Please answer! I need help ! We know that if pi n greaterthanorequalto 5 and (1
ID: 3232757 • Letter: P
Question
Please answer! I need help ! We know that if pi n greaterthanorequalto 5 and (1 - pi) n greaterthanorequalto 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 = 0.05? We can find that n pi = 15 times 0.3 = 4.5 lessthanorequalto 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 lessthanorequalto c_1 or x greaterthanorequalto c_2}. c_1 and c_2 are two critical values. Can you find c_1 and c_2 to get P-value and do the hypothesis testing?Explanation / Answer
In order to conduct the hypothesis test using binomial distribution, we need to perform two test i.e. left tailed and right tailed.
Below are the null and alternate hypothesis
H0: p = 0.3
H1: p < 0.3
Probability that less than 3 are with high quality
P(X<3) = 0.3828 This is the p-value and calculated using binomial distribution formula
P(X=r) = nCr*p^r*(1-p)^(n-r)
In another case for right tailed, Below are the null and alternate hypothesis
H0: p = 0.3
H1: p > 0.3
Probability that more than 3 are with high quality
P(X>3) = 0.3504 This is the p-value and calculated using binomial distribution formula
P(X=r) = nCr*p^r*(1-p)^(n-r)
The p-value in both the case is greater than 0.025 (significance level), we fail to reject the null hypothesis. This means there are not sufficient evidence to conclude that proportion of high quality products is different from 0.3.