Question

Suppose that a sequence of Bernoulli trials is to be carried out with an unknown probabilityθof success on each trial, and the following hypotheses are to be tested:

H₀ : θ = 0.1

H₁ : θ = 0.2

Let X denote the number of trials required to obtain success, and suppose that H₀ is to be rejected if X≤ 5. Determine the probabilities of errors of type I and type II.

Short answer

The probabilities of error of type I and type II are 0.41 and 0.33 respectively.

Step-by-step solution

Step 1:To determine the probabilities of errors of type I and type II

In the given setup, X~Geo(θ)

P(type I error) = P(H₀is rejected| H₀)

= P(X≤ 5 | θ = 0.1)

= 1-(0.9)⁵

=0.41

And,

P(type I error) = P(H₀is not rejected| H₁)

= P(X≥ 5| θ = 0.2)

= 1-(0.8)⁵

=0.33

Finally, we get the probabilities of error of type I and type II are o.41 and 0.33, respectively.