Statistics Cumulative Review

A mixed review for choosing statistical tools and explaining conclusions carefully.

This review connects the statistics workflow. The goal is not only to calculate; it is to choose an appropriate method, check conditions, and communicate a careful conclusion.

Learning objectives

After this review you should be able to:

  • choose among descriptive statistics, confidence intervals, hypothesis tests, group comparisons, chi-square tests, and regression;
  • explain why study design affects conclusions;
  • connect standard error to confidence intervals and tests;
  • interpret p-values and confidence intervals accurately;
  • avoid common overclaims about causation and certainty.

The statistical workflow

Use this checklist for most problems:

  1. Question. What is being asked?
  2. Data. Who or what was measured? How were data collected?
  3. Variables. Are variables quantitative or categorical?
  4. Summary. What do plots and descriptive statistics show?
  5. Method. Which inferential tool matches the question and data type?
  6. Conditions. Are sampling, independence, and model assumptions reasonable?
  7. Computation. Calculate the estimate, standard error, interval, test, or model.
  8. Conclusion. Answer in context, including uncertainty and limitations.

Method selection guide

Data/question Useful tool
one quantitative variable, describe shape histogram, boxplot, mean/median, SD/IQR
one categorical variable, estimate proportion one-proportion confidence interval
one mean, estimate population mean one-mean confidence interval
one mean or proportion, test claim one-sample hypothesis test
two independent means or proportions compare groups with difference estimate
before/after on same individuals paired differences
one categorical distribution vs claim chi-square goodness-of-fit
two categorical variables chi-square test of independence
two quantitative variables scatterplot, correlation, regression

Worked example 1: survey estimate

A random sample of 400 adults finds 220 support a policy.

Step 1: Identify data type.

Support is categorical yes/no.

Step 2: Identify goal.

Estimate a population proportion.

Step 3: Compute point estimate.

p-hat = 220/400 = 0.55

Step 4: Choose method.

Use a confidence interval for one proportion.

Step 5: Interpret.

The sample suggests support near 55%, with uncertainty from random sampling.

Worked example 2: test or interval?

A researcher asks whether a machine’s mean fill weight is different from 500 g.

Step 1: Notice the wording.

“Different from 500 g” is a claim-testing question.

Step 2: Write hypotheses.

H0: mu = 500
Ha: mu != 500

Step 3: Use a hypothesis test.

A confidence interval could also be informative, but the wording asks for a test of a baseline value.

Worked example 3: categorical association

A dataset counts people by exercise category and stress category.

Step 1: Identify variables.

Both variables are categorical.

Step 2: Choose method.

Use a chi-square test of independence.

Step 3: Interpret carefully.

A significant association does not automatically prove exercise causes stress changes, especially if the study is observational.

Worked example 4: quantitative relationship

A dataset contains house size and selling price.

Step 1: Identify variables.

Both are quantitative.

Step 2: Start with a scatterplot.

Look for linear form, outliers, and clusters.

Step 3: Choose method.

If roughly linear, use regression to predict price from size.

Step 4: Caution.

Avoid extrapolating to house sizes far outside the dataset.

Common final-exam pitfalls

  • Choosing a method before identifying variable types.
  • Ignoring whether samples are paired or independent.
  • Using a hypothesis test when the question asks for estimation.
  • Saying “accept the null” instead of “fail to reject the null.”
  • Claiming causation from observational data.
  • Reporting a number without context, units, or uncertainty.
  • Trusting a formula after biased sampling.

Practice exercises

  1. Which tool summarizes one quantitative variable before inference?
  2. Which tool estimates a population mean with uncertainty?
  3. Which tool tests association between two categorical variables?
  4. Which tool predicts a quantitative response from a quantitative explanatory variable?
  5. A study measures the same patients before and after treatment. Are samples paired or independent?
  6. A small p-value proves the alternative is true. True or false?
  1. Descriptive statistics and plots.
  2. Confidence interval for a mean.
  3. Chi-square test of independence.
  4. Regression.
  5. Paired.
  6. False. A small p-value is evidence against the null under assumptions; it is not proof.

Guided mixed-review checkpoints

Checkpoint 1: identify variable types

Guess first. Age is quantitative or categorical?

Guided answer. Age is quantitative when measured numerically. Age group, such as 18-24, is categorical.

Classify: favorite color, blood pressure, pass/fail result.

Checkpoint 2: choose a method

Guess first. A two-way table of major and graduation status needs regression or chi-square?

Guided answer. Both variables are categorical, so use a chi-square test of independence.

Choose a method: height and weight for the same people.

Checkpoint 3: explain uncertainty

Guess first. What connects standard error to both confidence intervals and hypothesis tests?

Guided answer. Standard error measures how much an estimate varies across samples. Intervals use it to form a margin of error; tests use it to standardize distance from the null value.

Why does increasing sample size usually narrow a confidence interval?

Checkpoint 4: write a conclusion

Guess first. Should a conclusion mention the context?

Guided answer. Yes. A statistical conclusion should answer the original question using the problem’s units and population, with uncertainty and limitations.

Rewrite carefully: “The p-value is small, so the treatment definitely works.”

  1. Favorite color: categorical; blood pressure: quantitative; pass/fail: categorical.
  2. Regression or correlation, after a scatterplot, because both variables are quantitative.
  3. Larger sample size reduces standard error, which reduces margin of error.
  4. Better: “The small p-value provides evidence against the null hypothesis and suggests the treatment may have an effect, assuming the study design and model conditions are appropriate.”

Cumulative statistics guessing game

Using this lesson with edumath and SymPy

Use edumath helpers to connect the workflow.

from edumath.statistics import (
    confidence_interval_proportion,
    describe,
    one_sample_z_test,
)

describe([12, 15, 17, 20, 21])
DescriptiveStats(mean=17.0, median=17.0, variance=13.5, standard_deviation=3.6742346141747673, minimum=12.0, maximum=21.0)
confidence_interval_proportion(220, 400)
ConfidenceInterval(estimate=0.55, lower=0.5012465114550304, upper=0.5987534885449697, margin_of_error=0.048753488544969664, confidence_level=0.95)
one_sample_z_test(105, 100, 2.5, alternative="greater")
HypothesisTestResult(statistic=2.0, p_value=0.02275013194817921, alternative='greater')

Use SymPy to show the shared structure behind intervals and tests.

import sympy as sp

estimate, null, se, z = sp.symbols("estimate null se z")
sp.solve(sp.Eq(z, (estimate - null) / se), null)
[estimate - se*z]