Use technology to solve the following problem: For a particular diamond mine, 80% of the diamonds fail to qualify as "gemstone grade". A random sample of 131 diamonds is analyzed. Find The probability that less than 75% of the sample diamonds fail to qualify as gemstone grade.
A) 0.9237
B) 0.0933
C) 0.0763
D) 0.9067

Question

Quizplus · Accepted Answer

We can use the binomial distribution to solve this problem. Let X be the number of diamonds in the sample that fail to qualify as gemstone grade. Then X has a binomial distribution with n = 131 and p = 0.8. We want to find P(X < 0.75(131)) = P(X < 98.25).

Using normal approximation to binomial, we can approximate X with a normal distribution with mean np = 131(0.8) = 104.8 and standard deviation sqrt(np(1-p)) = sqrt(131(0.8)(0.2)) = 4.29. Then,

z = (98.25 - 104.8)/4.29 = -1.51

Using a standard normal table or calculator, we find that P(Z < -1.51) = 0.0658. Therefore, the probability that less than 75% of the sample diamonds fail to qualify as gemstone grade is approximately 0.0763.

Hence, the answer is C.

Use Technology to Solve the Following Problem: for a Particular