# Hypothesis testing of sum of uniform distribution

I want to have some clarifications related to below question from Casella Berger

Regarding $$\phi_1(X_1),$$ I understand that $$\alpha$$-value is 0.05 and hence we are looking for a value of $$C$$ for which $$\phi_2(X_1,X_2)$$ is also 0.05

When I plot $$Y = X_1+X_2.$$ I get uniform distribution of $$(0,2),$$ assuming $$theta = 0.$$

Hence, question reduces to value of $$C$$ so $$Y(0,2) > C.$$ Since this is a uniform shouldn't $$C$$ be equal to $$0.95*2 = 1.9?$$ However, correct answer is $$C = 1.68$$ (approx)

• I have edited your question using JaX-notation. Please take a look to make sure I have not misunderstood your intended meaning. // Also because this is a textbook problem, please include the 'self-study' tag. – BruceET Jun 5 at 3:00
• Thanks @BruceET for your help !! – Dataist Jun 5 at 3:10

For example, try simulating the sum of two uniform random variates (say $$10^4$$ values or so) and see what a histogram (with say 100 bins) and an ecdf of that looks like (which will approximate the density and cdf in large samples)
• Correct. Do you get the right answer for the upper tail critical value $C$? – Glen_b -Reinstate Monica Jun 5 at 4:21
• Thanks @Glen_b, now I get the right answer of C = 1.684. I got it by solving $(2-C)^2 / 2 = 0.05$ – Dataist Jun 5 at 4:38