PSTAT 120B: Midterm Review Questions

February 26, 2025

Question 1: Qualities of Point Estimators

Suppose we have a sequence of n iid Normal Random Variables with parameters (0, n/1). denoted Y1, Y2, ..., Yn. Consider n¯Y

a.) State the bias of n¯Y

b.) State the efficiency of n¯Y

c.) Is n¯Y squared error consistent for the mean?

Question 2: Method of Moments and MLEs

Suppose we have a sequence of n iid Random Variables which are distributed Poisson(λ).

a.) Find a MOM estimator for λ (Method of Moments).

b.) Find the MLE for λ.

Question 3: Sufficient Statistics

Find a sufficient statistic for the exponential distribution. Is this statistic an MVUE? If not, find f(U) that is.

Question 4: Hypothesis Testing

The italics are for fun. The meat of the question is bolded. . .

IVStreets takes fit-check photos every Thursday in The Arbor. You think this causes an increase in foot traffic through The Arbor in early weeks of the quarter, and then levels out in the latter weeks as people stop caring about being put on his Instagram. You make it your life’s mission to get to the bottom of this. Over four years, you take two samples per quarter (3 quarters per year), one sample Week 2, and another Week 9. The average of the Week 2 samples was 480 people with a variance of 52. The average of the Week 9 samples was 389 people with a variance of 36. At α = 0.01, is there enough evidence to say there are less people total walking through The Arbor in Week 9 compared to Week 2? (Assume our samples come from a normal population, and use α = 0.01). Additionally, state and interpret the p-value.