Bonus Question (hypothesis test for the proportion on small sample size)

we know that if (pi)n>=5 and (1-(pi) )>=5, we can assume sample distribution is normal because of central limit theorem. then we can use z-test to do the hypothesis testing. but if the sample size is small, we cannot assume the sample distribution is normal. Here is the question:

One manufacturer’s high-quality product rate is 30%. recently, a sample with size 15 products is collected. among them, 3 are with high quality. Can we say the high-quality rate remains 30% using level of significance (alpha symbol)=0.05 ?

We can find that n(pi)=15 x 0.3 = 4.5 <= 5 (in reality, n(pi) is better be much larger than 5), so we do not take the assumption that it is normally distributed. we cannot use z-test. This hypothesis is still two-tail hypothesis and the rejection region is {x<= c1 or x>=c2}. C1 and C2 are two critical values. Can you find c1 and c2 to form the rejection region and do the hypothesis testing?

Ho: p=0.3 vs H1: p (does not equal) 0.3 (hint: binomial distribution)