Statistics for data science quiz

Question 1

What distinguishes a Type I error from a Type II error, and how does the significance level α relate to them?

Accepted Answer

Type I rejects a true null (prob α); Type II fails to reject a false null (prob β)

Answer

Type I fails to reject a false null; Type II rejects a true null

Answer

Both refer to the same error measured on different scales

Answer

Type I is bias and Type II is variance

Question 2

A p-value of 0.03 was obtained. Which interpretation is correct?

Accepted Answer

Assuming the null is true, there is a 3% chance of data this extreme or more

Answer

There is a 3% probability the null hypothesis is true

Answer

There is a 97% probability the alternative hypothesis is true

Answer

The effect size is 0.03

Question 3

What does the Central Limit Theorem guarantee, and what are its key conditions?

Accepted Answer

The sample mean's distribution approaches normal for i.i.d. data with finite variance

Answer

The population becomes normal as n grows

Answer

Any estimator becomes unbiased as n grows

Answer

The variance of the sample equals σ/√n

Question 4

Explain the difference between the standard deviation and the standard error of the mean.

Accepted Answer

SD measures spread of observations; SEM measures spread of the sample mean (σ/√n)

Answer

They are identical quantities

Answer

SEM measures spread of observations; SD measures spread of the mean

Answer

SEM is always larger than SD

Question 5

What is the correct interpretation of a 95% confidence interval?

Accepted Answer

Under repeated sampling, 95% of such intervals contain the true parameter

Answer

There is a 95% probability the parameter lies in this specific interval

Answer

95% of the data fall within the interval

Answer

The interval contains 95% of future predictions

Question 6

What is the bias-variance tradeoff and how does it relate to expected prediction error?

Accepted Answer

Error = bias² + variance + irreducible error

Answer

Error = bias × variance × noise

Answer

Bias and variance always decrease together with complexity

Answer

Variance is the only reducible component of error

Question 7

When should you use a t-test instead of a z-test for a population mean?

Accepted Answer

Use t-test when σ is unknown/estimated, particularly for small samples

Answer

Use t-test when σ is known; z-test when σ is unknown

Answer

t-tests are only for proportions

Answer

z-tests require non-normal data

Question 8

What is the multiple comparisons problem and how does the Bonferroni correction address it?

Accepted Answer

It tests each hypothesis at α/m to control family-wise error

Answer

It multiplies each p-value by α

Answer

It increases statistical power for each test

Answer

It controls the false discovery rate exactly

Question 9

Explain the difference between controlling the family-wise error rate (FWER) and the false discovery rate (FDR).

Accepted Answer

FWER controls probability of ≥1 false positive; FDR controls expected fraction of false positives among rejections

Answer

FWER controls expected proportion of false discoveries; FDR controls any false positive

Answer

They are mathematically equivalent

Answer

FDR is always more conservative than FWER

Question 10

What does it mean for an estimator to be unbiased, and is unbiasedness always desirable?

Accepted Answer

Unbiased means E[estimator]=parameter; it is not always optimal vs MSE

Answer

Unbiased means zero variance

Answer

Unbiased estimators always minimize MSE

Answer

Bias and variance cannot both be controlled

Question 11

Why is the sample variance divided by (n−1) rather than n (Bessel's correction)?

Accepted Answer

To produce an unbiased estimate of population variance (one df used by the mean)

Answer

To make the estimator larger artificially

Answer

Because n−1 minimizes variance of the estimator

Answer

It is an arbitrary convention with no effect

Question 12

What is the difference between covariance and correlation?

Accepted Answer

Correlation is scale-invariant in [−1,1]; covariance is unit-dependent

Answer

They are identical except correlation can be negative

Answer

Covariance is always between −1 and 1

Answer

Correlation measures causation, covariance does not

Question 13

What assumptions underlie ordinary least squares (OLS) linear regression for valid inference?

Accepted Answer

Linearity, independent and homoscedastic errors, exogeneity, no perfect collinearity

Answer

Predictors must be normally distributed

Answer

The response must be uniformly distributed

Answer

Errors must be perfectly correlated

Question 14

What is heteroscedasticity and how does it affect OLS estimates?

Accepted Answer

It biases standard errors but not coefficients; use robust SEs

Answer

It biases the coefficient estimates themselves

Answer

It only occurs in logistic regression

Answer

It eliminates the need for the intercept

Question 15

What is multicollinearity and how is the Variance Inflation Factor (VIF) used to detect it?

Accepted Answer

VIF=1/(1−Rj²); high values indicate inflated coefficient variance from collinearity

Answer

VIF measures residual normality

Answer

Multicollinearity biases prediction accuracy severely

Answer

VIF below 1 indicates collinearity

Question 16

How does L1 (Lasso) regularization differ from L2 (Ridge) in its statistical effect on coefficients?

Accepted Answer

L1 yields sparse coefficients (some exactly zero); L2 shrinks smoothly

Answer

L1 shrinks smoothly; L2 sets coefficients to zero

Answer

Both produce identical coefficient paths

Answer

L2 performs feature selection, L1 does not

Question 17

What is a Q-Q plot and what does it diagnose?

Accepted Answer

It compares sample quantiles to theoretical quantiles to assess distributional fit

Answer

It plots residuals against fitted values

Answer

It measures correlation between two variables

Answer

It is used only for time series stationarity

Question 18

What is the difference between the maximum likelihood estimate (MLE) and the maximum a posteriori (MAP) estimate?

Accepted Answer

MAP maximizes the posterior (likelihood × prior); MLE maximizes only the likelihood

Answer

MLE uses a prior; MAP does not

Answer

They always give identical estimates

Answer

MLE requires a Gaussian prior

Question 19

State Bayes' theorem and explain each component in an inference context.

Accepted Answer

Posterior ∝ likelihood × prior, normalized by the evidence

Answer

Posterior = likelihood / prior

Answer

Prior = posterior × likelihood

Answer

Evidence = posterior × prior

Question 20

A disease affects 1% of a population. A test has 99% sensitivity and 95% specificity. What is the approximate probability of disease given a positive test?

Accepted Answer

About 17%

Answer

About 99%

Answer

About 95%

Answer

About 50%

Question 21

What is the difference between a probability mass function (PMF), a probability density function (PDF), and a cumulative distribution function (CDF)?

Accepted Answer

PMF is for discrete values; PDF gives density for continuous variables; CDF is cumulative P(X≤x)

Answer

PDF gives probabilities directly at each point for continuous variables

Answer

CDF only applies to discrete variables

Answer

PMF and PDF are the same thing

Question 22

When is the Poisson distribution an appropriate model, and what is its key property?

Accepted Answer

It models event counts with mean equal to variance (both λ)

Answer

It models continuous waiting times; variance = mean²

Answer

It requires the mean to be much larger than the variance

Answer

It is only valid for binary outcomes

Question 23

What is the difference between parametric and non-parametric statistical methods?

Accepted Answer

Parametric assume a distributional form; non-parametric make minimal assumptions

Answer

Non-parametric methods assume normality; parametric do not

Answer

Non-parametric methods always have more power

Answer

They are interchangeable in all settings

Question 24

Why does correlation not imply causation, and what are common confounding mechanisms?

Accepted Answer

Confounders, reverse causation, or selection bias can create non-causal correlation

Answer

Correlation always implies causation in large samples

Answer

Causation requires only a high correlation coefficient

Answer

Randomization is unnecessary for causal claims

Question 25

What is Simpson's paradox and what causes it?

Accepted Answer

An aggregate trend that reverses when data are stratified, due to a confounder

Answer

A trend that strengthens identically across all subgroups

Answer

A paradox caused by small sample sizes only

Answer

A type of measurement error in surveys

Question 26

What is statistical power, and which factors increase it?

Accepted Answer

Power increases with larger n, larger effect size, lower variance, and higher α

Answer

Power decreases as sample size grows

Answer

Power is independent of effect size

Answer

Power equals the p-value

Question 27

How does the bootstrap method estimate the sampling distribution of a statistic?

Accepted Answer

It resamples observed data with replacement to approximate the sampling distribution

Answer

It samples without replacement from a known distribution

Answer

It requires the population to be normal

Answer

It is identical to cross-validation

Question 28

What is the difference between the chi-square goodness-of-fit test and the chi-square test of independence?

Accepted Answer

Goodness-of-fit checks one variable's distribution; independence tests association between two variables

Answer

Both test the mean of continuous variables

Answer

Goodness-of-fit requires two variables; independence requires one

Answer

They are identical tests with different names

Question 29

In A/B testing, what is the danger of 'peeking' at results, and how can it be mitigated?

Accepted Answer

Peeking inflates Type I error; use fixed n or sequential/alpha-spending methods

Answer

Peeking has no effect on error rates

Answer

Peeking only affects Type II error

Answer

Peeking increases statistical power safely

Question 30

What is the difference between covariate-adjusted (CUPED) variance reduction and stratification in experiments?

Accepted Answer

CUPED uses pre-experiment covariates to cut variance; stratification balances subgroups

Answer

Both methods introduce bias to gain power

Answer

CUPED increases variance; stratification reduces it

Answer

They are only applicable to observational studies

Test yourself