Statistical and Probabilistic Analysis Flashcards

A dataset has a mean of 50 and a standard deviation of 10. Using Chebyshev's inequality, what is the minimum proportion of data within 3 standard deviations of the mean?

Answer: At least 88.9%

Chebyshev's inequality states at least 1 - 1/k² of data lies within k standard deviations; for k=3, that's 1 - 1/9 ≈ 88.9%.

Which test is most appropriate to assess whether two continuous variables are linearly associated when both are normally distributed?

Answer: Pearson correlation test

Pearson correlation tests linear association between two normally distributed continuous variables and is parametric.

In a Type II error scenario, a researcher fails to reject a null hypothesis that is actually false. Which factor, if increased, most directly reduces this error?

Answer: Sample size

Increasing sample size reduces the standard error, increasing statistical power and reducing the probability of a Type II error (β).

A random variable X follows a Poisson distribution with λ = 4. What is P(X = 0)?

Answer: e⁻⁴

For Poisson, P(X=k) = (λᵏ e⁻λ)/k!; at k=0, this simplifies to e⁻⁴.

The joint probability P(A ∩ B) = 0.12, P(A) = 0.4, and P(B) = 0.3. Are events A and B independent?

Answer: Yes, because P(A)·P(B) = 0.12

Independence holds when P(A ∩ B) = P(A)·P(B); here 0.4 × 0.3 = 0.12, which matches.

Which property of estimators means the estimator's expected value equals the true parameter value?

Answer: Unbiasedness

An unbiased estimator has E[θ̂] = θ, meaning on average it hits the true parameter without systematic error.

A 95% confidence interval for a population mean is (42, 58). Which interpretation is correct?

Answer: If the procedure were repeated many times, 95% of such intervals would contain the true mean

Confidence intervals reflect the long-run frequency of the procedure, not the probability that a specific interval contains the true parameter.