Question

A fabric manufacturer believes that the proportion of orders for raw material arriving late is p = 0.6. If a random sample of 10 orders shows that 3 or fewer arrived late, the hypothesis that p = 0.6 should be
rejected in favor of the alternative p < 0.6. Use the binomial distribution.

Find the probability of committing a Type I error if the true proportion is p = 0.6.

Recall, a Type I error is committed when the 'Null Hypothesis' (what we assume) is rejected when in fact it is true. In other words, using the criteria that we reject the claim of p=0.6, when 3 or fewer orders arrive late out of a sample of 10, how often will we commit a Type I error when the true probability of a order arriving late is actually p=0.6.

You can use the rules of probability based on the binomial distribution or you can simulate a bunch of samples of size 10 and estimate the Type I error rate empirically.

*** Use R to solve***

Answer 1

Answer late Let number of orders arring n: 10, x N Binominal (n=1024) He : P:06 rls Hii P20.6 Rejection Region ; RR = { *\x3} Now , P : proportion of prder waving lute (1) P (type I error) =P ( rejecting to be there) = PL*33) Ho: P=0.6) P(x=31&NB110,0.6) =P(x=0)+P(x=1) +P (1-2) +P(1-3) - = 0.0001 +0.0016+0.0106+0.0125 0.0548 (b) PC type I enter) -P ( azeepting Ho ) un true) = P(x3) H: p=0.3 1-P (x+3)NB (10,0.3) --[P(2:0)=P(x=1) +P/7=2) +P (x-3)] 1- [0.0282 +0.12.11 +0.2335+0.2668 1-0.6496 1.3504