A little intellectual exercise

Statistics practice

33 questions about what statistics actually means. Choose your answers, then check the explanations. No account, no grades, and no results sent anywhere.

Ungraded practice: This is an introductory self-check, not a final examination, a credit-bearing assessment, or proof of mastery. If an answer surprises you, revisit the accompanying lesson.

Question 1 · The population is not the sample

A researcher draws 250 names from the roster of 8,000 students at a university to estimate average weekly study time. What is the population?

The 250 students drawnAll 8,000 enrolled studentsEvery college student in the countryOnly the students who respond

Question 2 · Numbers can be labels

Postal ZIP codes consist of digits. What kind of variable are they?

A ratio-scale measurementA continuous quantitative measurementA categorical identifierA random variable that must be normally distributed

Question 3 · An outlier moves the mean

For five observations, the largest value increases by 100 while the other four stay fixed. What happens to the mean?

It increases by 100It increases by 20It stays the sameIt increases by 5

Question 4 · The same data, a different chart

A student changes the number of equal-width histogram bins but does not change any underlying observations. Which statement is correct?

The sample mean must changeThe data are no longer quantitativeThe sample median must changeThe apparent distribution shape may change, but the mean and median do not

Question 5 · The central half

Using the linear-interpolation (type-7) convention, what is the interquartile range of 2, 4, 6, 8, 10?

2468

Question 6 · What zero correlation does not mean

Pearson correlation is close to zero. Which conclusion is justified without further information?

The variables must be independentThere is little evidence of a linear association in this sampleThere cannot be any curved relationshipOne variable must be constant

Question 7 · More of the wrong thing

A city asks only attendees at a rally whether they support the rally’s cause. What is the main problem with using responses to estimate citywide support?

Selection biasThe sample size may be evenThe variance must equal zeroAn estimator cannot be calculated

Question 8 · The price of greater precision

Under the independent identically distributed finite-variance model, increasing sample size from 100 to 400 changes the standard error of the sample mean by which factor?

It doublesIt is unchangedIt is halvedIt becomes one-quarter as large

Question 9 · What the CLT actually describes

Under its standard conditions, the central limit theorem describes the limiting distribution of what?

Every individual observationThe original population, which becomes normalA suitably standardized sample meanEvery possible sample, which must be symmetric

Question 10 · Probability is an area

For a standard normal random variable Z, approximately what is P(−1.96 < Z < 1.96)?

50%68%90%95%

Question 11 · Density versus probability

For a continuous normal variable, what is the probability that X equals exactly its mean?

ZeroOne-halfThe height of the density at its meanOne

Question 12 · Association is not causation

Students who sleep more get higher average exam scores. Which statement best characterizes the finding?

More sleep has been proved to cause higher scoresThe variables are associated; further design and evidence are needed to establish causationExam performance has been proved to cause sleepThe study cannot calculate any descriptive statistic

Question 13 · The complement rule

Three independent fair coin tosses are made. What is the probability of at least one tail?

1/83/87/81

Question 14 · Independent or exclusive?

Can two positive-probability events be both independent and mutually exclusive?

Yes, alwaysIf their probabilities are equalNo, independence requires positive intersection probabilityOnly when their probabilities sum to one

Question 15 · The base-rate trap

A fictional condition has prevalence 1%; a test has sensitivity 90% and specificity 95%. What share of positive results are true positives?

90%95%15.4%1%

Question 16 · The model matters

Which characteristic would violate the standard binomial model?

A fixed number of trialsTwo outcomes per trialChanging success probability between trialsIndependent trials

Question 17 · The parameter stays still

A properly calibrated frequentist 95% confidence procedure means what?

95% of observations fall in each intervalThe parameter moves with each sampleAbout 95% of intervals made repeatedly contain the fixed true parameterExactly 19 of every 20 intervals must contain it

Question 18 · What a p-value means

A pre-specified test reports p=0.03. Which interpretation is valid?

The null is 3% likely to be trueThe alternative is 97% likely to be trueThere is a 3% chance the data were caused by chanceUnder the null and test conditions, results at least this extreme have probability 3%

Question 19 · The missing variable

A treatment has a higher pooled recorded success proportion but a lower recorded proportion within each of two stone-size groups. What explains how this is mathematically possible?

A numerical calculation error is inevitableDifferent numbers of easier and harder cases can change the weighted averageIt proves the pooled comparison is causalConfidence intervals always reverse the result

Question 20 · How assignments become probabilities

Eight people are assigned four each to two treatments using every balanced allocation with equal probability. How many allocations are possible?

81670256

Question 21 · The sharp null

What does the sharp null hypothesis mean in the simplest Fisher randomization test?

All participants have zero individual treatment effectsOnly the population average effect is zeroTreatment assignment was randomThe observed difference must be zero

Question 22 · Why replacement matters

If a sample has 12 observations, what happens if we draw bootstrap samples of size 12 without replacement from those same 12?

The mean varies widelyThe same data appear in every resample and its mean never variesThe sample size doublesIt corrects sampling bias

Question 23 · Resampling is not a new survey

Does raising the number of bootstrap replicates from 1,000 to one million make a highly biased convenience sample representative?

Yes, because the bootstrap creates new observationsYes, if a percentile interval is reportedNo; it improves Monte Carlo approximation but not the original sampling processOnly for binary data

Question 24 · The observed value minus the prediction

A least-squares line predicts 12 for an observation whose measured response is 17. What is the residual?

5−512/1729

Question 25 · Understand the slope

A fitted line is y-hat = 12 + 3x. What does the slope say?

Increasing x by one causes y to increase by threeOne higher x unit is associated with a fitted y value three units higherEvery observed y equals the fitted yThe correlation must equal three

Question 26 · What R squared cannot say

For an in-sample OLS regression with intercept, R² = 0.9. Which conclusion is supported?

The model has a 90% probability of being truex causes 90% of yThe fitted line accounts for 90% of the sample total sum of squares under the standard definitionThe next prediction has 90% accuracy

Question 27 · Sampling and assignment

Which is the principal distinction between random sampling and random treatment assignment?

They are synonymsSampling supports generalization to a population; assignment helps identify treatment effects within the studySampling guarantees causal effectsAssignment ensures every participant represents the population

Question 28 · Two independent means

Two independent groups have sample variances 2.5 and 22 and sample sizes 5 and 8. What is the estimated standard error of the difference in their means?

2.51.800.5024.5

Question 29 · Not every comparison is paired

Which design justifies a paired t procedure rather than a two-independent-sample t procedure?

Two independent randomly selected classesMeasurements on the same people before and after a procedureGroups with equal sample sizes but unrelated participantsOne group with more variation

Question 30 · Watch the denominator

A historical dataset has 153 daily records and 37 ozone observations are missing. How many days contribute to the ozone mean if R excludes missing readings?

15337116190

Question 31 · What a complete-case model omits

Why should a regression analyst report that 111 of 153 dataset rows were used?

To show that the model has a known causal effectBecause missing predictors or responses changed the analysis sampleBecause missingness guarantees unbiased estimatesBecause every spreadsheet must have 111 rows

Question 32 · Holding included predictors fixed

A multiple regression relates ozone to temperature, wind, and solar radiation. What does the fitted temperature coefficient describe?

The causal effect of changing the weatherAn adjusted association in expected ozone per temperature unit, holding the other included predictors fixedThe probability ozone is positiveThe percentage of days with no missing observations

Question 33 · When more responses are misleading

A town has stable opinions, but one demographic group is far more likely to answer a survey. What can change?

The true townwide support proportion necessarily changesOnly the estimate can change, as respondents come from an altered mixture of groupsThe variance of every variable must become zeroThe survey becomes randomized

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