Honors Exam 2014: Statistics

1. A market research company employs a large number of typists to enter data into a computer. The time taken for new typists to learn the computer system is well-approximated by a normal distribution with a mean of 90 minutes and a standard deviation of 20 minutes.

a. Calculate the proportion of new typists that take more than two hours (120 minutes) to learn the computer system.

b. Calculate the time below which which 25% of new typists take to learn the computer system.

c. Two typists start learning the computer system at the same time. What is the probability they both learn the system before 70 minutes have passed? Explain any assumptions required for your calculation.

2. Nobel Laureate Linus Pauling (1901-1994) conducted a randomized experiment to study whether taking vitamin C supplements helps prevent the common cold. The results were reported in the Proceedings of the National Academy of Sciences. He randomly assigned 279 French skiers to two groups, group C (that took vitamin C supplements) and group S (that took a sugar pill placebo). Here are the results:

The ultimate question: is there evidence that Vitamin C helps reduce the incidence rate of colds in this population? Let pc denote the (unknown) population incidence rate of colds for people taking vitamin C supplements, and let ps denote the (unknown) population incidence rate of colds for people taking the sugar placebo.

a. State the null (H0) and alternative (HA) hypotheses for an appropriate test.

b. What test statistic will you use and what is its (approximate) sampling distribution, assuming your null hypothesis H0 is true? Explain your assumptions, and draw a rough picture of the sampling distribution (but clearly label the picture). Please carefully define any notation you introduce.

c. Find or approximate the p-value of the test and state and justify your conclusions.

3. The total lifetime in days of a certain very delicate mechanical component of a machine is known to be approximately N(µ = 100, σ² = 100). After 95 days, your component is still working. How much longer do you expect the component to work?

4. Suppose that X₁ and X₂ are independent from the N(0, θ) distribution (here θ is the variance), where 0 < θ < ∞ is unknown.

a. Find the maximum likelihood estimator of the variance θ.

b. You are told to conduct a hypothesis test of H₀: θ = 1 versus the alternative H_A: θ > 1. A sample of size n = 2 yields the MLE θ^{^} = 2.12. What do you conclude, and why?

c. A student recommends using "an unbiased estimator of θ, the sample variance

s² = 1/n - 1 _i=1∑ⁿ(X_i - X^-)².

And when H₀: θ = 1 with a sample of size n = 2 the sampling distribution of s² is known to be χ²₁ d.f. and we can use this for conducting our hypothesis test." What do you think of this proposal compared to your solution in part (b), and why?

5. Suppose X₁, X₁, ..., X_n-1 are independent, identically distributed N(µ, σ²) for fixed, unknown parameters µ and σ². Assume X_n is independent N(τ, σ²) where τ is very large (with respect to µ) so that X_n acts as an outlier. It's so large, in fact, that you may treat X_n as a constant taking the value τ. Explore and describe the statistical properties of the standard t-test statistic and/or 95% confidence interval based on the full sample of size n for conducting inference on µ in the presence of such an outlier.

6. Consider an independent, identically distributed set of random variables X₁, ..., X_n that are known to be uniform on the interval [0, 1]. Let X^- denote the sample mean, and X^~ denote the sample median. You are familiar with the Central Limit Theorem for the sample mean X^-. However, there is also a Central Limit Theorem for the sample median X^~.

Under certain conditions (satisfied here for the median of the X_i), stated casually:

X^~∼ N(0.5, σ²_X~ = 1/4n),

approximately, for large enough n. Or in general, if f(m) > 0, F(m) = 1/2, and F is differentiable at m then

√n(m^{^} - m) →_d N (0, 1/[2f(m)]²)

where m is the population median, mc is the sample median from a sample of size n, and f and F are the population density and cumulative distribution functions, respectively. The conditions essentially ensure the uniqueness of the median.

Now suppose that Y₁, ..., Y_n are independent from the uniform distribution on the set [-2, -1] ∪ [1, 2]. In this case the conditions above do not apply to the median of the Y_i. You should attempt parts (a) through (e) on this exam. You may choose to answer parts (f) through (j); if you choose not to do so, please come to the oral exam prepared to discuss them.

a. Show that the variance of X₁ is 1/12.

b. What is E(X²₁)?

c. What is E ((1 + X₁)²)?

d. Find the variance of Y₁.

e. Suppose n = 100. What is P(X^- < 0.45), approximately?

f. Suppose n = 100. What is P(X^~ < f 0.45), approximately?

g. Suppose n = 100. Can you find P(Y^- < -0.05), approximately? If so, do it.

h. Suppose n = 100. Can you find P(Y^~ < -0.05), approximately? If so, do it.

i. Suppose n = 100. Can you find P(Y^- < -1.05), approximately? If so, do it.

j. Suppose n = 100. Can you find P(Y^~ < -1.05), approximately? If so, do it.