It is known that the random variable 𝑌 has a probability density function 𝑓(𝑦; 𝜃) = {

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It is known that the random variable 𝑌 has a probability density function

𝑓(𝑦; 𝜃) = {

𝜃𝑦

𝜃−1

, 0 < 𝑦 < 1

0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

It is desired to test the null hypothesis 𝐻0: 𝜃 = 2 against the alternative 𝐻1: 𝜃 = 1.

A random sample 𝑌1 of size 𝑛 = 1 is to be used.

i) Calculate the level of significance or type I error which rejects 𝐻0 if 𝑌1 ≤

1

2

. (4)