Explanation
1. Understand Type I Error
A Type I error occurs when we reject the null hypothesis (H0) even though it is actually true. The probability of committing a Type I error, denoted by α, is defined as:
α=P(Reject H0∣H0 is true)=P(X∈Critical Region∣θ=1)
2. Probability Density Function (PDF) under H0
Under H0, θ=1. The random variable X follows a uniform distribution U(0,1). The PDF of X is:
f(x)={1−01=10amp;if 0≤x≤1amp;otherwise
3. Calculate the Probability of the Critical Region
The critical region is given as 0.8≤X≤1.3. However, since the support of X under H0 is limited to [0,1], we only consider the portion of the critical region that falls within this interval:
Effective Critical Region={x:0.8≤x≤1}
Now, compute the probability α:
α=∫0.81f(x)dx
α=∫0.811dx
α=[x]0.81
α=1−0.8=0.2
Conclusion
The probability of a Type I error (α) is 0.2.
This directly matches option (b).
