: Simulated Sample Means for Books Read
Histogram of 200 simulated sample means, each the mean of 40 values drawn with replacement from the survey data in the table (sample mean 87/40 = 2.175 books, the dashed line). The shaded band is the rounded interval 2.2 ± 0.5 and holds 190 of the 200 simulated means. Bars are 0.1 book wide and drawn to scale. The simulated means cluster much more tightly than the individual answers, which range from 0 to 6 books.
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Homework Assignment
~30 min
HSS.IC.B.4 Homework: Estimates and Margins of Error
Directions: Show all work. For every estimate, name the statistic you computed and the population value it estimates. Round proportions to two decimal places. Interpret every margin of error in a sentence that uses the context.
Part 1: Estimating from a Sample Survey (Problems 1-3)
A school district randomly selects 250 of its 3,000 high school students. Of these, 95 say they walk or bike to school. (a) Estimate the proportion of all the district's high school students who walk or bike. (b) Estimate the number of students who walk or bike.
A library randomly selects 10 of its card holders and asks how many books each checked out last month. The answers are 3, 0, 5, 2, 8, 1, 4, 2, 6, 4. Estimate the mean number of books checked out per card holder last month, and explain why a second random sample would probably give a different estimate.
A town wants to estimate the proportion of households that would pay for curbside composting. Plan A posts a link on the town's social media page and gets 900 responses. Plan B randomly selects 200 addresses and visits them. Which plan's estimate should the town trust more? Explain in terms of how the sample was chosen.
Part 2: Margin of Error from Simulation (Problems 4-6)
For the district in Problem 1, a student simulates 100 random samples of 250 students from a model in which each student walks or bikes with probability equal to your estimate from Problem 1(a). The 100 simulated proportions have a standard deviation of 0.031. (a) Find the margin of error. (b) Write the interval of plausible values for the population proportion. (c) Would it be surprising to learn that 45% of all the district's students walk or bike? Explain.
A random sample of 45 students at a college reports a mean of 5.2 hours of exercise per week. The 45 values are put in a model population and 200 samples of 45 are drawn with replacement. The simulated means have a standard deviation of 0.35 hours. Find the margin of error, write the interval, and interpret it.
A class simulates 100 samples of size 64 from a model with p = 0.20 and finds that the simulated proportions have a standard deviation of 0.05. (a) Predict the standard deviation and the margin of error for samples of size 576. (b) Describe the simulation you would run to check your prediction. (c) A classmate says that running 900 simulations instead of 100 would also cut the margin of error to a third. Is the classmate right? Explain.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Point Estimates
Correct p̂ or x̄, named, with the parameter it estimates
Correct value, but statistic or parameter not named
Missing or incorrect
Simulation Model
Model matches the survey (same n, probability p̂ or the sample values) and is repeated many times
Model has one mismatch, such as the wrong sample size
No workable model
Margin of Error
Found from the spread of simulated results and written as estimate ± margin
Correct value with an arithmetic slip or missing interval
Missing or based on the number of simulations
Interpretation
Interval interpreted in context as plausible population values, sample-size effect explained
Interpretation vague or partly incorrect
No interpretation
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Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A random sample of 320 residents of a town is asked about a new park; 208 are in favor. What is the best estimate of the proportion of all residents who favor the park?
Answer: C
The sample proportion is p̂ = 208/320 = 0.65, and it is the estimate of the population proportion. Choice B is the proportion who are not in favor (112/320). Choice A divides by 1,000 instead of by the sample size, and D uses the sample size itself as if it were a proportion.
Question 2 of 20 · Multiple Choice
A random sample of 8 employees at a company reports these commute times, in minutes: 12, 25, 18, 30, 22, 15, 28, 26. What is the estimate of the mean commute time for all employees?
Answer: A
The sum is 176 minutes, and 176/8 = 22 minutes. Choice B is the median (the mean of 22 and 25 after sorting), which is a different measure of center. Choice D divides the sum by 7 instead of 8.
Question 3 of 20 · Multiple Choice
A random sample of 150 of the 1,800 students at a school finds that 42 have a part-time job. About how many students at the school have a part-time job?
Answer: B
p̂ = 42/150 = 0.28, and 0.28 × 1,800 = 504 students. Choice A reports only the sample count. Choice C estimates the students without a job (0.72 × 1,800). Choice D multiplies 0.28 by 1,000 instead of by the population size.
Question 4 of 20 · Multiple Choice
Why do statisticians simulate many random samples from a model population when they already have one real sample?
Answer: A
The simulated samples show the chance variation that random sampling produces, and the width of that variation is the margin of error. Choice C confuses simulated samples with real data: simulation never adds new information about the population. Choice D is wrong because a simulation assumes random sampling and cannot detect a biased sample.
Question 5 of 20 · Multiple Choice
A pollster simulates 100 samples of 500 people from a model with p = 0.44. The simulated proportions have a standard deviation of 0.022. What is the approximate margin of error?
Answer: D
The margin of error is about 2 standard deviations of the simulated results: 2 × 0.022 = 0.044. Choice A uses one standard deviation, which covers only about two thirds of the simulated results. Choice B halves the standard deviation instead of doubling it.
Question 6 of 20 · Multiple Choice
A survey's sample proportion is 0.47. In a simulation, the middle 95% of the simulated sample proportions runs from 0.41 to 0.53. What is the margin of error?
Answer: B
The margin of error is half the width of the middle 95%: (0.53 - 0.41)/2 = 0.06, so the estimate is 0.47 ± 0.06. Choice A is the full width of the interval, not half of it. Choice D is the upper end of the interval.
Question 7 of 20 · Multiple Choice
A survey of 100 randomly chosen people is replaced by a survey of 900 randomly chosen people. What happens to the margin of error?
Answer: C
The spread of a sample statistic shrinks with the square root of the sample size. Multiplying n by 9 divides the margin of error by √9 = 3. Choice D assumes the margin of error shrinks in proportion to n, which a simulation quickly shows is false.
Question 8 of 20 · Multiple Choice
A website asks visitors to click yes or no on "Should the city build a new stadium?" and 3,000 visitors respond. Why is the result not a good estimate of the proportion of all city residents in favor?
Answer: C
An estimate of a population value needs a random sample. Here people with strong opinions choose to respond, so the sample can be biased, and no margin of error fixes that. Choice A is wrong: 3,000 is a large sample, but size does not cure bias. Choice D is too absolute: the bias could go either way.
Question 9 of 20 · Multiple Choice
A random sample of 60 shoppers spent a mean of 34 minutes in a store. Resampling the 60 values 200 times gives simulated means with a standard deviation of 1.5 minutes. Which interval gives the plausible values of the mean time for all shoppers?
Answer: B
Margin of error ≈ 2 × 1.5 = 3 minutes, so 34 ± 3 gives 31 to 37 minutes. Choice A uses only one standard deviation. Choice C uses four standard deviations. Choice D halves the standard deviation.
Question 10 of 20 · Multiple Choice
From a random sample of likely voters, a poll reports that 53% support a ballot measure, with a margin of error of 4 percentage points. Which conclusion is justified?
Answer: C
The interval 53% ± 4% runs from 49% to 57%. Because it includes values below 50%, the poll does not rule out that less than half of likely voters support the measure. Choice D is a common error: the sample result is exactly 53%, and the interval describes the population, not the sample.
Question 11 of 20 · Multiple Choice
A random sample of 60 students found p̂ = 0.30 for students who own a bike. Which simulation correctly models one random sample for finding the margin of error?
Answer: A
The model must match the survey: probability 3/10 = 0.30 of "owns a bike" and the same sample size, 60. Choice B uses probability 0.4. Choice C uses the wrong sample size, which would make the margin of error too large. Choice D does not record a sample proportion.
Question 12 of 20 · Multiple Choice
Two simulations estimate the spread of a sample mean from the same model population: one uses samples of 16 and the other samples of 144. How do the spreads of the two sets of simulated means compare?
Answer: D
Larger samples give sample means closer to the population mean. The spread shrinks with √n, and √144/√16 = 12/4 = 3, so samples of 16 give means that vary about three times as much. Choice B ignores the effect of sample size. Choices A and C reverse the direction.
Question 13 of 20 · Multiple Choice
For random samples of 100 from a population with p = 0.5, about what standard deviation should a simulation give for the sample proportions?
Answer: B
A simulation gives a standard deviation close to √(p(1 - p)/n) = √(0.25/100) = √0.0025 = 0.05. Choice A divides by n but forgets the square root. Choice C is p(1 - p) without dividing by n, and choice D is √(p(1 - p)), which ignores the sample size.
Question 14 of 20 · Multiple Choice
A random sample of 20 households records the number of pets: 5 households have 0, 8 have 1, 4 have 2 and 3 have 3. What is the estimate of the mean number of pets per household?
Answer: D
Multiply each value by its count: 0·5 + 1·8 + 2·4 + 3·3 = 25 pets in 20 households, so x̄ = 25/20 = 1.25. Choice B averages the four values 0, 1, 2, 3 without weighting by the counts. Choice C is the median. Choice A divides 25 by 4 instead of by 20.
Question 15 of 20 · Short Answer
A random sample of 800 teenagers in a state is asked whether they have a driver's license; 248 say yes. A simulation of 200 samples of 800 from a model with p = 0.31 gives simulated proportions with a standard deviation of 0.016. Estimate the population proportion, find the margin of error, and interpret the result.
p̂ = 248/800 = 0.31. Margin of error ≈ 2 × 0.016 ≈ 0.03. Interval: 0.31 ± 0.03, or 0.28 to 0.34. Interpretation: based on this random sample, it is plausible that between 28% and 34% of all teenagers in the state have a driver's license.
Question 16 of 20 · Short Answer
A random sample of 6 students at a school reports the minutes they spent reading yesterday: 20, 0, 35, 15, 40, 10. Estimate the mean for all students at the school, and explain why the estimate is uncertain.
x̄ = (20 + 0 + 35 + 15 + 40 + 10)/6 = 120/6 = 20 minutes. The estimate is uncertain because a different random sample of 6 students would give a different mean; with only 6 students, sample means can vary a lot, so the margin of error would be large.
Question 17 of 20 · Short Answer
A random sample of 120 customers finds that 62% would use a store's new app. Describe a simulation that gives a margin of error for this estimate. Say what one trial is, how many trials to run and what to do with the results.
Model: each simulated customer says yes with probability 0.62 (for example, random numbers from 1 to 100, with 1-62 meaning yes). One trial: simulate 120 customers and record the proportion who say yes. Repeat at least 100 times, ideally 200 or more. Results: plot the simulated proportions, find the middle 95% or their standard deviation, and use half the width of the middle 95% (about 2 standard deviations, close to 0.09 here) as the margin of error. The estimate is 0.62 ± that margin.
Question 18 of 20 · Short Answer
A random sample of 50 students at a university reports a mean of 4.5 hours of studying per week. In 100 resampled means, the middle 95% runs from 4.1 to 4.9 hours. State the margin of error and interpret it.
Margin of error = (4.9 - 4.1)/2 = 0.4 hours. The estimate is 4.5 ± 0.4 hours. It is plausible that the mean weekly study time for all students at the university is between 4.1 and 4.9 hours. It does not mean that 95% of students study between 4.1 and 4.9 hours a week: individual students vary much more than sample means do.
Question 19 of 20 · Short Answer
A student says, "The margin of error is 3 percentage points, so the true population percent is guaranteed to be within 3 points of the poll result." Explain what is wrong with this statement.
The margin of error comes from the middle 95% of the simulated results, so about 95% of random samples give an estimate within the margin of error of the truth, and about 5% do not. This sample could be one of the unlucky ones. Also, the margin of error covers only random sampling variation: if the sample was biased (for example, by poor question wording or non-response), the true value could be even farther off.
Question 20 of 20 · Short Answer
A student simulates only 10 samples to find the margin of error for a poll of 400 people. Another student simulates 1,000 samples. Whose margin of error is more trustworthy, and whose is smaller? Explain.
The student with 1,000 simulations has a more trustworthy margin of error, because 10 simulated values give a rough and unstable picture of the spread. The two margins of error should be about the same size, though, because both simulate samples of 400. Only the sample size of the survey, not the number of simulations, controls how large the margin of error is.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSS.IC.B.4 mean?
HSS.IC.B.4 means students use a random sample survey to estimate a population mean or proportion and use simulation to find the margin of error. In practice, students compute p̂ or x̄ from the survey, simulate many random samples of the same size from a model that matches the survey, and use the spread of the simulated results to say how far the estimate could plausibly be from the true population value.
Is HSS.IC.B.4 taught in Algebra 2 or in Statistics?
It is usually taught in Algebra II in schools that follow the Common Core course sequence, and it also appears in introductory statistics courses. The Common Core lists it in the high school Statistics and Probability category, in the domain Making Inferences and Justifying Conclusions. Algebra II treatments tend to stay with simulation; statistics courses later add confidence interval formulas.
What is a margin of error in simple terms?
The margin of error is how far a survey estimate is likely to be from the true population value because of random sampling alone. If a poll says 57% ± 5%, values from 52% to 62% are plausible for the whole population. In this lesson it is found as half the width of the middle 95% of simulated sample results, which is about 2 standard deviations of those results.
Why use a simulation instead of a formula for the margin of error?
The standard asks students to develop the margin of error through simulation, because simulation shows where the number comes from: many possible samples, each giving a slightly different result. Once students see that, the formula √(p̂(1 - p̂)/n) for the spread of p̂ becomes a shortcut that matches what they already observed, rather than a rule to memorize. Simulation also works for means and for situations where no simple formula applies.
How many simulated samples should students run?
At least 100, and 200 to 1,000 when technology is available. More simulations give a steadier picture of the spread, so two students' margins of error agree more closely. They do not make the margin of error smaller: that depends on the sample size of the survey. Keeping these two numbers apart is one of the main goals of the lesson.
Does a larger population need a larger sample?
Not much, as long as the population is much larger than the sample. The margin of error depends mainly on the sample size n and the proportion p, not on the population size. A random sample of 1,000 gives about the same margin of error for a city of 100,000 as for a country of 300 million. Students can check this in a simulation, because the model never uses the population size.
What mistakes do students make with margin of error?
A common error is to read "estimate ± margin of error" as the range of individual answers, for example saying that 95% of the fans spent between $16.30 and $20.70 on food when the interval is about the mean spending. Other frequent mistakes are using the number of simulations as if it were the sample size, using one standard deviation instead of two, and believing that a large margin of error is caused by a small population. Many students also think the margin of error accounts for biased questions or voluntary responses; it does not.
What is the difference between a statistic and a parameter?
A parameter describes the whole population and is usually unknown, such as the true proportion p of voters who support a bond or the true mean μ. A statistic describes the sample and is known, such as p̂ = 0.57 or x̄ = 2.2. HSS.IC.B.4 uses the statistic to estimate the parameter and the simulation to show how much the statistic varies from sample to sample.
Why does the sample have to be random?
Only a random sample lets you use chance to describe the error. The simulation models random sampling, so its margin of error describes random sampling error and nothing else. If the sample was chosen by convenience or people chose to respond, the estimate may be biased in ways no simulation can measure. That is why a large online poll can be less trustworthy than a small random sample.
How does HSS.IC.B.4 connect to the SAT and to later courses?
The digital SAT Math section includes questions on inference from sample statistics and margin of error in its Problem-Solving and Data Analysis domain, so students who can interpret "estimate ± margin of error" are prepared for them. In later statistics courses, the simulation idea becomes the sampling distribution, and the margin of error becomes part of a confidence interval. The next standards in the cluster, HSS.IC.B.5 and HSS.IC.B.6, use the same simulation reasoning to compare treatments and to judge reports.
07
Related Standards
6 standards
These standards connect to HSS.IC.B.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.SP.A.2Prerequisite
Use data from a random sample to draw inferences about a population and gauge variation
