1. To determine whether glaucoma affects the corneal thickness, measurements were made in 8 people affected by glaucoma in one eye but not the other. The corneal thickness (in microns) were as follows:

(a) Explain why the samples should be considered matched pairs.

(b) Conduct the appropriate t-test to determine whether eyes affected by glaucoma differ from normal eyes in terms of corneal thickness. Use α = 0.10.

(c) For the equivalent sign test for matched pairs, compute the p-value.

2. Suppose x₁, x₂, ..., x_n are iid observations from a distribution with the following density:

f(x|θ,Φ) = θe^-θ(x-Φ) for x ≥ Φ.

(a) Write the likelihood function.
(b) Find the mle of Φ assuming θ is known.

3. Let X₁, X₂, ..., X_n be a random sample from the Uniform(0, ??) distribution. Show that X(n) = max{X_i; 1 ≤ ?? ≤ ??} is sufficient for ??.

4. An urn is known to contain 12 balls, some black, some white. Six will be drawn without replacement. It is conjectured that the number of white balls is 6, that is P(W) =6/12. The alternative is that there are less than 6 balls. A test is proposed that would reject the null hypothesis if 1 or less balls are selected among the 6.

a) What is the significance level of this test?
b) What is the power associated with a true number of white balls being 0, 1, 2, 3, 4, 5, 6?

5. The following sample data, recorded in days, represent the length of time to recovery for patients randomly treated with one of two medications to clear up severe bladder infections.

Find a 99% confidence interval for the difference in means μ₂ - μ₁ assuming normal populations and equal variances.

6. A taxi company is trying to decide whether to purchase brand A or brand B for its fleet of taxis. To estimate the difference in the two brands, an experiment is conducted using a random sample of 12 of each brand. The results show that the sample standard deviation of brand A is 5,000; of brand B is 6,100. Before proceeding with the test of means, the company decided to compute a 95% confidence interval on the ratio of variances σ_B²/σ_A². Assume that the populations are normal. Develop this confidence interval.

7. Three different design configurations are being considered for a particular component. There are four possible failure modes for the component. An engineer obtained the following data on number of failures in each mode for each of the three configurations. Does the configuration appear to have an effect on type of failure? Use α = 0.05.

8. Consider the density f (x|β) = β / X^β+1; x>1.

(a) Determine the method of moments estimator for β.

(b) Simulate the sampling distribution of the moments estimator if β = 4 and n=10. You will need to simulate random x values from the distribution. A single random deviate X can be obtained by (1-u)^-1/4 if ?? = 4, where u is a uniform (0,1) random number. Simulate 100 estimates based on n-10 and comment on whether the estimator appears unbiased.

9. Let X have an exponential distribution with a mean θ; that is, the pdf of X is f (x) = (1/θ)e^-X/??, 0 < x < ∞. Show the GLT for H₀:θ=3 versus H₁:θ=5 can be based on n the statistic _i=1Σⁿ x_i , the sum of the random sample. This is actually a simple null hypothesis and alterative, a situation in which the GLT will deliver a most powerful test.