Reference no: EM131105792

Honors Exam 2012: Statistics

1. A basketball coach wants to devise an innovative way to test whether her players have improved by practicing over the summer. The previous season, her star, Julie, made 70% of her free throws. Julie claims to have improved over the summer; she now thinks she is an 80% free throw shooter. The coach doubts Julie has improved; she asks Julie to start shooting free throws. Let X be a random variable corresponding to the number of consecutive free throws Julie makes before her first miss. If we assume Julie's free throws are independent and that she makes each shot with some fixed, unknown probability p, the geometric distribution would seem appropriate:

P (X = x) = p^x(1 - p), for x = 0, 1, 2, . . .

The following table presents these probabilities for two values of p, 0.8 and 0.7. The last row gives the probabilities of making 20 or more shots before the first miss.

x	P(X = x/p = 0.8)	P(X = x/p = 0.7)
0	0.2000	0.3000
1	0.1600	0.2100
2	0.1280	0.1470
3	0.1024	0.1029
4	0.0819	0.0720
5	0.0655	0.0504
6	0.0524	0.0353
7	0.0419	0.0247
8	0.0336	0.0173
9	0.0268	0.0121
10	0.0215	0.0085
11	0.0172	0.0059
12	0.0137	0.0042
13	0.0110	0.0029
14	0.0088	0.0020
15	0.0070	0.0014
16	0.0056	0.0010
17	0.0045	0.0007
18	0.0036	0.0005
19	0.0029	0.0003
≥ 20	0.0115	0.0009

a. Consider testing H₀: p = 0.7 versus H_a: p > 0.7. Derive the rejection region for the test corresponding to significance level α = 0.05.

b. What is the probability of a Type I error? Explain this concept in the context of this problem to Julie's coach, who has never studied statistics.

c. What is the power of this test against Julie's proposal that she improved and is actually an 80% free throw shooter? Again, explain this concept in the context of this problem to Julie's coach, who has never studied statistics.

2. Suppose {X_i} is a set of n ≥ 1 independent identically distributed (iid) random variables from the uniform distribution on the interval (0, θ) for 0 < θ < ∞.

a. Find the maximum likelihood estimator (MLE) for θ.

b. Prove that the MLE is a biased estimator for θ.

c. Prove that the MLE is a consistent estimator for θ.

d. Propose an unbiased estimator for θ. Is this estimator preferable to the MLE? Include a discussion of the basis for your preference, as if you were presenting this solution to a fellow student of mathematical statistics.

3. Suppose X₁, X₂, . . . are iid Bernoulli random variables such that P (X_i = 1) = p, with p some fixed value in (0, 1). Define L₁ and L₂ to be the lengths of the first and second "runs," respectively, in the sequence generated by the Xi's. A "run" is a collection of consecutive common outcomes. So, for example, the sequence 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, . . . would produce  L₁  = 3,  L₂  = 2,  L₃  = 1,  and  L₄ = 5.

a. What is the distribution of L₁?

b. What is the distribution of L₂?

c. Are L₁ and L₂ independent? Discuss.

4. A coin will be tossed once, and we are interested in estimating the probability of heads, p. A Bayesian will propose using a uniform prior distribution on p,

g(p) = 1,  0 ≤ p ≤ 1,

and will estimate p using the mean of the posterior distribution. A frequentist will use the maximum likelihood estimator (MLE) for p. Show how to derive both the MLE and the Bayes estimator. Using squared error loss, describe (exactly or approximately, with justification) the range of values of p for which the MLE is preferable to the Bayes estimator.

5. This problem examines data showing the effect of two soporific (sleep-inducing) drugs. The variable extra shows the increase in hours of sleep for each of n = 20 patients who had been randomly assigned to two groups for the study. A few observations are shown here:

	extra	group
1	3.4	0
2	0.8	0
3	1.9	1
4	4.4	1
5	-1.2	0

There are n₀ = n₁ = 10 subjects in each group. In terms of underlying probability models, you may assume EXTRA_i ∼ N (µ₀, σ²) for subject i in group 0, and EXTRA_j ∼ N (µ₁, σ²) for subject j in group 1. The sample variance of all 20 measurements is

s²_total = 1/n - 1 _i=1Σⁿ(extra_i - (extra)^-)² = 4.0720,

where the overall mean is

(extra)^- = 1/n_i=1Σⁿ extra_i = 1.5400.

The sample variance of measurements in group 0 is s₀² =3.2006, and the variance of measurements in group 1 is s₁² =4.009. A pooled 2-sample t-test is performed, making use of the pooled estimate of the variance,

s²_pooled = ((n₀ - 1) ∗ s₀² + (n₁ - 1) ∗ s₁²/n₀+n₁-2) = 3.6048,

and giving the following result:

a. Show how to obtain the confidence interval (-3.363874, 0.203874) from the information provided, above.  Clearly define any quantities used.

b. Critique the following statement: The 95% confidence interval (-3.363874, 0.203874) contains the true difference in drug effectiveness with probability 0.95.

Attachment:- Assignment.rar