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L1079 Mathematical Statistics

Workshop 10 exercises

1. Suppose we have a random sample (X1,X2,X3,......, Xn) from a population with mean θ. Consider an estimator for θ which is formed by averaging the first two observations in the sample. Is this estimator unbiased? Is it consistent?

Solution: The estimator is unbiased since it is basically a sample average (albeit only formed from two observations). But it cannot be consistent since the sampling distribution does not change as the sample size increases.

2. The random variable X follows a uniform distribution on the interval (0,q).  A single observation on X is used for testing H₀: q = 1 versus H_a: q = 2 and the decision rule is to accept H₀ if X < 0.5.  Find the probabilities of Type I and II errors.

A: The probabilities of Type 1 and Type II error are as follows:

P(Type I error) = P(X > 0.5|θ = 1) = 0.5

P(Type II error|θ = 2) = P(X < 0.5|θ = 2) = 0.25

3. We have a random sample of observations from a uniform distribution over the interval (0,θ). We want to test the null hypothesis that θ=1 against the alternative that θ<1. Our decision rule is to reject the null if the sample mean is less than 0.5.

Is the power of the test greater if the true θ=0.9 or if the true θ=0.6?

A: Intuitively, the test will have greater power against θ=0.6 than against θ=0.9, because hypotheses that are very similar are harder to distinguish than hypotheses that are very different.