Purpose and Overview

Learning Objectives

1. 1Describe the complete statistical investigation process from question to conclusion.
2. 2Formulate a clear, answerable statistical question.
3. 3Plan appropriate data collection for a given question and context.
4. 4Organize, clean, and prepare a data set for analysis.
5. 5Choose appropriate displays and numerical summaries for a given distribution or relationship.
6. 6Select a statistical model appropriate to the pattern in the data.
7. 7Evaluate model fit using residuals, graphs, and contextual reasoning.
8. 8Draw conclusions from evidence with appropriate acknowledgment of uncertainty.
9. 9Explain uncertainty using confidence intervals, margins of error, and limitations.
10. 10Communicate statistical findings clearly using graphs, tables, and written explanation.
11. 11Identify ethical concerns in data collection, analysis, and reporting.
12. 12Recognize and correct common mistakes in data-based reasoning.
13. 13Apply the full statistical investigation process in a capstone project.
14. 14Describe how statistics connects to college courses, careers, and civic life.

Sections at a Glance

• 14.1The Full Statistical Investigation ProcessSeven stages: Ask, Plan, Collect, Organize, Analyze, Interpret, Communicate; why each stage matters
• 14.2Asking a Statistical QuestionWhat makes a question statistical; anticipating variability; specifying population and variables; revising weak questions
• 14.3Planning Data CollectionPopulation, sampling method, operationalizing variables, minimizing bias; evaluating a collection plan
• 14.4Organizing and Cleaning DataMissing values, outliers, inconsistent categories, wrong types, duplicates; creating new variables; documentation
• 14.5Choosing Displays and SummariesMatching display to variable type; mean vs. median; what to look for in shape, center, spread
• 14.6Choosing a ModelLinear, quadratic, exponential, logarithmic, power, probability, sampling distribution models; choosing by shape
• 14.7Evaluating Model FitResidual plots; patterns in residuals; R²; context checks; interpolation vs. extrapolation
• 14.8Drawing Conclusions from EvidenceLanguage of conclusion; association vs. causation; scope of inference; weak vs. strong conclusions
• 14.9Explaining UncertaintySources of uncertainty; confidence intervals; margin of error; p-values; scope of generalization
• 14.10Communicating with Graphs and TablesPrinciples of effective communication; labeling; describing displays; avoiding misleading presentations
• 14.11Data Ethics and Misuse of StatisticsCherry-picking, misleading graphs, overgeneralization, p-hacking, privacy, algorithmic bias; ethical practice
• 14.12Capstone Project DesignChoosing a question; designing collection; conducting analysis; writing a full statistical report
• 14.13Statistics in Public Policy, Science, Finance, and TechnologyGovernment data, scientific research, financial models, machine learning; statistical literacy in civic life
• 14.14Common Data-Based Decision MistakesTen recurring errors; explanations and corrections; prevention strategies
• 14.15Readiness for College Statistics and Data ScienceWhat comes next; habits of mind; the value of statistical literacy in every field

The Full Statistical Investigation Process

Asking a Statistical Question

1. Identify each as a statistical question (S) or not a statistical question (N). Explain each answer.
   
   a. What is the height of Mount Everest?
   b. How much do adult males in the United States weigh?
   c. How long does it take a specific ball to fall from a ten-meter height?
   d. How does the average price of gasoline vary by region across the country?
2. Revise this question into a strong statistical question: "Are students stressed?"
3. Write your own statistical question about sleep and academic performance. Include the population, the variables, and what comparison or relationship the question addresses.

Planning Data Collection

1. For each study description, identify at least one source of bias and suggest how to address it.
   
   a. A television station asks viewers to call in to express their opinion about a proposed city park.
   b. A researcher studying exercise habits surveys members of three local gyms.
   c. A school principal reads a survey aloud in class and collects anonymous responses immediately.
2. A researcher wants to compare reading levels between students in urban and rural schools in a state. Describe a stratified sampling plan that would allow fair comparison.
3. A student wants to study whether the amount of fertilizer affects plant height in a controlled experiment. List the variables, describe how you would operationalize each, and explain how treatments would be assigned.

Organizing and Cleaning Data

1. A data set of 50 survey responses includes these problems. Describe how you would handle each.
   
   a. Five entries in "grade level" say "11th" while all others say "11."
   b. One student reported a GPA of 5.8 on a 4.0 scale.
   c. The "hours of screen time" column has three blank entries.
   d. Two rows have identical student IDs and identical responses on all questions.
2. Explain in your own words why data cleaning matters. What can happen to a statistical conclusion if data are not cleaned?
3. A data set includes height (in inches) and weight (in pounds). Describe how to create a new variable for BMI. Formula: BMI = 703 × weight / height²

Choosing Displays and Summaries

For each scenario, state the appropriate display and numerical summary. Justify your choices.

1. A teacher records final exam scores (0–100) for 32 students. She wants to describe the distribution of scores.
2. A researcher surveys 200 adults and records political affiliation (four categories) and opinion on a tax policy (three categories). He wants to compare opinion across political groups.
3. A biologist records the length (in cm) and weight (in grams) of 45 fish. She wants to explore whether longer fish tend to be heavier.
4. A student records daily high temperatures for 60 days during summer and fall. The data appear to have two clusters: one around 85°F and one around 55°F.

Choosing a Model

For each scenario, identify the most appropriate type of model and explain your reasoning.

1. A car's value depreciates by approximately 15% of its current value each year.
2. A student's reading speed improves rapidly in the first few months of a reading program and then levels off.
3. Weekly sales revenue plotted against menu items offered shows a roughly straight-line pattern with positive slope.
4. A ball thrown upward reaches its highest point and then comes back down. Height is measured at one-second intervals.
5. A savings account earns 4% annual interest compounded annually. Balance recorded at end of each year.

Evaluating Model Fit

1. A residual plot shows residuals all close to zero for small values of x, but large and positive for large values of x. What does this tell you about the model?
2. A residual plot for a fitted exponential model shows residuals scattered randomly above and below zero with no pattern. What does this tell you?
3. A student fits a linear model with r = 0.72. A classmate says R² = 0.518 is too low. A second classmate says r = 0.72 is strong enough. Write a paragraph explaining how to evaluate model fit more completely, using both the residual plot and R².
4. Explain why extrapolation is risky, using a specific example from everyday life.

Drawing Conclusions from Evidence

1. A survey finds students who own more than 50 books at home score 18 points higher on a reading test. A politician concludes the city should buy every student 50 books. Write a paragraph explaining the problem with this conclusion and what additional evidence would be needed.
2. An observational study finds r = –0.81 between fast-food restaurants per capita and average life expectancy in counties. Is it appropriate to conclude that fast food causes shorter lives? Explain what other explanation might account for this correlation.
3. A randomized experiment with 200 participants finds no statistically significant difference in weight loss between a new diet and a standard diet (p = 0.21). Write a complete conclusion including what the data showed and what can and cannot be concluded.

Explaining Uncertainty

1. A 95% confidence interval for average homework hours per week among high school students is (5.2, 7.8). Write a sentence interpreting this interval in plain language.
2. A hypothesis test produces p = 0.04. A student concludes: "There is a 4% probability that the null hypothesis is true." Explain the error in this interpretation and state the correct interpretation.
3. A study of 18 patients produces a very wide confidence interval. A study of 180 patients produces a much narrower interval. Explain why sample size affects interval width.
4. A national poll reports 61% support a new infrastructure bill, margin of error ±2.5%. Explain what the margin of error means and identify one source of uncertainty it does not capture.

Communicating with Graphs and Tables

1. A histogram is displayed without a title, without labeled axes, and without units. What information is missing, and why does each missing element matter?
2. A student presents opinion data using raw counts in a two-way table where group sizes are very different. A classmate says it is hard to interpret. Explain what the student should do differently and why.
3. Write a one-paragraph description of imaginary data: a dot plot of test scores for 25 students ranging from 62 to 98, most clustered between 80 and 90, one outlier at 62, one at 98, slightly left-skewed.
4. Describe two ways a graph could be designed to make a result look more impressive than it is. For each, explain what an honest alternative would look like.

Data Ethics and Misuse of Statistics

1. A newspaper headline reads: "Study finds chocolate lovers live longer." The study found that people who ate chocolate at least weekly had lower risk of heart disease than non-chocolate-eaters. Identify at least two problems with the headline and suggest a more accurate one.
2. A school district reports that average test scores increased 12 points after a new curriculum was adopted, comparing this year's students to last year's students. A critic says this is a flawed comparison. Explain the problem and describe what evidence would be stronger.
3. A social media company analyzes posts to predict which users are at risk for depression and sells this information to insurance companies. Identify at least two ethical concerns.
4. Explain what p-hacking is and why it is a problem for scientific research.

Capstone Project Design

1. Write a capstone project plan for a topic of your own choosing. Follow the Ask → Plan → Collect → Organize → Analyze → Interpret → Communicate structure. Specify the statistical question, population, sampling method, variables, and types of analysis.
2. Identify three potential sources of bias in the sleep and quiz score project in Example 14.12.1 and describe how each might be addressed.
3. A classmate finds r = 0.61 between hours of daily physical activity and self-reported happiness score. Write the conclusion section for this project, including what the data show, what cannot be concluded, and what further investigation might explore.

Statistics in Public Policy, Science, Finance, and Technology

1. A randomized controlled trial evaluates a new blood pressure drug. Explain why random assignment of patients to treatment and control groups is essential for concluding that the drug causes reduced blood pressure.
2. A polling organization reports 48% support for a ballot measure, with a margin of error of ±4 points at the 95% confidence level. Explain what this means and describe what would happen to the margin of error if the sample size were doubled.
3. A city uses historical crime data to train an algorithm that predicts which neighborhoods are at high risk for crimes. Identify at least two ways that using historical data in this way could introduce bias or cause harm.
4. Explain how statistical thinking is relevant to at least one career that interests you personally. Be specific about which concepts from this book are directly relevant.

Common Data-Based Decision Mistakes

For each scenario, identify the mistake and explain how the claim should be qualified.

1. A school principal reports that after adopting a new attendance policy, the average GPA rose from 2.8 to 3.0. She concludes the new attendance policy caused the improvement.
2. A survey of 15 customers finds 80% are satisfied with a new product. The marketing team concludes that most customers nationwide love the product.
3. A scatter plot shows r = 0.92 between temperature and ice cream sales. A columnist concludes that eating ice cream raises body temperature.
4. A company reports their drug reduces the duration of a cold by an average of 4 hours (p = 0.03). The press release announces: "New drug dramatically shortens colds."

Readiness for College Statistics and Data Science

1. List three specific ways that statistical reasoning appears in a career area of your choice. Be specific about which statistical concepts from this book are directly relevant.
2. Explain the difference between this course and a college Probability and Statistics course taken after Calculus. What new mathematical tools would be added?
3. A classmate says: "I'm going to study English literature, so I'll never need statistics." Write a paragraph explaining at least two ways that statistical literacy is relevant to someone working in the humanities or social sciences.
4. Reflect on the statistical investigation process (Ask → Plan → Collect → Organize → Analyze → Interpret → Communicate). Which stage do you find most challenging? Which do you find most interesting? Why?

Chapter Summary

• Sec 14.1The statistical investigation process has seven stages: Ask, Plan, Collect, Organize, Analyze, Interpret, and Communicate. Every stage matters. A flawed plan cannot be rescued by sophisticated analysis.
• Sec 14.2A statistical question anticipates variability, can be answered with data, and specifies the population of interest. Weak questions are too vague, too narrow, or already answered by a known fact.
• Sec 14.3A data collection plan specifies who will be measured, how they will be selected, what will be measured, and how bias will be minimized. Poor collection cannot be fixed by analysis.
• Sec 14.4Data cleaning addresses missing values, impossible values, inconsistent categories, wrong variable types, and duplicates. All cleaning decisions should be documented. In real investigations, cleaning often takes more time than the analysis itself.
• Sec 14.5Displays and summaries should match the variable type and the distribution shape. Compare proportions rather than raw counts when group sizes differ. Describe shape, center, spread, and unusual features in every quantitative display.
• Sec 14.6Model selection requires matching the model shape to the pattern in the data. Look at a scatter plot or graph first, then try models whose shape matches. Check fit with residual plots.
• Sec 14.7A well-fitting model produces a residual plot with random scatter around zero. Curved patterns indicate the wrong model shape. Fan shapes indicate non-constant variance. R² alone is not enough to confirm a good fit.
• Sec 14.8Statistical conclusions should be specific, grounded in evidence, and honest about uncertainty. Association does not prove causation. Conclusions are only as broad as the data allow — scope of inference is determined by the study design and the sample.
• Sec 14.9Uncertainty is communicated through confidence intervals, margins of error, p-values, and honest statements of limitations. A 95% CI reflects the reliability of the method, not the probability that any one interval contains the true value.
• Sec 14.10Effective statistical communication labels everything, describes what displays show, reports numerical summaries alongside graphs, acknowledges limitations, and avoids presentations that distort the message visually.
• Sec 14.11Statistical misuse includes cherry-picking, misleading graphs, overgeneralization, p-hacking, suppressing uncertainty, and algorithmic bias. Ethical practice requires honest reporting, protection of privacy, and acknowledgment of limitations.
• Sec 14.12A capstone project follows all seven stages and produces a report including an introduction, methods, analysis (graphs and summaries), inference where applicable, and a conclusion that acknowledges uncertainty and limitations.
• Sec 14.13Statistical reasoning appears in public policy, science, finance, and technology. Statistical literacy — the ability to evaluate evidence, communicate uncertainty, and recognize misuse — is an essential skill for civic life and most professional fields.
• Sec 14.14Ten recurring errors in data-based decisions: confusing significance with importance; treating observational studies as experiments; overgeneralizing; ignoring variability; extrapolating; ignoring missing data; misleading graphs; wrong summary for the shape; drawing conclusions from one study; failing to state limitations.
• Sec 14.15College statistics, probability, and data science all build on the reasoning developed in this course. The most important outcomes are habits of mind: question how data were collected, consider variability, state uncertainty honestly, and protect the people whose data you use.

Key Terms Review

Concept Review Questions

1. List the seven stages of the statistical investigation process in order. For each stage, write one sentence describing what it involves.
2. What makes a question a statistical question? Give one example of a non-statistical question and revise it into a statistical question.
3. Why is the data collection stage more important than the analysis stage in determining what conclusions can be drawn?
4. What is data cleaning, and why is it necessary before analysis?
5. Match each combination of variable types and comparison goals to an appropriate display (bar graph, two-way table, histogram, box plot, scatter plot, or side-by-side box plots).
6. Describe three different types of statistical models and the data patterns that suggest each one.
7. What does a well-fitting residual plot look like? What does a poorly fitting residual plot suggest about the model?
8. What is the difference between statistical significance and practical significance? Give an example where a result is statistically significant but not practically important.
9. What is the correct interpretation of a 95% confidence interval?
10. List five forms of statistical misuse and explain why each one is a problem.

Skill Practice

For each scenario, identify the stage of the investigation process being described and evaluate whether it is being carried out responsibly.

1. A researcher decides to study whether a new teaching method improves test scores. She plans to measure scores before and after the program for students whose teachers volunteer to participate.
2. After collecting survey responses, a student notices that 12 of 80 surveys have a question left blank. He excludes those 12 surveys and proceeds without noting this in his report.
3. A data analyst produces a bar graph with a y-axis starting at 80 rather than 0. The visual suggests one response option is twice as popular as another, when the actual difference is 4 percentage points.
4. A student fits a linear model to data showing rapid exponential growth. R² = 0.78, so she concludes the model is adequate.
5. A pharmaceutical company conducts 25 tests of their drug against 25 different conditions. One test shows p = 0.04. They publish only that result.

For each scenario, state the type of model that is most appropriate and explain your reasoning.

1. City housing starts grew rapidly for ten years but have been leveling off in recent years.
2. Distance fallen by a dropped object is recorded at each second. Distances increase at an accelerating rate.
3. Hours studied vs. quiz score for 30 students shows a straight-line pattern with moderate positive association.
4. A company's revenue doubles every three years. Fifteen years of revenue data are available.
5. Drug concentration in the bloodstream decreases rapidly at first, then more slowly, approaching zero.

1. A study compares blood pressure in regular meditators vs. non-meditators. The meditation group has a mean systolic blood pressure of 118 mmHg; the non-meditation group 127 mmHg. The study was observational.
   
   a. Describe what the data show.
   b. Explain whether a causal conclusion is appropriate.
   c. Identify one confounding variable that might explain the difference.
2. A polling organization surveys 1,200 adults and reports 38% support a proposed policy change, with margin of error ±2.8 points at the 95% confidence level.
   
   a. Calculate the confidence interval.
   b. Interpret the interval in plain language.
   c. Identify one source of uncertainty the margin of error does not address.
3. A scatter plot of years of education and annual income for 500 adults shows r = 0.58. A linear regression is fit; the residual plot shows no systematic pattern.
   
   a. Is a linear model appropriate? How do you know?
   b. Interpret r = 0.58 in context.
   c. Explain why it would not be appropriate to conclude that more education directly causes higher income.

For each student statement, identify the statistical error and explain the correct reasoning.

1. "My 95% confidence interval is (12.3, 17.6). This means there is a 95% chance the true mean is between 12.3 and 17.6."
2. "The correlation between shoe size and vocabulary score in children is r = 0.72. Therefore, having bigger feet helps children learn more words."
3. "My p-value is 0.08, which is just slightly above 0.05. The result is almost significant, so the evidence is fairly strong."
4. "I surveyed 20 students in my lunch period and found 14 prefer online classes. I conclude most students in the country prefer online classes."
5. "My residual plot shows a U-shaped curve, but R² = 0.85, so the linear model fits well."
6. "The study tested a new reading curriculum on 500 students at one school and found a statistically significant improvement. This proves the curriculum should be adopted nationwide."

Chapter 14 Checkpoint

Answers may be placed in the answer key.