Consider a radar system that uses radio waves to detect aircraft. The system receives a signal and, based on the received signal, it needs to decide whether an aircraft is present or not. Let X be the received signal. Suppose that we know 

where W ∼ N(0,σ² = 1/9). Thus, we can write X = θ +W, where θ = 0 if there is no aircraft, and θ = 1 if there is an aircraft. Suppose that we define H₀ and H₁ as follows:

H₀ (null hypothesis): No aircraft is present.

H₁ (alternative hypothesis): An aircraft is present.

a. Write the null hypothesis, H₀, and the alternative hypothesis, H₁, in terms of possible values of θ.

b. Design a level 0.05 test (α = 0.05) to decide between H₀ and H₁.

c. Find the probability of type II error, β, for the above test. Note that this is the probability of missing a present aircraft.

d. If we observe X = 0.6, is there enough evidence to reject H₀ at significance level α = 0.01?

e. If we would like the probability of missing a present aircraft to be less than 5%, what is the smallest significance level that we can achieve?

Transcribed Image Text:

X = W, X = 1 +W, if no aircraft is present. if an aircraft is present.