Each of two independent observations is 0 or 4 with equal probability. The four ordered samples are equally likely. What is the expected sample mean?

Answer: ______________

A sample of three readings is 4, 6 and 8. Compute the sample mean.

Answer: ______________

For the readings 11, 13 and 15 with mean 13, the squared deviations total 8. With the n minus 1 divisor, what is the sample variance?

• 4
• 8 over 3
• 8

Priya says the expected sample mean is greater than the population mean for independent observations from the same population with finite mean. Is this claim correct?

Circle one:   True   False

Let Q be the sum of squared deviations from the sample mean. A sample has only one observation. Why is the sample variance Q/(n - 1) unavailable?

• The single observation is always an outlier, so variance has to be discarded.
• Both Q and n - 1 are zero, so their quotient is undefined.
• The denominator is one, so the quotient equals the observation itself.
• A single observation makes the population variance infinite in every case.

For five observations, dividing the sum of squared deviations from their sample mean by n gives 12. What value results when the same total is divided by n - 1?

Answer: ______________

If an estimator's bias and variance both tend to zero as the sample size grows, these conditions are sufficient for consistency.

Circle one:   True   False

For n >= 2 independent, identically distributed observations with finite positive variance, why does the unbiased sample variance divide by n - 1 instead of n?

• It makes all squared deviations equal to each other
• It corrects downward bias in the variance estimate
• It cancels any unusually large values in the sample

For every sample size n, estimator A has bias 1 and variance 1/n; estimator B has bias 0 and variance 4/n. Both estimate the same target theta. Which statement is correct?

• At n = 1, A has lower MSE, but only B is consistent.
• At n = 1, B has lower MSE, but only A is consistent.
• At n = 1, A has lower MSE, and both estimators are consistent.
• At n = 1, B has lower MSE, and neither estimator is consistent.

A class takes one sample with mean 6 and declares the population mean is exactly 6 with no uncertainty. Which statement explains the problem?

• One sample mean equals the population mean because the estimation rule is unbiased.
• Unbiasedness fixes the expected sample mean, while one observed mean can differ.
• The sample mean must use n - 1 as its divisor whenever the population mean is unknown.