STOR 155 Introduction to Data Models and Inference Help for UNC-Chapel Hill Students

Course overview

STOR 155 introduces statistical reasoning as a connected process rather than a collection of unrelated calculations. A useful analysis begins with a question about a population, a process, or a relationship. It then asks what data could address that question, how those data were generated, and what limitations entered before any arithmetic began. Descriptive summaries reveal patterns in the observed values, while probability models provide language for the variation that would be expected across samples or repeated outcomes.

That foundation supports the move from description to inference. Correlation and regression model relationships between variables. Confidence intervals describe the precision of an estimate, and hypothesis tests evaluate how compatible the data are with a stated null model. Each result still has to be translated into the context that motivated the analysis. A technically correct number can be unhelpful when the observational units, variables, assumptions, or target population are unclear.

The course therefore moves through a reasoning chain: collect and describe data, examine how the data arose, represent uncertainty, model a relationship or population quantity, estimate or test, and explain what the evidence supports. The aim is not to press a statistical button. It is to build an argument in which the question, design, method, output, and conclusion agree with one another.

Current STOR 155 course facts

UNC-Chapel Hill lists STOR 155 in the Department of Statistics and Operations Research. The current catalog identifies it as a three-credit, letter-grade course with MATH 110 as the prerequisite. An honors version is available. The course carries FC-QUANT, while FY-LAUNCH applies only to designated sections.

These catalog facts describe the course identity, not the details of a particular class meeting. Section schedules, instructors, materials, assessment formats, and grading weights can vary and are not needed to understand the statistical scope presented here.

Course code

STOR 155

Title

Introduction to Data Models and Inference

Department

Statistics and Operations Research

Credits

3

Prerequisite

MATH 110

Grading status

Letter grade

How the course topics fit together

A statistical conclusion is only as strong as the links that lead to it. The following flow is a practical map for the topics in STOR 155. It is not a rigid recipe; analysts may revisit earlier steps when a graph, assumption check, or design limitation changes the question they can responsibly answer.

1. 1.Question
2. 2.Data
3. 3.Study design
4. 4.Description
5. 5.Probability model
6. 6.Statistical model
7. 7.Estimate or test
8. 8.Interpretation

The question identifies the population, quantity, comparison, or relationship of interest. Data supply observations, but their meaning depends on study design: who or what was measured, how observations were selected, and whether a treatment was assigned. Description comes next because distributions, unusual values, missingness, and visible relationships may reveal that the initial plan needs revision.

A probability model represents the random mechanism or uncertainty relevant to the problem. A statistical model connects that uncertainty to an unknown population parameter or relationship. Estimates and tests then summarize what the observed sample says under those assumptions. Interpretation returns to ordinary language, limits the claim to the design, and distinguishes numerical evidence from practical importance. Treating these steps as connected prevents a polished calculation from outrunning the data that produced it.

Data and variable types

Every row in a data set should represent a defined observational unit: a person, location, device, transaction, day, or another entity measured in a consistent way. Variables are the characteristics recorded for those units. A categorical variable places an observation into a group, such as a service category or preferred study setting. A quantitative variable records a numerical amount for which arithmetic has contextual meaning, such as minutes waited, distance traveled, or number of attempts.

The distinction matters because summaries and graphs should match the variable. Counts and proportions often describe categorical distributions. Centers, spreads, percentiles, and shape help describe quantitative distributions. A bar display answers a different question from a histogram, and a histogram answers a different question from a scatterplot. The graph should make the structure visible without implying precision or comparisons that the data do not support.

In questions about relationships, it is also useful to name explanatory and response variables. The explanatory variable is used to account for or predict variation in the response; that role does not automatically make it a cause. For example, a study might compare hours of optional practice with time needed to complete a neutral puzzle. Practice time can be the explanatory variable and completion time the response, but self-selection, prior experience, and other factors may influence both.

Before choosing a method, inspect distributions and data quality. Look for impossible values, inconsistent units, missing observations, clusters, skewness, and points far from the rest. Decide whether a summary should describe a typical value, total variation, group difference, or relationship. Clear variable definitions make later probability and inference statements much easier to audit.

Sampling and experimental design

A population is the full group or process named by the research question; a sample is the set of units actually observed. The sample statistic is known after data are collected, while the corresponding population parameter remains unknown. Inference uses the statistic to learn about the parameter, but the quality of that inference depends on how the observations entered the sample.

Random sampling gives eligible population members a known chance of selection and helps protect population estimates from systematic selection patterns. It does not guarantee that one particular sample perfectly mirrors the population, but it supplies a defensible basis for quantifying sampling variability. A larger convenience sample can produce a narrow calculation around the wrong target if certain kinds of units are consistently missing. More observations reduce random error under appropriate conditions; they do not automatically remove selection bias.

An observational study records what occurs without assigning the explanatory condition. Such a study can identify patterns and associations, but an apparent relationship may reflect confounding: another variable connected to both the explanatory and response variables. An experiment deliberately assigns conditions. Random assignment helps balance measured and unmeasured background characteristics across treatment groups over repeated experiments, strengthening a causal comparison when the experiment is otherwise well conducted.

Random sampling and random assignment solve different problems. Sampling concerns which population can be represented. Assignment concerns whether a difference between conditions can reasonably be attributed to the treatment. A study may have one, both, or neither. The conclusion should match the design actually used: population generalization needs a defensible selection process, and a causal interpretation needs a defensible treatment comparison rather than correlation alone.

Basic probability models

Probability provides a model for outcomes that are not known in advance. A random variable assigns a numerical value to each possible outcome of a random process. Its expected value describes the long-run center of that model, not a promise about the next observation. Variability describes how widely outcomes tend to differ from the center. Two processes can have the same expected value while having very different levels of uncertainty.

The normal distribution is a continuous, symmetric model described by a center and spread. It can be useful for measurement variation and for sampling distributions under suitable conditions, but not every variable is normally distributed. Strong skewness, bounds, outliers, mixtures, or a small sample may make a normal approximation questionable. The model should be justified by the process, the data, and the inferential method rather than selected because its curve is familiar.

The binomial distribution models a count of successes across a fixed number of trials when each trial has two defined outcomes, the success probability stays constant, and the trials satisfy the needed independence condition. “Success” is only a label; it need not mean a desirable event. A sample proportion is the success count divided by the number of trials, so binomial reasoning connects probability to inference about a population proportion.

Assumptions are the bridge between a real question and a probability calculation. State what each trial or measurement represents, what is treated as random, and why the chosen model is a reasonable approximation. If those assumptions do not fit, a precise probability output can still be misleading.

Correlation and regression

A scatterplot is the first place to examine a relationship between two quantitative variables. Look for direction, form, strength, clusters, gaps, and unusual points. An upward pattern indicates a positive association, while a downward pattern indicates a negative association. A straight-line summary is useful only when a roughly linear pattern is a defensible description of the observed range.

Correlation summarizes the direction and strength of a linear relationship in a unitless number. It can be strongly affected by unusual observations and does not describe curved relationships well. A correlation near zero means little linear association; it does not prove that the variables are unrelated. Correlation also does not establish causation because lurking variables, reverse direction, selection, or coincidence may explain the pattern.

A simple regression line expresses a predicted response as an intercept plus a slope multiplied by the explanatory value. The slope is the predicted change in the response for a one-unit increase in the explanatory variable, within the model and context. The intercept is the predicted response when the explanatory variable equals zero, but that interpretation is useful only if zero is meaningful and reasonably close to the observed range.

A residual is the observed response minus the model prediction. Residual plots help reveal curvature, changing spread, clusters, and individual points that a line does not capture. A strong fit statistic by itself does not validate a model. The pattern of residuals, study design, measurement quality, and purpose of the analysis all matter.

Prediction should remain within a range where the data support the relationship. Extrapolation applies the line beyond the observed explanatory values, where the relationship may change or cease to make sense. Even a useful prediction is not a causal statement. Explain what the model predicts, how uncertain that prediction is, and what conditions limit its use.

Confidence intervals

A parameter is a fixed but unknown feature of a population or model, such as a mean, proportion, or regression slope. A sample produces a point estimate of that parameter. Because a different random sample would usually produce a different estimate, the standard error describes the method’s sampling variability. A confidence interval combines the point estimate with a margin of error to display a range of values compatible with the data and model at a stated confidence level.

For a familiar two-sided interval, the margin of error is a critical multiplier times the standard error. Higher confidence generally requires a wider interval when the data and method stay the same. More independent information generally reduces the standard error, while greater variability generally increases it. These relationships help explain why precision cannot be judged from sample size alone.

A 95% confidence procedure is designed so that, over many repetitions under its assumptions, about 95% of the intervals it produces contain the fixed parameter. After one interval has been calculated, the parameter is not randomly moving in and out of it. Avoid saying there is a 95% probability that the fixed parameter lies inside the completed interval. Instead, report the interval in context and explain which parameter values remain plausible under the procedure.

An interval should also be read for practical meaning. A narrow interval around a trivial change can be statistically precise without being important. A wide interval may include effects that would lead to different decisions. Design bias, measurement problems, and model misspecification are not repaired by a small margin of error.

Hypothesis testing

A hypothesis test begins with a null hypothesis about a population parameter or model and an alternative that represents the competing possibility. The test statistic measures how far the observed estimate is from the null expectation relative to its modeled variability. The p-value is the probability, assuming the null model and other test assumptions, of obtaining a test statistic at least as incompatible with that model as the one observed.

That definition conditions on the null model; it does not assign odds to the truth of a hypothesis. Nor does the p-value report how much of a result came from chance or describe the effect's magnitude. Smaller values indicate greater incompatibility between the observations and the stated null model, but they do not identify the cause of that incompatibility or validate every assumption used in the calculation.

A decision threshold such as a significance level should be selected as part of the analysis plan rather than adjusted after seeing the result. If the p-value falls below the threshold, the conventional decision is to reject the null hypothesis. If it does not, the decision is to fail to reject. Failing to reject does not prove the null; the data may be too limited or variable to distinguish meaningful alternatives.

Evidence should not be reduced to a bright line. Report the estimate, uncertainty, p-value when relevant, assumptions, and study limitations. Statistical significance does not establish practical significance, and a result that misses a threshold is not automatically unimportant. The strongest conclusion states what the evidence suggests, how large the estimated effect is, and what remains uncertain.

Means, proportions, and regression parameters

The official STOR 155 scope includes inference for means, proportions, and regression parameters. These targets share a common reasoning structure, but the appropriate method still depends on variable types, study design, sampling behavior, and assumptions.

• Means

  A population mean describes the long-run average of a quantitative variable. The sample mean estimates it, while an interval or test accounts for sampling variability and the design that produced the observations.

• Proportions

  A population proportion describes the share of units with a defined categorical outcome. The sample proportion estimates it, with uncertainty shaped by the sample size, observed balance, selection process, and model conditions.

• Regression parameters

  A regression coefficient describes a modeled relationship, such as the expected response change associated with a one-unit explanatory change. Inference for that coefficient depends on the form of the model, residual behavior, and study design.

Spreadsheet-based statistical work

The course scope includes the use of spreadsheet software. A reliable workflow keeps the original observations separate from calculations and makes each transformation understandable to another reader. The output should support reasoning, not replace it.

1. 1

   Preserve raw data

   Keep an unchanged source table and perform cleaning or recoding in a documented working copy.

2. 2

   Define variables

   Give each column a clear meaning, unit, allowed range, and role in the question.

3. 3

   Check data types

   Confirm that categorical labels, numeric values, dates, and identifiers are represented consistently.

4. 4

   Inspect missing values

   Distinguish a true zero from unavailable, invalid, or not-applicable data before calculating summaries.

5. 5

   Calculate summaries

   Choose counts, proportions, centers, or spreads that match the variables and research question.

6. 6

   Create appropriate graphs

   Use plots to inspect distributions, comparisons, and relationships before relying on a model.

7. 7

   Document formulas

   Make ranges, references, transformations, and exclusions visible enough to review.

8. 8

   Verify outputs

   Check formulas, ranges, units, sample counts, and a few calculations independently.

9. 9

   Interpret in context

   Translate the result back to the population, parameter, design, and limitations in the original question.

Common STOR 155 difficulties

Many statistics errors occur in the explanation rather than the arithmetic. These checkpoints help identify where the reasoning chain has broken.

• Choosing the correct summary

  Start with the variable type and question. A mean can be distorted by a skewed distribution or unusual values; a proportion is meaningful only after the category and denominator are clearly defined.

• Separating the sample from the population

  The sample is observed and produces a statistic. The population is the broader target and has a parameter. A sample result does not automatically describe every possible population.

• Interpreting a scatterplot

  Describe direction, form, strength, and unusual points before quoting correlation or fitting a line. A curved or clustered pattern may make a single linear summary inadequate.

• Keeping correlation separate from causation

  Association shows that variables move together in the observed data. A causal conclusion needs design support and careful attention to confounding, selection, and alternative explanations.

• Explaining a p-value

  State the probability in relation to the null model and results at least as extreme as observed. Do not turn it into the probability that either hypothesis is true.

• Connecting an interval to the question

  Name the parameter and units, then explain which values are compatible with the data and method. Do not describe the interval as containing most individual observations.

• Recognizing design limits

  A precise analysis of a biased or confounded study remains limited. State which population and which type of conclusion the design can support.

• Writing beyond the output

  A table of coefficients or test results is not a conclusion. Translate direction, magnitude, uncertainty, evidence, assumptions, and limitations into ordinary language.

How to approach STOR 155 course work

Use this ten-step workflow as a reasoning checklist. It is a study method, not a UNC-Chapel Hill rubric, and the order may loop when new evidence reveals a data or model problem.

1. 1

   1. Identify the question

   Write the population, outcome, comparison or relationship, and the decision the analysis is meant to inform.

2. 2

   2. Identify the observational units

   State what each row represents and check that repeated measurements or grouped units are recognized.

3. 3

   3. Classify the variables

   Mark variables as categorical or quantitative and identify explanatory and response roles when relevant.

4. 4

   4. Determine how the data were generated

   Review sampling, assignment, timing, measurement, missingness, and possible confounding before choosing a conclusion.

5. 5

   5. Inspect the data

   Use summaries and graphs to find shape, spread, relationships, unusual values, invalid entries, and useful comparisons.

6. 6

   6. Choose a defensible model or method

   Match the target parameter, variable types, design, and analytical purpose rather than selecting from a keyword alone.

7. 7

   7. Check assumptions

   Explain which conditions matter, how they were evaluated, and how a violation would change the method or claim.

8. 8

   8. Perform the calculation

   Keep units, sample sizes, formulas, and intermediate quantities clear enough to reproduce and inspect.

9. 9

   9. Interpret in context

   Report the estimate, uncertainty, evidence, and practical meaning using the language of the original question.

10. 10

    10. Verify the conclusion

    Check that the words do not claim a broader population, stronger causal result, or greater certainty than the design and model support.

Featured STOR 155 guides

Use the conceptual guide when individual topics feel disconnected, then use the study guide to organize Course topics, review the STOR 120 comparison, and plan exam preparation without relying on section-specific materials.

Related statistics courses at other institutions

These DomyClass resources cover related statistical ideas in courses at other institutions. They can provide another explanation of a concept, but they do not establish course equivalence, shared requirements, transfer credit, or prerequisite substitution for STOR 155.

Related statistics guides

Use these focused guides to revisit sampling quality, testing language, regression output, and the relationship between uncertainty and practical importance.

Sources

Course identity and scope are based on the current UNC-Chapel Hill catalog, the UNC Statistics and Operations Research course page, and the department catalog page.

• Statistics and Operations Research (STOR)

  University of North Carolina at Chapel Hill

• Undergraduate Courses

  UNC Statistics and Operations Research

• Department of Statistics and Operations Research

  University of North Carolina at Chapel Hill