Answer keys and reasoning
Questions are useful only when the answer makes sense and the explanation exposes the relevant assumption.
Unit 1: Thinking with and describing data
- The population is not the sample
A city interviews 400 of its 20,000 residents. The target is citywide transit opinion. What is the population?
Answer: All 20,000 city residents
The target population is all city residents. The sampled 400 are the selected subset.
- A parameter or a statistic?
The mean study time calculated from 150 sampled students is which of these?
Answer: A sample statistic
A numerical summary computed from the sampled observations is a statistic. The corresponding population mean is a parameter.
- Digits can still be labels
A student's postal ZIP code is best treated as which variable type?
Answer: A nominal categorical identifier
The digits label geographic areas; adding or averaging the codes has no meaningful measurement interpretation.
- Resistant center
Which statistic stays at 50 if data 40, 45, 50, 55, 60 change to 40, 45, 50, 55, 600?
Answer: Median
The middle ordered value remains 50. Mean and spread change substantially.
- A sample-standard-deviation detail
Why does the usual sample variance use n−1 in the denominator?
Answer: To correct the estimated population variance under standard i.i.d. assumptions
The familiar n−1 correction yields an unbiased estimate of population variance for independent identically distributed observations with finite variance.
- Bin choice versus data
Changing a histogram from 5 to 20 equal-width bins while holding the raw observations fixed can change what?
Answer: The histogram's apparent shape
Binning changes how the same observations are displayed, not the underlying numeric summaries.
- Quartiles require a convention
For the ordered values 2,4,6,8,10, what is the type-7 interpolated IQR?
Answer: 4
Q1=4 and Q3=8 under the stated type-7 convention, so IQR=4.
- Beware the truncated bar chart
A bar chart displaying 98 and 102 uses a vertical axis beginning at 95. What is the primary concern?
Answer: Truncated bar baselines can exaggerate visual differences
Bars conventionally encode magnitude through length. A truncated baseline can exaggerate the relative appearance of a small absolute difference.
- Sampling versus assigning
Randomly assigning volunteers to two treatments primarily helps with which goal?
Answer: Causal comparison under the experimental design
Random assignment supports treatment comparisons; who volunteered and how they were selected remains a generalization issue.
- A blank is not a zero
A temperature record is missing on one day. What should a reproducible analysis generally avoid?
Answer: Silently replacing it with zero
Zero is a substantive measurement; missing indicates unknown information and must be handled and reported deliberately.
- Correlation is not enough
Pearson's r is near zero. Which inference is justified?
Answer: A strong curved relationship may still exist
Pearson correlation summarizes linear association. Nonlinear dependencies may remain even when it is near zero.
- The research needs a source
Which submission is most reproducible?
Answer: A documented source, code, units, transformations and missingness
Provenance, a data dictionary, transformation log and executable analysis allow readers to retrace the computation.
Unit 2: Sampling, probability, and distributions
- Simple random sampling
What characterizes a simple random sample of n from N?
Answer: Every possible size-n subset has equal selection probability
Each size-n subset has equal selection probability; realized composition can still vary by chance.
- Stratified design
What is a defining feature of stratified sampling?
Answer: The population is partitioned into strata and sampled within them
Samples are selected within defined population strata; weighting may be needed for overall estimates.
- Cluster dependence
Why can 500 respondents from two schools offer less independent information than 500 separately selected students?
Answer: Students in a school may be correlated
Within-cluster correlation can reduce effective information and require design-aware standard errors.
- At least one success
In three independent fair coin tosses, what is the probability of at least one head?
Answer: 7/8
Use the complement: 1 minus probability of all tails, 1−(1/2)^3 =7/8.
- Independence versus exclusion
Can two positive-probability events be both independent and mutually exclusive?
Answer: No, since the product of positive probabilities cannot equal their zero intersection
Mutual exclusivity makes the intersection probability zero; independence makes it equal to a strictly positive product.
- Selecting participants
How many four-person groups can be formed from eight distinct people without regard to order?
Answer: 70
There are 8 choose 4 equally sized subsets. The factorial formula 8! divided by (4!×4!) equals 70, because the order of members in the selected group does not matter.
- Law of total probability
60% use transit. Early commute rates are 30% among users and 10% among nonusers. Overall early commute rate?
Answer: 22%
Weighted by the population partitions: .6(.3)+.4(.1)=.22.
- Base-rate reasoning
A condition has prevalence 1%, test sensitivity 90%, specificity 95%. What is PPV approximately?
Answer: 15.4%
True-positive probability .009 divided by all-positive probability .009+.0495, about .154.
- The expected outcome
A reward is 8 with probability .25 and 0 otherwise. What is its expected value?
Answer: 2
The probability-weighted mean is .25×8 + .75×0 = 2.
- Independent Bernoulli trials
For X~Binomial(40,.25), what is E[X]?
Answer: 10
The expectation of a binomial success count is np. With 40 trials and success probability .25 per independent trial, that expectation is 40×.25=10 successes.
- A count of no events
If X is Poisson with mean 2, P(X=0) is closest to which?
Answer: .1353
Poisson probability at zero is e^(−2), about .1353.
- What becomes normal?
The central limit theorem typically concerns the limiting normality of which object under suitable conditions?
Answer: A suitably standardized sample mean
The CLT describes standardized sums/means; it does not make the original population normal.
- Probability under a density
For X~Uniform(0,2), what is P(.5<X<1.5)?
Answer: .5
The interval has width 1 and density 1/2; area equals .5.
Unit 3: Statistical estimation and inference
- The 95% interval claim
What does a properly calibrated 95% frequentist confidence procedure mean?
Answer: About 95% of intervals constructed by repetition contain a fixed parameter
Coverage is a repeated-sampling property of a random interval construction procedure under its assumptions.
- Unknown population variance
A one-sample t test from 20 independent normal observations has how many degrees of freedom?
Answer: 19
The one-sample t reference has n−1=19 degrees of freedom.
- The zero-event problem
Why is a Wald confidence interval [0,0] from zero successes in a small sample suspect?
Answer: The event's true probability could still be positive
A small observed count of zero does not remove uncertainty; Wilson or other better-calibrated procedures convey that.
- Two sources of variance
Independent groups: sA²=2.5, nA=5; sB²=22, nB=8. Estimate the SE of mean A−mean B.
Answer: 1.80
The independent Welch standard error is sqrt(2.5/5+22/8)=sqrt(3.25)≈1.803.
- What is paired?
What justifies applying a paired t procedure?
Answer: The same persons measured before and after an event
Paired differences require a meaningful matching or repeated-measurement design.
- Compare two outcome rates
18 of 30 versus 12 of 30 experienced an outcome. What is the risk difference in percentage points?
Answer: 20
Group A's observed proportion is 18/30=.60 and B's is 12/30=.40. The risk difference is .60−.40=.20, meaning 20 percentage points, not 20% of the baseline.
- The p-value is conditional
For a pre-specified test, p=.03 primarily means what?
Answer: Assuming the null and other test conditions, results at least this extreme have probability 3%
A p-value is a probability of data extremeness conditional on the null model, not a probability of the null itself.
- Fair die goodness of fit
Six fair-die expected counts are ten each; observed counts are 15,5,10,10,10,10. Pearson's X²?
Answer: 5
Two cells contribute 25/10 each. Total chi-square statistic is 5.
- Table degrees of freedom
For a 3-by-4 independent-count contingency table, the usual chi-square reference has what degrees of freedom?
Answer: 6
The conventional chi-square degrees of freedom in an r-by-c contingency table are (r−1)(c−1); with three rows and four columns that gives (3−1)(4−1)=6.
- Power is conditional on an effect
Power 80% for a given alternative means?
Answer: The test rejects the null in 80% of repetitions under that specified alternative
Power is the probability of rejecting under the stated alternative and design.
- Many tests
Twenty independent tests at alpha .05 with all nulls true have what chance of at least one false rejection?
Answer: About 64%
1−(.95)^20≈.642; the nominal single-test error does not apply to the selected familywide event.
- Small randomization experiment
For balanced random assignment of four of eight participants to A, how many possible allocations?
Answer: 70
The exact count is binomial coefficient 8 choose 4=70.
- Why replacement is essential
A standard nonparametric bootstrap for a sample mean draws how?
Answer: From the empirical sample with replacement
Sampling with replacement from the observed empirical distribution generates variation in the sample statistic.
Unit 4: Models, causal limits, and research practice
- The aggregate result reverses
A success rate ranks treatment A higher within each severity group, but B higher overall. What is a possible explanation?
Answer: Different severity-group weights in the two treatments
Aggregated rates are weighted subgroup averages, and weights can differ across treatments.
- Interpret a fitted slope
For y-hat = 12+3x, the fitted difference between x=5 and x=6 is?
Answer: 3
Holding the model fixed, an increase of one x unit changes fitted y by the slope, 3 units.
- Prediction mistake
The observed response is 17 and fitted response is 12. The residual equals?
Answer: 5
A regression residual is the observed value minus the fitted prediction. For observed 17 and predicted 12, this is 17−12=5, so the observation lies above the fitted line.
- What R² is not
For ordinary least squares with an intercept, R²=.9 tells us what?
Answer: In sample, the model reduces squared error by 90% relative to the mean-only model under the usual definition
R²=1−SSE/SST for this model; it is an in-sample descriptive fit ratio.
- A slope's estimated uncertainty
A simple OLS model with intercept has n=5. What are the residual degrees of freedom for a slope t statistic?
Answer: 3
Estimating an intercept and a slope uses two degrees of freedom, leaving n−2=3 residual degrees.
- One-way ANOVA
Three groups of three values have means 3,5,7 and grand mean 5. What is between-group SS?
Answer: 24
With three values per group, 3[(3−5)²+(5−5)²+(7−5)²]=24.
- Interpret the global F rejection
A properly conducted one-way ANOVA rejects equality of three population means. What follows?
Answer: At least one population mean differs under the tested model
The global alternative says not all group means are equal; it doesn't identify every pair.
- Logistic regression scale
In a logistic model, exp(beta1) represents what?
Answer: An odds ratio per x unit under the model
Exponentiating the log-odds slope yields a conditional odds ratio, not a probability ratio.
- Association versus causation
A multivariable observational regression estimates an adjusted association. What else is required for a causal claim?
Answer: Strong justified design and identification assumptions about confounding, selection and intervention
Covariate adjustment alone does not prove exchangeability or control unmeasured confounding.
- Air-quality data audit
Historical R airquality has 153 rows, with 37 missing Ozone readings. How many observed ozone values?
Answer: 116
153−37=116 ozone observations. A multivariable complete-case model can use fewer.
- Compare different regressions
Why fit simple and multivariable airquality models on the same complete-case sample?
Answer: To distinguish changed covariates from changed observation sets
Otherwise coefficients can change because both included predictors and analytic records changed.
- Reproducibility
Which report is most reproducible?
Answer: Documented source, cleaning, code, software assumptions and complete results
A reader must be able to retrace data provenance and rerun a documented analysis.
- Capstone claim
Which conclusion is appropriate for a historical observational air-quality regression?
Answer: Ozone readings in observed complete cases were associated with recorded temperature conditional on included covariates
Observational regression describes conditional association; causal, missingness, and seasonal limitations remain.
Scoring and reuse
The browser tests give one point per correct multiple-choice answer and show the rationale for every response. An instructor should assess justification, numerical work, and interpretation separately; these questions alone do not validate mastery.
For a deeper summative task, use the capstone research report with its 100-point rubric and the laboratory teaching guide.