Answers to all questions should be handed in to the postbox in the statistics corridor by 2pm on
Thursday 22nd February. Please remember to fill the assignment form in all the required
parts. You can find the form in the Statistics corridor.
1. Answer the questions in exercise 1.9 of section 10 (Confidence Intervals) of the textbook (by Roussas).
[6 marks]
2. Suppose that you are playing a coin tossing game with a friend, where you get a point when coin
falls on Heads and your friend gets a point when coin falls on Tails. Whoever has more points after
n tosses wins. You have suspicions that the coin is loaded and you want to use the data of the first
n tosses, x = (x1, . . . , xn) to decide whether you have an advantage or a disadvantage.
(a) (i) Describe an appropriate statistical model for your data. [2 marks]
(ii) Construct two hypotheses corresponding to you having an advantage or a disadvantage in
the game. [1 marks]
(iii) Suppose that you want to construct a test that choses one of the two hypotheses, making
sure that the probability of not detecting a disadvantage is very small. Which hypothesis
are you going to choose for your null hypothesis and which for your alternative hypothesis?
[2 marks]
(b) Now suppose that you are testing the hypothesis H0 : θ < 1
2
vs. H1 : θ > 1
2
for (X1, . . . , Xn)
i.i.d. Bernoulli(θ), using the following test:
φ(x) =
1, if 1
n
Pn
i=1 xi > c
0, otherwise .
(i) Compute the size of test φ as a function of c, for c ∈ (0, 1) and n = 10. Can you choose c
to achieve any given size α? [2 marks]
(ii) Choose constant c so that the size gets as close as possible to α = 0.05 without exceeding
it. [1 mark]
(iii) Now consider the following test
φ˜(x) =
1, if 1
n
Pn
i=1 xi > c
γ, if 1
n
Pn
i=1 = c
0, otherwise
.
for c the constant you computed in (b.ii). Compute its size as a function of γ, for γ ∈ (0, 1).
As before, n = 10. [2 marks]
(iv) Choose (c, γ) so that the size of test φ˜ is equal to α = 0.05. [1 mark]
(v) Plot the power function of test φ˜ on the alternative hypothesis. [2 marks]
(vi) If θ =
2
3
, what is the probability of not rejecting θ < 1
2
? [1 mark]
