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Turn a sample size into a margin of error, read a poll as an interval rather than a number, tell an experiment from an observational study, and judge a difference against a simulation of chance.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you ask what a sample can tell you about the population it came from. A margin of error of about $\frac{1}{\sqrt{n}}$ turns a sample size into a width, so a poll becomes an interval rather than a single number, and a conclusion counts only if it holds everywhere in that interval. You then separate two questions that are easy to confuse: whether two things go together, and whether one causes the other — where only random assignment settles it — and you use a reshuffling simulation to decide whether a difference is more than chance.
You can work out a percent of a number, you have just met the normal distribution and its standard deviation, and you know that $\sqrt{400} = 20$. That is the whole tool kit. What is new is the question it is pointed at: not "what did this group do?" but "what does this group tell me about the far larger group I did not ask?"
Population: everyone you want to know about. Sample: the few you ask.
Parameter: a number describing the population, usually unknown. Statistic: the matching number from the sample.
Random sample: every member of the population had the same chance of being picked.
Margin of error: how far the statistic may plausibly sit from the parameter. Confidence interval: the statistic give or take that margin.
Random assignment: deciding by chance which units are treated. Not the same as random sampling, and it is what licenses a claim about cause.
Ask a different random sample and you get a different answer: that spread is sampling variability, and it shrinks as the sample grows. A useful rough rule for a percentage from a random sample of $n$ people is
$$\text{margin of error} \approx \frac{1}{\sqrt{n}},$$
read as a proportion, so $n = 400$ gives $\frac{1}{20} = 5\%$ and $n = 2500$ gives $\frac{1}{50} = 2\%$. Note the cost: to halve the margin you must quadruple the sample. The result is reported as an interval — $47\%$ support with a margin of $4\%$ means the true figure is plausibly anywhere from $43\%$ to $51\%$ — and a conclusion is safe only if it holds at every point of that interval.
Causation is a second question. An observational study records what people already do, so any difference it finds may be caused by whatever made them different in the first place. An experiment assigns the treatment at random, balancing those hidden differences on average and leaving the treatment as the only systematic one. Only that supports "causes".
How big must a difference be before it means anything? Simulate the chance explanation: shuffle the group labels at random many times and record the difference each shuffle produces. If chance beats the real difference only rarely — under about $5\%$ of shuffles is the usual line — the difference is called statistically significant. If chance produces it easily, you have found nothing.
Another way: picture
A dot plot of the percentages from two hundred samples of the same population. They pile up around the true value in a bell-shaped heap, most within one margin of it. A single poll is one dot from that heap, which is why it comes with a width attached.
Another way: story
A margin of error is not a mistake. It is the price of asking $1{,}000$ people instead of everyone, and it can be quoted in advance because $n$ alone fixes it.
"A bigger population needs a bigger sample." The margin depends on $n$, not on the size of the population: $1{,}000$ people pin down a city and a country about equally well.
"$52\%$ with a margin of $3\%$ proves a majority." The interval runs from $49\%$ to $55\%$, and $49\%$ is not a majority. Report the interval, not the headline.
"They found a correlation, so it causes it." People who use a revision app may simply be the people who were already revising. Only random assignment rules that out.
$\sqrt{900} = 30$, so the margin is $\frac{1}{30}$ as a proportion.
The sample size goes under the root first.
$\frac{1}{30} = 0.0333\ldots$, which is about $3.3\%$.
Multiply a proportion by $100$ to read it as a percent.
So a poll of $900$ carries a margin of roughly $3\%$.
The interval is $53 - 4 = 49\%$ to $53 + 4 = 57\%$.
Both ends, always.
$49\%$ is not a majority, so the interval contains values that are not majorities.
One end below the line is enough.
The poll leans that way but does not settle it.
A larger sample narrows the interval.
$\sqrt{1600} = 40$, so the margin is $\frac{1}{40} = 2.5\%$.
The interval is $49.5\%$ to $54.5\%$, which reaches below $50\%$: not settled.
A random sample of $1600$ people is taken. Using $\frac{1}{\sqrt{n}}$, what is the margin of error, as a percent?
Answer:
A poll finds $47\%$ support with a margin of error of $5\%$. Can you conclude that a majority supports it?
A researcher wants to claim that a new protective coating causes better working life, not merely that the two go together. She has $44$ machine parts to work with. Which design supports the causal claim?
Two groups differ by some amount. Reshuffling the group labels at random produces a difference that large or larger $8\%$ of the time. What does that suggest?
A council will accept a margin of error of $\frac{5}{2}\%$ on its survey, and no more. Using $\frac{1}{\sqrt{n}}$, how many people must it sample?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A random sample of $900$ people is taken. Using $\frac{1}{\sqrt{n}}$, what is the margin of error, as a percent?
Answer:
You can turn a sample size into a margin of error and say what a study does and does not establish. Give the margin for a sample of $2500$, say whether $51\%$ with a margin of $3\%$ shows a majority, and explain why a survey of app users cannot show the app causes higher marks.
8. Your turn: $1600$ people polled, $52\%$ in favour. Is a majority in favour?, step 2