DSE Statistical Inference and Hypothesis Testing 5

Question 1

A permutation test determines the p-value by:

Accepted Answer

Repeatedly shuffling group labels and recomputing the test statistic to build a null distribution

Answer

Permutation tests approximate the null distribution empirically by randomly reassigning labels many times, making no distributional assumptions.

Question 2

In the context of maximum likelihood estimation, the score function is:

Accepted Answer

The first derivative of the log-likelihood with respect to θ

Answer

The score function S(θ) = ∂ℓ/∂θ equals zero at the MLE and has expectation zero under the true parameter.

Question 3

Which statement about bootstrap confidence intervals is TRUE?

Accepted Answer

They approximate the sampling distribution using resampling from observed data

Answer

Bootstrap intervals use repeated resampling with replacement to estimate the sampling distribution without relying on parametric assumptions.

Question 4

A z-test for a proportion is valid when:

Accepted Answer

Both np and n(1−p) are at least 10, ensuring the normal approximation holds

Answer

The normal approximation to the binomial is reliable when np ≥ 10 and n(1−p) ≥ 10, ensuring enough expected successes and failures.

Question 5

The Cramér-Rao lower bound gives the minimum variance achievable by:

Accepted Answer

Any unbiased estimator of a parameter

Answer

The CRLB states that Var(θ̂) ≥ 1/I(θ) for any unbiased estimator, where I(θ) is the Fisher information.

DSE - Data Science Practice Test