How do you distinguish surveys, experiments, and observational studies, and what makes a sample trustworthy?
Distinguish sample surveys, experiments, and observational studies; recognize random sampling and sources of bias; understand the role of randomization and a control group; and use simulation to model a sampling distribution and estimate a margin of error.
A NY Regents Algebra II answer on study design: surveys, experiments, and observational studies, random sampling and bias, randomization and control groups, and using simulation to estimate a margin of error.
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What this topic is asking
The Regents Algebra II exam (the Making Inferences and Justifying Conclusions, S-IC, cluster) wants you to distinguish sample surveys, experiments, and observational studies, recognize random sampling and sources of bias, understand the role of randomization and a control group, and use simulation to model a sampling distribution and estimate a margin of error. This topic asks for clear reasoning about how data is collected, not just calculation.
Three kinds of studies
The Regents distinguishes three designs by how data is gathered.
- A sample survey asks questions of a sample to estimate a population characteristic (a poll of likely voters).
- An observational study observes and measures existing groups without imposing any treatment (comparing smokers and non-smokers as they already are).
- An experiment actively imposes a treatment on subjects and compares outcomes (giving one group a drug and another a placebo).
Only a well-designed experiment with random assignment can support a claim that a treatment causes an effect; surveys and observational studies can show association but not causation.
Random sampling and bias
A sample is trustworthy only if it is representative of the population, and random sampling (every member having a known, nonzero chance of selection) is the safeguard. Bias is any systematic tendency for a sample to misrepresent the population: convenience samples, voluntary-response samples, and leading questions all introduce it. The Regents often asks you to identify why a sampling method is biased and how it would skew the conclusion.
Randomization and control in experiments
A good experiment uses two devices. Randomization (randomly assigning subjects to treatment and control groups) balances out other variables so the groups are comparable. A control group (which receives no treatment or a placebo) provides the baseline to compare the treatment against. Together they isolate the treatment's effect, which is why an experiment can establish cause where an observational study cannot.
Simulation and margin of error
A simulation uses random digits or technology to repeatedly draw samples and record a statistic, building up a sampling distribution of how that statistic varies from sample to sample.
Reasoning about it for credit
A clarifying point worth stressing is that causation requires a randomized experiment: the Regents tests whether you know that an observational study or survey, however large, can show only association. A second idea is that the margin of error reflects sampling variability, so a larger sample produces a smaller margin (the sampling distribution tightens). When a question asks for the plausible range of the true value, build the interval as the sample statistic plus or minus the margin of error; reporting only the margin, or only the standard deviation, leaves out the interval the question wants. Explaining a source of bias or the role of randomization in words is frequently where the constructed-response credit sits.
Try this
Q1. A study compares people who already exercise with those who do not, without assigning anyone. What type of study is it? [1 credit]
- Cue. No treatment is imposed, so it is an observational study.
Q2. A simulation gives sample proportions with standard deviation 0.04. Estimate the margin of error (two standard deviations). [1 credit]
- Cue. .
Exam-style practice questions
Practice questions written in the style of NYSED exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
Regents (style)2 marksPart I (multiple choice). A researcher randomly assigns volunteers to take a new medication or a placebo and compares the outcomes. This is best described as (1) an observational study (2) a survey (3) an experiment (4) a censusShow worked answer →
The correct answer is (3).
Because the researcher actively imposes a treatment (medication or placebo) and randomly assigns subjects, this is an experiment. An observational study would only observe existing groups without assigning treatment; a survey gathers responses without intervention. The random assignment to treatment is the defining feature of an experiment.
Regents (style)2 marksPart II (constructed response). A simulation of a poll produces 200 sample proportions with a mean of 0.52 and a standard deviation of 0.03. Estimate the margin of error (using two standard deviations) and state the interval of plausible values for the true proportion.Show worked answer →
A 2-credit question: 1 credit for the margin of error, 1 for the interval.
The margin of error at about two standard deviations is . The interval of plausible values is , that is, from to . Reporting only the standard deviation as the margin, or forgetting to build the interval around the sample mean, costs a credit.
Related dot points
- Recognize the properties of a normal distribution; use the empirical (68-95-99.7) rule; compute a z-score; and use z-scores (with a calculator or table) to find the proportion of data in an interval.
A NY Regents Algebra II answer on the normal distribution: the bell-curve properties, the 68-95-99.7 empirical rule, computing z-scores, and using them to find the proportion of data in an interval.
- Compute conditional probability from two-way tables; apply the addition rule for the probability of A or B; apply the multiplication rule for A and B; and test for independence of two events.
A NY Regents Algebra II answer on probability: conditional probability from two-way tables, the addition rule for A or B, the multiplication rule for A and B, and testing two events for independence.
- Fit linear, exponential, and other regression models to data; interpret the parameters and the correlation coefficient in context; use a residual plot to judge whether a model is appropriate; and use a model to make predictions.
A NY Regents Algebra II answer on regression: fitting linear and exponential models, interpreting parameters and the correlation coefficient, reading a residual plot to judge model fit, and making predictions.
- Construct and interpret scatter plots; fit a linear (or exponential) model to bivariate data; interpret the slope and intercept in context; compute and interpret residuals; and distinguish the correlation coefficient from causation.
A NY Regents Algebra I answer on bivariate data: scatter plots, fitting a line of best fit, interpreting slope and intercept, computing residuals, reading the correlation coefficient, and the correlation-versus-causation distinction.
- Represent and interpret one-variable data with dot plots, histograms, and box plots; compute and interpret measures of center (mean, median) and spread (range, interquartile range, standard deviation informally); identify outliers; and compare two distributions.
A NY Regents Algebra I answer on one-variable data: dot plots, histograms, and box plots, the mean and median, range, interquartile range and standard deviation, the 1.5 times IQR outlier rule, and comparing distributions.
Sources & how we know this
- Educator Guide to the Regents Examination in Algebra II — NYSED (2025)
- New York State Next Generation Mathematics Learning Standards — NYSED (2017)