According to the Central Limit Theorem, the sampling distribution of the sample mean X̄ for a large sample size n from any population with finite mean μ and variance σ^2 tends to:

A. Approximately normal with mean μ and variance σ^2/n B. A t-distribution with n−1 degrees of freedom C. A Poisson distribution with parameter μ D. A uniform distribution on [0,1]

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2

For a random sample X1,...,Xn drawn from a Poisson(λ) distribution, the maximum likelihood estimator (MLE) of λ is:

A. The sample median B. The sample mean (ΣXi/n) C. The sample variance (Σ(Xi−X̄)²/(n−1)) D. Zero if any Xi=0

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3

Which estimator attains the Cramér–Rao lower bound (is efficient) for estimating the mean μ of N(μ,σ^2) when σ^2 is known?

A. Sample median B. Sample variance C. Sample mean X̄ D. Any unbiased linear combination of observations

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4

Which of the following is the correct interpretation of a p-value in hypothesis testing?

A. The probability that the null hypothesis is true given the data B. The probability that the alternative hypothesis is true C. The probability of committing a Type II error D. The probability of obtaining data as extreme as observed or more so, assuming the null hypothesis is true

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5

Which of the following is NOT an assumption required for one-way ANOVA?

A. Equal sample sizes across the groups B. Independence of observations C. Normality of errors within groups D. Homogeneity of variances across groups

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6

Under the Gauss–Markov theorem, the ordinary least squares (OLS) estimator is BLUE (best linear unbiased estimator) provided which set of conditions holds?

A. Errors are normally distributed and predictors are independent B. Errors have zero mean, are homoscedastic and uncorrelated, and regressors have full rank C. Errors are heteroscedastic but mean zero D. Predictors are fixed at observed values and errors have any finite variance

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7

With a Binomial(n,p) likelihood and a Beta(α,β) conjugate prior for p, the posterior distribution for p is:

A. Normal with mean α/(α+β) B. Gamma with parameters α+n and β C. Beta(α+number of successes, β+number of failures) D. Binomial with updated sample size n+α+β

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8

A weakly stationary (covariance-stationary) time series requires which of the following properties?

A. Autocovariance depends on time t and lag k separately B. Mean can be a deterministic trend but variance must be infinite C. Only that the series be ergodic D. Constant mean and autocovariance that depends only on the lag (not on time)

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9

In a chi-square goodness-of-fit test for k categories where m parameters are estimated from the data, the appropriate degrees of freedom is:

A. k B. k − 1 − m C. k − m D. k + m − 1

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10

Which condition for central limit theorems is stronger (i.e., implies) the Lindeberg condition for triangular arrays of independent variables?

A. Kolmogorov condition B. Cramér condition C. Lyapunov condition D. Markov condition

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According to the Central Limit Theorem, the sampling distribution of the sample mean X̄ for a large sample size n from any population with finite mean μ and variance σ^2 tends to:

For a random sample X1,...,Xn drawn from a Poisson(λ) distribution, the maximum likelihood estimator (MLE) of λ is:

Which estimator attains the Cramér–Rao lower bound (is efficient) for estimating the mean μ of N(μ,σ^2) when σ^2 is known?

Which of the following is the correct interpretation of a p-value in hypothesis testing?

Which of the following is NOT an assumption required for one-way ANOVA?

Under the Gauss–Markov theorem, the ordinary least squares (OLS) estimator is BLUE (best linear unbiased estimator) provided which set of conditions holds?

With a Binomial(n,p) likelihood and a Beta(α,β) conjugate prior for p, the posterior distribution for p is:

A weakly stationary (covariance-stationary) time series requires which of the following properties?

In a chi-square goodness-of-fit test for k categories where m parameters are estimated from the data, the appropriate degrees of freedom is:

Which condition for central limit theorems is stronger (i.e., implies) the Lindeberg condition for triangular arrays of independent variables?

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