EXAMINATION: Tentamensskrivning i Matematisk Statistik (TMS061)

Time: Tuesday 29 May 2007

Jour: Daniel Ahlberg, mobile: 073 6175831, Eric Jacobsson mobile: 0737353153, Anastassia Baxevani, mobile: 0702972910

Aid: You are allowed to use a scientific calculator and a half page (both sides) of hand written notes

Grade: You need 42 points for 5, 34 points for 4 and 26 points for 3.

Motivate all your answers. Good Luck!

1) Determine the constant c so that the following function is a probability mass function: f(x) = cx for x = 1, 2, 3, 4. (4p)

2) Let X_{1}, X_{2}, …, X_{n} be a random sample.

a) What conditions do X_{1}, X_{2}, …, X_{n} have to satisfy? (1p)

b) If X_{1}, X_{2}, …, X_{n} are additionally normally distributed N(μ, σ^{2}) what can you say about the distribution of X̄? (2p)

c) What is E(X̄), V ar(X̄)? (2p)

d) Is X̄ an unbiased estimator of the true mean? (1p)

3)

−0.19, 0.73, 2.18, −0.14, 0.11, 1.07, 0.04, −0.10, −0.83, 0.29

are 10 computer generated N(0, 1) observations. Test the hypothesis μ = 1 with alternative μ ≠ 1 when

a) σ = 1 is known (3p)

b) σ is unknown (3p)

at significance level α = 0.1.

4) Answer the following questions:

a) Can a null hypothesis be rejected at α = 0.01 level when the p-value of the test was 0.007? (1p)

b) In an experiment if P (A) = 0.6 and P (B) = 0.7 is it possible that P (A ∩ B) = 0.2? (2p)

c) Do X and 2 · X have the same variance? (1p)

d) If P (A) = 0.3, P (B) = 0.5 and P (A ∪ B) = 0.65, are the events A and B independent? (2p)

5) Consider the following frequency table:

Values 0 1 2 3 4 5

Observed frequency 75 140 108 66 9 2

Based on 400 observations is a binomial distribution B(5, 0.3) an appropriate model? Perform a χ^{2} test with α = 0.05. (5p)

6) The performance of a mettalic device is to be tested. A sample of 14 specimen is taken to give x̄ = 876.5 and s_{x} = 91.4. After some adjustment has been performed a new sample of 16 specimen was taken to give ȳ = 975.3 and s_{y} = 116.6.

a) Formulate a suitable model to test the hypothesis that no change has occured with the introduction of the adjustment. (3p)

b) Provide with a confidence interval for the difference of the true means. (3p)

Use α = 0.05.

7) The Rayleigh distribution has probability density function

f(x) = \frac{x}{θ}e^{− \frac{x^{2}}{2θ}} , x > 0, 0 < θ < ∞

Find the maximum likelihood estimator of θ. (5p)

8) Suppose that the random variable X denotes the number of arrivals in a bus station and is distributed as a Poisson random variable with mean 4 arrivals per hour. Compute the following probabilities:

a) P (X = 0) (1p)

b) P (X ≥ 2) (1p)

c) P (X ≤ 4|X ≥ 2) (2p)

d) What is the probability that the first bus arrives after one hour? (3p)

9) Assume you are asked 4 questions to which you answer by guessing. For each correct answer you give, you win 1 point and for each wrong answer you loose 1 point. Let X be the maximum between your score and zero. What is the E(X)? (5p)