Inferences and Conclusions from Data
Unit 4 of Algebra II. 14 questions below, each with the working. Every answer was checked by a second pass before it was published.
Normal distributions, sampling and study design, estimating population parameters, margin of error, simulation, comparing treatments, evaluating reports.
How this unit is tested
Start by sorting every problem into one of two buckets: is it about a single normally distributed variable (use the empirical rule and z-scores), or is it about drawing conclusions from a sample (use study design, margin of error, and simulation ideas)? Most exam questions signal which bucket they belong to with a keyword — 'normally distributed,' 'mean and standard deviation' point to the first; 'survey,' 'sample,' 'randomly assigned,' 'margin of error' point to the second.
For normal distribution problems, sketch the bell curve, mark the mean and one, two, and three standard deviations on each side, and shade the region the question asks about before doing any arithmetic. Convert raw values to z-scores only when the empirical rule's fixed percentages (68, 95, 99.7) don't line up exactly with the boundary you need.
For sampling and study-design problems, first decide what kind of study it is — sample survey, observational study, or controlled experiment — because that decision alone tells you whether a causal claim is even allowed. Then check how subjects were chosen (random sample vs. convenience sample) and, for experiments, whether treatments were randomly assigned; these two kinds of randomness solve two different problems (generalizability vs. causality) and students constantly mix them up.
For margin of error, confidence intervals, and simulation, remember that these tools quantify sampling variability, not the quality of the study itself. A huge sample size shrinks the margin of error but does nothing to fix a biased sampling method. When evaluating a published report, work through the same checklist every time: who was sampled, how were they chosen, was there a control group, and does the conclusion claim causation when only correlation was shown.
What you have to know
Empirical Rule (68-95-99.7 Rule)
For a normal distribution, about 68% of values lie within 1 standard deviation of the mean, about 95% lie within 2 standard deviations, and about 99.7% lie within 3 standard deviations.
Z-score
$z=\dfrac{x-\mu}{\sigma}$ converts a value $x$ from a normal distribution with mean $\mu$ and standard deviation $\sigma$ into the number of standard deviations it lies from the mean, allowing comparison to standard normal percentages.
Margin of Error Approximation
For a sample of size $n$ used to estimate a population proportion or mean, the margin of error can be approximated by $\text{MOE}\approx \dfrac{1}{\sqrt{n}}$; the true population value is estimated to lie within the interval (sample statistic) ± MOE.
Random Sampling vs. Random Assignment
Random sampling (choosing subjects randomly from a population) allows results to be generalized to that population; random assignment (randomly placing subjects into treatment groups) allows differences in outcomes to be attributed to the treatment, i.e. supports causal claims.
Study Types
A sample survey collects data by asking members of a sample questions with no treatment applied; an observational study measures variables without imposing any treatment; a controlled experiment randomly assigns treatments to subjects and compares outcomes.
Simulation for Inference
Repeatedly resampling or re-randomizing data (by hand or with technology) and recording a statistic each time builds a simulated sampling distribution, which can be used to estimate margin of error or to judge whether an observed difference between treatment groups is larger than would occur by chance alone.
14 practice questions
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Scores on a standardized test are normally distributed with mean 500 and standard deviation 100. What percentage of test-takers scored between 400 and 600?
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Answer. About 68%
400 and 600 are exactly 1 standard deviation below and above the mean of 500. The empirical rule states that about 68% of normally distributed data falls within 1 standard deviation of the mean. -
Using the same test (mean 500, standard deviation 100), what is the z-score for a student who scored 650?
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Answer. z = 1.5
$z=\dfrac{x-\mu}{\sigma}=\dfrac{650-500}{100}=1.5$. This tells you the score is 1.5 standard deviations above the mean. -
About what percentage of test-takers scored above 700 on the same test (mean 500, SD 100)?
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Answer. About 2.5%
700 is exactly 2 standard deviations above the mean. By the empirical rule, 95% of scores fall within 2 SDs, leaving 5% in the two tails combined; by symmetry, 2.5% lies above 700. -
A researcher mails a survey to every household in a city and only tallies the responses that are returned. What is the main flaw in this sampling method?
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Answer. The sample is self-selected (voluntary response), not randomly chosen, so it likely overrepresents people with strong opinions and underrepresents others.
People who feel strongly about the topic are more likely to return the survey, producing a biased sample even if every household received it. A truly random sample requires selecting respondents randomly, not letting them opt in. -
A pharmaceutical company gives a new drug to volunteer patients and compares their recovery to patients who did not volunteer for the study. Why can't the company conclude the drug caused faster recovery?
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Answer. Because subjects were not randomly assigned to treatment and control groups, the two groups may differ in ways (e.g., health habits) that actually caused the difference, not the drug.
Without random assignment, this is an observational comparison, not a controlled experiment. Volunteers may differ systematically from non-volunteers, confounding the result. Only random assignment isolates the treatment's effect. -
In the statement 'the sample mean commute time was 27 minutes,' is 27 minutes a population parameter or a sample statistic?
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Answer. A sample statistic
27 minutes was calculated from a sample of commuters, not from every commuter in the population, so it is a sample statistic used to estimate the (unknown) population parameter. -
A poll of 1,600 randomly selected voters found that 48% favor a candidate. Estimate the margin of error using $\text{MOE}\approx 1/\sqrt{n}$.
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Answer. About 2.5%
$\text{MOE}\approx \dfrac{1}{\sqrt{1600}}=\dfrac{1}{40}=0.025$, or 2.5%. -
Using the poll above (48% favor, MOE ≈ 2.5%), give the interval likely to contain the true population percentage in favor.
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Answer. About 45.5% to 50.5%
Add and subtract the margin of error from the point estimate: $48\%-2.5\%=45.5\%$ and $48\%+2.5\%=50.5\%$. -
If the sample size in a poll is increased from 400 to 1,600 (quadrupled), what happens to the margin of error, assuming the same approximation formula?
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Answer. It is cut in half
Since $\text{MOE}\approx 1/\sqrt{n}$, quadrupling $n$ multiplies the MOE by $1/\sqrt{4}=1/2$, so the margin of error is halved, not quartered. -
A teacher wants to know if a new study method improves test scores. She randomly assigns half her class to use the new method and half to use the old method, then compares average scores. What type of study is this, and why does it allow a causal conclusion?
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Answer. A controlled experiment; random assignment of treatments means any score difference can be attributed to the study method rather than pre-existing differences between students.
Because students were randomly placed into groups, the two groups should be similar in every other respect on average, isolating the study method as the cause of any difference observed. -
In a simulation to test whether a treatment difference is significant, researchers repeatedly reshuffle subjects into two random groups (ignoring the real treatment) and record the difference in group means each time. If the actual observed difference falls far in the tail of this simulated distribution, what should they conclude?
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Answer. The observed difference is statistically significant — unlikely to be due to chance alone, suggesting the treatment had a real effect.
If random reshuffling rarely produces a difference as large as the one actually observed, that is evidence the real treatment assignment (not chance) produced the difference. -
A news article reports 'Students who eat breakfast get better grades' based on a survey asking students whether they eat breakfast and what their GPA is. Which conclusion is best supported by this study design?
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Answer. There is an association between eating breakfast and higher grades, but the study cannot show that breakfast causes better grades.
This is an observational study with no random assignment of a 'breakfast treatment,' so other factors (family income, sleep habits, etc.) could explain both variables. Correlation does not imply causation here. -
A city's park department wants to estimate what fraction of all residents support building a new pool. Which sampling method would best allow them to generalize their results to the entire city population?
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Answer. Selecting a random sample of residents from the full city population, such as with a random number generator applied to a complete resident list.
Random sampling from the entire population, rather than convenience methods like surveying people at the existing pool, gives every resident an equal chance of selection, avoiding bias tied to who happens to be sampled. -
A report claims 'Since sales increased after the store repainted its walls, the new paint color caused higher sales,' based only on comparing sales before and after repainting, with no control group. What is the key weakness in this report's reasoning?
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Answer. With no control group or random assignment, other factors (season, promotions, new products) could explain the sales increase; correlation over time does not establish that the paint caused it.
Evaluating a report requires checking whether a proper controlled comparison was used. Here there's no comparison group that didn't get the new paint, so confounding variables cannot be ruled out.
What people get wrong
- Applying the empirical rule to find a percentage at a boundary that isn't exactly 1, 2, or 3 standard deviations from the mean — instead convert to a z-score and use the standard normal table or software for exact boundaries.
- Concluding that an observational study 'proves' one variable causes another — instead remember that only a randomized controlled experiment supports a causal claim; an observational study can only show association.
- Treating margin of error as if it fixes bias — instead recognize that MOE only measures variability from random sampling; a biased sampling method (e.g., a self-selected online poll) produces a wrong estimate no matter how small the MOE is.
- Confusing random sampling with random assignment — instead ask separately 'were subjects chosen randomly from the population?' (affects generalizability) and 'were subjects randomly placed into groups?' (affects causal claims).
- Forgetting that increasing sample size shrinks margin of error according to $\dfrac{1}{\sqrt{n}}$, not linearly — instead recall that to cut the margin of error in half, the sample size must be roughly quadrupled.
Drill this unit until it sticks
These questions come back on a schedule built from what you get wrong, alongside the rest of Algebra II. Free, and no account needed to start.