Populations and samples, the four common sampling methods, recognising a biased sample, describing a procedure for drawing a random sample, computing a margin of error from the sample size, and finding the sample size needed for a given margin of error.
Subject: Algebra 2 · 65 slides · symbolic lesson
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Title
Algebra 2 · Chapter 11 — Data Analysis and Statistics
Select and Draw Conclusions from Samples
Objectives
Five outcomes. Who you ask decides what you can conclude.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 766-769 — the lesson these objectives are drawn from
Warm-up
Lessons 11.1 to 11.3 described data once you had it. This lesson asks how you get it.
Discussion prompt
You want to know how often people in your town attend concerts. You ask 50 people queueing outside a rock concert. What is wrong with that?
Hint: Who ends up in the queue?
Answer:
Everyone in that queue is at a concert right now, so they attend far more concerts than the average resident. The answer will come out far too high.
The problem is not the number 50 — asking 5000 people in the same queue would be just as wrong. It is who gets into the sample.
This lesson names that problem, describes how to avoid it, and gives a formula for how much a fair sample can still be off by.
Concept
A sample is a subset of a population. A random sample, in which every member has an equal chance of selection, is representative and supports conclusions about the population. Its margin of error depends on the sample size.
margin of error — A limit on how much a random sample's responses are likely to differ from the population's, approximated by plus or minus one over the square root of the sample size.
\[ \text{margin of error} = \pm\frac{1}{\sqrt{n}} \]
A biased sample cannot be rescued by making it bigger. Only once a sample is unbiased does its size start to matter.
Figure (svg): Two columns comparing biased and unbiased samples
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 766-768
Section
Section 1
Concept
A population is the whole group you want information about; a sample is a subset of it. Members may volunteer, be chosen for convenience, be picked by a rule, or be selected at random.
\[ \text{sample} \subset \text{population} \]
Only a random sample gives every member of the population an equal chance of being selected, which is why it is the one preferred.
Figure (svg): Four ways of choosing a sample from a population
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 766-766 — Sampling methods
Picture it
How each kind of sample is chosen.
Figure (svg): Four ways of choosing a sample from a population
The first three are quicker and the fourth is fairer. Which one was used is the first thing to establish about any survey.
Worked example
Example 1, both parts.
\[ \text{A writer contacts only coaches whose numbers he has; then he mails all coaches and uses the replies. Classify each.} \]
First: who was chosen
Why: The coaches easiest for him to reach.
First: name the method
Why: Selecting the accessible members.
Second: who decides
Why: Each coach chooses whether to reply.
Second: name the method
Why: Members volunteer to be included.
Figure (svg): Four ways of choosing a sample from a population
\[ \text{convenience}; \qquad \text{self-selected} \]
Verify: ask who did the choosing
Why: In the first the researcher chose, using ease of access as the criterion. In the second the respondents chose, by deciding whether to reply. That single question — who decided who is in the sample — separates the four methods quickly.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 766-766
Sorting
Ask who did the choosing.
Sort into buckets
Sort each situation.
Two of the five are convenience samples, which is by far the commonest method in practice and the one most likely to mislead.
Worked example
Guided Practice 1 and lesson exercises 3 to 5.
\[ \text{A teacher surveys one of his own classes; drivers survey every tenth customer; names are drawn from a hat.} \]
First: the teacher's own class
Why: The easiest students for him to reach.
Second: every tenth customer
Why: A rule selects the members.
Third: names from a hat
Why: Every name has an equal chance.
Note the pattern
Why: The method is named by how members were chosen, not by how many.
Figure (svg): The solution to Worked example three more classifications shown as a ladder of expressions, one row per algebraic move
\[ \text{convenience}, \; \text{systematic}, \; \text{random} \]
Verify: check the systematic one carefully
Why: Every tenth customer is chosen by a rule rather than at random, but it can still be reasonably representative provided nothing about the tenth position is special. Systematic sampling sits between convenience and random: fairer than the first, less clearly fair than the second.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 767-769
Trap
\[ 5000 \text{ replies to an online poll} \]
Call it a random sample
Why: The large number is taken as evidence of fairness.
\[ \text{random} \quad \text{(wrong)} \]
Everyone in it chose to respond, so it is a self-selected sample — and the people who bother to reply differ systematically from those who do not.
\[ \text{random requires equal chance for every member} \]
Ask how members were selected, not how many there are
Why: Randomness is a property of the process.
\[ \text{self-selected, whatever its size} \]
Online polls with hundreds of thousands of responses are still self-selected, which is why news outlets label them as unscientific.
Matching
How the members got in.
Match the pairs
Why: Only the last guarantees that the sample can represent the population. The other three may happen to be representative, but nothing in the method ensures it.
Fill the middle
Example 1.
Fill in the blanks
\textcoaches ___
Why: The population is every college baseball coach, and the sample is whichever subset he actually reaches. Naming the population first makes it obvious when a sample fails to represent it.
Prediction
Commit before reasoning.
Predict first
If a sample can mislead, why not simply survey the whole population?
Correct: Because it is usually too expensive, slow or impossible, and a good sample answers the question well enough.
\[ 1011 \text{ people give } \pm 3.1\% \]
Why: A national census costs hundreds of millions and takes years, while a well-chosen sample of a thousand gives a margin of about three percent in a few days. Sometimes surveying everyone is impossible outright: testing every light bulb's lifetime would destroy the entire stock. Sampling is a deliberate trade of a little precision for an enormous saving, and the margin of error is what quantifies the trade.
Section
Section 2
Concept
An unbiased sample is representative of the population. A biased sample over- or underrepresents part of it, so conclusions drawn from it do not transfer to the population.
\[ \text{biased} \;\Longrightarrow\; \text{conclusions do not transfer} \]
Bias is caused by the selection process, not by the sample size, so a larger biased sample is simply a more confidently wrong one.
Figure (svg): A biased sample and an unbiased one drawn from the same population
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 767-767 — Bias in sampling
Picture it
Example 2: asking concert-goers about concert attendance.
Figure (svg): A biased sample and an unbiased one drawn from the same population
The left sample is packed with people who attend concerts, because they were selected while attending one. The right reflects the community's actual mix.
Worked example
Example 2.
\[ \text{A manager asks } 50 \text{ people queueing for a rock concert how many concerts they attend a year. Is the sample biased?} \]
Name the population
Why: Everyone in the community.
Name the sample
Why: Fifty people at a concert.
Compare them
Why: Concert-goers attend more concerts than people in general.
Conclude
Why: The estimate will be far too high.
Figure (svg): A biased sample and an unbiased one drawn from the same population
\[ \text{biased: overrepresents concert-goers} \]
Verify: check the direction of the error
Why: Every person in the sample attends at least one concert a year, which is not true of the population — so the sample's average must exceed the population's. Predicting which way a bias pushes the answer is often possible, and it is more useful than merely noting that bias exists.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 767-767
Sorting
Ask whether the sample resembles the population.
Sort into buckets
Sort each sample.
Every biased sample here was chosen for convenience or by self-selection. The two unbiased ones came from a rule and from randomness.
Worked example
Guided Practice 1 and lesson exercises 3 to 5.
\[ \text{A teacher surveys his computer science class about a website; dog owners at a park are asked about an off-leash area; names are drawn from a hat.} \]
First: who is in the sample
Why: Students who chose to study computer science.
First: which way
Why: They are more comfortable with websites than students in general.
Second: dog owners at a park
Why: They are far more likely to want an off-leash area.
Third: names from a hat
Why: Every student has an equal chance.
Figure (svg): The solution to Worked example three more judgements shown as a ladder of expressions, one row per algebraic move
\[ \text{biased}, \; \text{biased}, \; \text{unbiased} \]
Verify: check the second's direction
Why: Dog owners who bring dogs to the park would use an off-leash area, while residents without dogs might object to losing park space. Surveying only the first group guarantees support well above the town's, so the council would be misled into thinking the measure more popular than it is.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 767-769
Error analysis
A researcher is told a sample of 50 concert-goers is biased.
Annotate
On: \( \text{survey } 500 \text{ concert-goers instead} \)
Size and bias are independent problems. Only changing HOW the sample is selected can remove a bias; changing how many only narrows the margin around the wrong answer.
Prediction
Commit before reasoning.
Predict first
Dog owners at a park are asked whether the town should create an off-leash area. What will the survey show?
Correct: Support far above the town's true level.
\[ \text{sample} = \text{beneficiaries only} \;\Longrightarrow\; \text{support overstated} \]
Why: The sample consists entirely of people who bring dogs to that park, exactly the group an off-leash area would benefit. Residents without dogs, who might object to losing shared space, are absent. So the reported support will exceed the town's. Predicting the direction of a bias is often possible from a moment's thought about who is missing.
Two truths and a lie
All three are about bias.
Eliminate the wrong options
Two of these are true. Knock those out and keep the false one.
Survives elimination: B
Why: The survivor is false. An online poll with a million self-selected responses is still self-selected, and the famous 1936 Literary Digest poll surveyed over two million people and predicted the wrong president — because its sample came from car and telephone owners in the middle of a depression. Size narrows the margin around whatever the sample measures; it cannot change what the sample measures.
Fill the middle
Example 2.
Fill in the blanks
\textconcert-goers ___
Why: Everyone in the queue is a concert-goer, so they appear far more often in the sample than in the town. Naming the overrepresented group is the clearest way to explain a bias.
Section
Section 3
Concept
To draw a random sample, list every member of the population and assign each a number, generate the required count of distinct random numbers in that range, and take the members those numbers name.
\[ 1 \text{ to } 324; \; 40 \text{ distinct draws} \]
Duplicates are discarded and replaced so that no member appears twice, and so that the sample really has the size it claims.
Figure (svg): The three steps of drawing a random sample from a listed population
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 767-767 — Choose an unbiased sample
Picture it
Example 3: 40 seniors from a class of 324.
Figure (svg): The three steps of drawing a random sample from a listed population
The list makes the population explicit, the random numbers make the selection fair, and the third step is bookkeeping.
Worked example
Example 3.
\[ \text{Describe a method for polling } 40 \text{ of } 324 \text{ seniors about the prom.} \]
List and number the population
Why: Every senior gets a distinct integer.
\[ 1\text{ to } 324 \]
Generate random integers
Why: Forty of them, in that range.
\[ 184, 1, 106, 215, 67, 213,... \]
Handle duplicates
Why: Discard any repeat and draw a replacement.
\[ 40\text{ distinct numbers} \]
Take the matching seniors
Why: Poll exactly those 40.
Figure (svg): The three steps of drawing a random sample from a listed population
\[ \text{number, draw, match} \]
Verify: check that every senior had an equal chance
Why: Each of the 324 numbers is equally likely on each draw, and discarding duplicates does not favour anyone — so every senior has the same chance of ending up in the sample. That is exactly the definition of a random sample, and the numbering step is what makes it checkable.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 767-767
Ranking
Drawing a random sample.
Put in order
Why: Step one is easy to skip and decides everything after it: a sample can only be random with respect to a stated population. Step two is where most real surveys fail, because the available list is not the whole population.
Worked example
Guided Practice 2.
\[ \text{Give another way to select } 40 \text{ seniors at random.} \]
Write every name on identical slips
Why: One slip per senior, all the same size.
\[ 324\text{ slips} \]
Mix them thoroughly
Why: So that no slip is easier to reach.
Draw 40 without looking
Why: Removing each slip after it is drawn.
\[ 40\text{ names} \]
Check the fairness condition
Why: Every slip was equally likely at every draw.
Figure (svg): The solution to Worked example an alternative method shown as a ladder of expressions, one row per algebraic move
\[ \text{a hat, properly mixed} \]
Verify: identify what could go wrong
Why: If some slips were folded differently, or the later ones sat on top, the draw would favour them and the sample would not be random. The requirement is not merely that no one chose deliberately, but that every member had a genuinely equal chance — which is why thorough mixing matters.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 767-767
Trap
\[ \text{number the seniors whose emails you have} \]
Draw at random from that list
Why: Random numbers are used, so the sample is called random.
\[ \text{a random sample} \quad \text{(wrong)} \]
Seniors without email on file had no chance of selection at all, so the sample is random within a subgroup rather than within the population.
\[ \text{number all } 324 \text{ seniors first} \]
Make the list complete before drawing
Why: Equal chance means equal for every member of the population.
\[ \text{anyone excluded from the list has probability } 0 \]
This is where most real surveys go wrong: the randomisation is fine and the list it draws from is not. Telephone polls that miss households without landlines fail in exactly this way.
Fill the middle
Example 3.
Fill in the blanks
\text324 1 \text___ ___
Why: The range must cover the whole population, one integer per senior. A range that stopped short would give the last seniors no chance of selection.
Sorting
Did every member have an equal chance?
Sort into buckets
Sort each procedure.
The third is the subtle one: the randomisation is genuine, but it is applied to a list that is not the population. That is the commonest failure in real surveys.
Prediction
Commit before reasoning.
Predict first
Why is a repeated random number discarded and replaced rather than kept?
Correct: Because one senior would otherwise be surveyed twice, and the sample would hold fewer than 40 people.
\[ 40 \text{ draws} \neq 40 \text{ people, unless duplicates are replaced} \]
Why: Keeping a duplicate would double-count one opinion and leave the sample with only 39 distinct respondents, so the effective sample size falls and one person's view gets extra weight. Discarding and replacing keeps the sample at 40 distinct members without favouring anyone. This is sampling without replacement, exactly as in Lesson 10.5.
Section
Section 4
Concept
For a random sample of size n from a large population, the margin of error is about plus or minus one over the square root of n. The population's true percent is likely to lie within that distance of the sample's.
\[ \pm\frac{1}{\sqrt{n}} \]
The result of a survey is therefore an interval rather than a single figure, and quoting the figure alone overstates what the sample can support.
Figure (svg): The margin of error formula and the interval it produces around a survey result
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 768-768 — Margin of Error Formula
Picture it
Example 4: 52 percent from a sample of 1011.
Figure (svg): The margin of error formula and the interval it produces around a survey result
The margin is about 3.1 percent, so the true figure is likely between 48.9 and 55.1 percent — which includes half, so the survey cannot claim a majority.
Worked example
Example 4, both parts.
\[ \text{In a survey of } 1011 \text{ people, } 52\% \text{ named television. Find the margin of error and the likely interval.} \]
Write the formula
Why: Plus or minus one over the square root of n.
\[ 1 / \sqrt{1011} \]
Compute the square root
Why: The square root of 1011 is about 31.8.
\[ 31.8 \]
Divide
Why: One over 31.8.
\[ \text{about } 0.031 \]
Build the interval
Why: Fifty-two percent minus and plus 3.1 percent.
\[ 48.9 \%\text{ to } 55.1 \% \]
Figure (svg): The margin of error formula and the interval it produces around a survey result
\[ \pm 3.1\%; \quad 48.9\% \text{ to } 55.1\% \]
Verify: notice what the interval includes
Why: The interval runs from 48.9 to 55.1 percent, which contains 50 percent — so the survey does not establish that television is the main source for a majority. Reporting only the 52 percent would suggest a majority the data does not support, which is exactly what the margin exists to prevent.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 768-768
Fill the middle
Example 4.
Fill in the blanks
\frac31.8___} = \frac______} \approx 0.031
Why: The square root of 1011 is about 31.8, and 1 over that is about 0.031, or 3.1 percent. The square root is what makes the margin fall so slowly with sample size.
Worked example
Guided Practice 3.
\[ \text{In a survey of } 1202 \text{ people, } 11\% \text{ used the internet over ten hours a week. Find the margin and the interval.} \]
Take the square root
Why: The square root of 1202 is about 34.7.
\[ 34.7 \]
Divide
Why: One over 34.7.
\[ \text{about } 0.029 \]
State the margin
Why: About 2.9 percent.
\[ +- 2.9 \% \]
Build the interval
Why: Eleven percent minus and plus 2.9.
\[ 8.1 \%\text{ to } 13.9 \% \]
Figure (svg): The solution to Worked example a larger survey shown as a ladder of expressions, one row per algebraic move
\[ \pm 2.9\%; \quad 8.1\% \text{ to } 13.9\% \]
Verify: compare with the earlier survey
Why: Nineteen percent more respondents cut the margin only from 3.1 to 2.9 percent, a reduction of about 6 percent. That is the square root at work: sizeable increases in effort buy small increases in precision, which is the subject of the next idea.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 769-769
Error analysis
A student computes the margin of error for a survey of 1011 people.
Annotate
On: \( \text{margin} = \pm\frac{1}{1011} \approx \pm 0.001 \)
A margin under one percent would need a sample of over ten thousand. Checking the answer against what surveys actually report catches this immediately.
Matching
One over the square root.
Match the pairs
Why: Quadrupling the sample from 100 to 400 halved the margin, and quadrupling again would halve it once more. The pattern is exactly what a square root produces.
Comparison
Fill the blanks. A result plus a margin gives an interval.
Comparison matrix
| Quantity | Television survey | Internet survey |
|---|---|---|
| Sample size | 1011 | 1202 |
| Reported percent | 52 | 11 |
| Margin of error | about 3.1 percent | about 2.9 percent |
| Likely interval | 48.9 to 55.1 percent | 8.1 to 13.9 percent |
Both intervals are about six percentage points wide, since both samples are near a thousand — the margin depends on the sample size and not at all on the reported percent.
Prediction
Commit before reasoning.
Predict first
One survey of 1011 reports 52 percent and another reports 11 percent. How do their margins compare?
Correct: The margins are the same, since this formula depends only on n.
\[ \pm\frac{1}{\sqrt{n}} \text{ mentions only } n \]
Why: One over the square root of n contains no reference to the reported percent, so any two surveys of the same size get the same margin under this approximation. More refined formulas do depend slightly on the percent, giving a narrower margin for results near 0 or 100 — but the simple version taught here treats all results alike, which is why it is stated as an approximation.
Section
Section 5
Concept
Setting the margin of error equal to a target and solving for n gives the sample size needed. Because the formula involves a square root, halving the margin requires four times the sample.
\[ 0.06 = \frac{1}{\sqrt{n}} \;\Longrightarrow\; n \approx 278 \]
That relationship is why national polls settle around a thousand respondents: the margin is about three percent, and pushing it much lower gets expensive very quickly.
Figure (svg): Margin of error falling as the sample size grows, with two sample sizes marked
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 769-769 — Find a sample size
Picture it
Example 5: the margin falling as the sample grows.
Figure (svg): Margin of error falling as the sample size grows, with two sample sizes marked
The curve drops steeply at first and then flattens. Going from 278 to 1112 respondents halves the margin; going from 1112 to 2500 barely improves it.
Worked example
Example 5, a multiple-choice item.
\[ \text{How many people must be surveyed for a margin of error of about } \pm 6\%? \]
Set the formula equal to the target
Why: Point zero six equals one over the square root of n.
\[ 0.06 = 1 / \sqrt{n} \]
Square both sides
Why: Point zero zero three six equals one over n.
\[ 0.0036 = \frac{1}{n} \]
Solve for n
Why: Take the reciprocal.
\[ n = \frac{1}{0.0036} \]
Compute
Why: About 277.8, rounded up.
\[ \text{about } 278 \]
Figure (svg): Margin of error falling as the sample size grows, with two sample sizes marked
\[ n \approx 278 \]
Verify: check the answer forwards
Why: One over the square root of 278 is 1 over 16.67, which is 0.06 — the target margin. Working the formula in both directions is the natural check, and it costs one square root.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 769-769
Fill the middle
Example 5.
Fill in the blanks
0.06 = \frac278___} \;\Longrightarrow\; 0.0036 = \frac______ \;\Longrightarrow\; n \approx ___
Why: One over 0.0036 is about 277.8, rounded up to 278. Squaring both sides first is what turns the square root into something solvable.
Worked example
Guided Practice 3, second part.
\[ \text{A survey of } 1202 \text{ gave a margin of about } 2.9\%. \text{ How many are needed for } \pm 2\%? \]
Set up the equation
Why: Point zero two equals one over the square root of n.
\[ 0.02 = 1 / \sqrt{n} \]
Take reciprocals
Why: The square root of n is 50.
\[ \sqrt{n} = 50 \]
Square
Why: Fifty squared.
\[ n = 2500 \]
Compare with the original
Why: Twenty-five hundred against 1202.
Figure (svg): The solution to Worked example cut a margin to two percent shown as a ladder of expressions, one row per algebraic move
\[ n = 2500 \]
Verify: check the ratio of effort to gain
Why: Doubling the sample from 1202 to 2500 cut the margin only from 2.9 to 2.0 percent, a gain of under a percentage point. To reach one percent would need ten thousand respondents, and half a percent forty thousand. Each halving costs four times as much.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 769-769
Trap
\[ n = 278 \text{ gives } 6\%; \text{ want } 3\% \]
Double the sample
Why: The margin is halving, so the effort is doubled.
\[ n = 556 \quad \text{(wrong)} \]
One over the square root of 556 is about 4.2 percent, not 3. Doubling the sample cuts the margin by a factor of only the square root of 2.
\[ 0.03 = \frac{1}{\sqrt{n}} \;\Longrightarrow\; n \approx 1112 \]
Quadruple the sample to halve the margin
Why: The square root means n changes by the square of the factor.
\[ 4 \times 278 = 1112 \quad \checkmark \]
Every extra decimal place of precision costs a hundredfold increase in sample size, which is why polls report to the nearest percent rather than the nearest tenth.
Ranking
Smallest sample first.
Put in order
Why: The sample sizes are 100, 278, 1011, 2500 and 10,000. Notice the last step: halving the margin from 2 to 1 percent costs 7500 extra respondents, more than the first four targets combined.
Comparison
Fill the blanks. The square root sets the price.
Comparison matrix
| Margin | Sample needed | Factor from the row above |
|---|---|---|
| 8 percent | about 156 | — |
| 4 percent | about 625 | 4 times |
| 2 percent | 2500 | 4 times |
| 1 percent | 10,000 | 4 times |
Every halving of the margin quadruples the sample, so the cost of precision grows as the square of the improvement wanted.
Prediction
Commit before reasoning.
Predict first
National opinion polls almost always survey around a thousand people. Why that number?
Correct: Because it gives a margin of about 3 percent, and improving on that gets expensive very fast.
\[ \frac{1}{\sqrt{1000}} \approx 3.2\% \]
Why: One over the square root of a thousand is about 3.2 percent, which is precise enough to distinguish a clear lead from a close race. Getting to 1.5 percent would need four thousand respondents and getting to 1 percent ten thousand — several times the cost for a gain that rarely changes any conclusion. A thousand is where the curve of diminishing returns bends, which is why the figure recurs across countries and decades.
Comparison
Fill the blanks. Two separate questions about a survey.
Comparison matrix
| Question | Controlled by | Fixed by |
|---|---|---|
| Is the answer aimed at the right target? | how the sample was selected | using a random sample |
| How precise is the answer? | the sample size | surveying more people |
| A biased sample of 5000 | wrong target, narrow margin | confidently wrong |
| A random sample of 50 | right target, wide margin | correct but imprecise |
The two problems are independent, and only one of them can be fixed by asking more people.
Pattern
Define, select, then quantify.
A biased sample cannot be repaired by enlarging it. Fix the selection first, then worry about the size.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 766-769
Check
Who did the choosing?
Check your understanding
A writer mails surveys to all coaches and uses only the ones returned. What kind of sample is that?
Answer: A
Why: Each coach decides whether to reply, so the members select themselves.
Check
One over the square root.
Check your understanding
What is the margin of error for a random sample of 1011 people?
Answer: A
Why: One over the square root of 1011 is about 0.031.
Check
Square both sides.
Check your understanding
How many people must be surveyed for a margin of error of about 6 percent?
Answer: A
Why: 0.06 equals 1 over the square root of n, so n is 1 over 0.0036.
Real world
In 1936 a magazine mailed ten million ballots to names taken from car registrations and telephone directories, and received 2.4 million replies. It predicted a landslide for Landon over Roosevelt. Roosevelt won 46 of 48 states.
Discussion prompt
Compute the margin of error for a sample of 2.4 million, and explain how the poll went so badly wrong.
Hint: The margin formula does not measure bias.
Answer:
\[ \pm\frac{1}{\sqrt{2{,}400{,}000}} \approx \pm 0.00065, \text{ about } 0.06\% \]
The margin of error was under a tenth of one percent — the most precise poll ever conducted at the time, and wrong by nearly twenty points.
Two failures, both from this lesson. The list was drawn from car and telephone owners, who in the depths of the Depression were far richer than average and far more likely to favour Landon — a biased frame. And of the ten million contacted only 24 percent replied, so the sample was self-selected on top of that. Meanwhile George Gallup predicted the result correctly from a random sample of about fifty thousand, and even predicted what the magazine's poll would say. The magazine folded within two years. The lesson stands: a margin of error measures precision, never accuracy, and no sample size repairs a biased selection.
Commit first
Answer, then rate your confidence honestly.
Predict first
A sample of 50 concert-goers is biased. Does surveying 500 of them instead fix the problem?
Correct: No — every respondent is still a concert-goer, so the same group is overrepresented.
\[ \text{margin} \to 0, \text{ but the bias stays} \]
Why: Bias comes from the selection process and size has no bearing on it. Enlarging a biased sample narrows the margin of error around the wrong answer, producing a more confident mistake rather than a better estimate. The 1936 Literary Digest poll surveyed 2.4 million people with a margin under a tenth of a percent and still called the election wrong by twenty points. The only repair is to change how the sample is chosen.
Explain it
They think a survey with more responses is automatically more trustworthy.
Discussion prompt
In four sentences or fewer, explain why a huge survey can still be wrong.
Hint: Think about who answers.
Answer:
Suppose you want to know how often people in town go to concerts, and you ask everyone in the queue outside a concert. Ask fifty of them or five thousand — every single one is at a concert, so your answer comes out far too high either way.
The problem is who you asked, not how many. Asking more of the wrong people just makes you more confident about a wrong answer.
Exit ticket
Name the weakest spot before you close the deck.
Predict first
Which of these would you least want handed to you cold?
Correct: Whichever you picked is tonight's ten minutes, and each has a one-line fix.
Why: For the methods, ask who did the choosing: the researcher, the members, a rule, or chance. For bias, name the group that is over- or underrepresented. For the margin, take the square root of n first and then the reciprocal. For sample size, square both sides before solving.
Connect it up
Paper. Fifteen minutes.
Draw it
Build a sampling page. Top: list the four sampling methods with a one-line description and an example of each, and mark which one is preferred and why. Middle left: describe three surveys of your own invention, judge each for bias, and for the biased ones name the overrepresented group and predict which way the error runs. Middle right: write out the three steps for drawing a random sample of 40 from 324, and note beside step two what a repeated number means. Bottom left: compute the margin of error for samples of 100, 400, 1011 and 2500, and plot the four points. Bottom right: solve for the sample size needed for margins of 6, 3 and 2 percent, and write one sentence on the pattern.
If your plotted margins fall in a straight line, recheck them: the relationship is a square root, so the curve should drop steeply and then flatten.
Recap
Five things, and two separate questions about any survey.
| If you see | Then |
|---|---|
| Members volunteering | A self-selected sample |
| The easiest members chosen | A convenience sample |
| A rule such as every tenth | A systematic sample |
| Equal chance for every member | A random sample |
| A survey result quoted alone | Ask for its margin of error |
| A target margin | Square both sides and solve for n |
Lesson 11.5 turns from one variable to two, and asks which kind of model best fits a set of paired data.
McDougal Littell Algebra 2 (Texas Edition), Ch. 11 Data Analysis and Statistics — Lesson 11.4 Select and Draw Conclusions from Samples §11.4, pp. 766-769 — everything on these slides traces back here
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