For a particular diamond mine, 80% of the diamonds fail to qualify as "gemstone grade". A random sample of 94 diamonds is analyzed. Find the probability that the proportion of the sample diamonds That fail to qualify as gemstone grade is between 0.76 and 0.83.
A) 0.6013
B) 0.1660
C) 0.8340
D) 0.7673

Question

Quizplus · Accepted Answer

This problem can be solved using the Central Limit Theorem (CLT) for proportions. Given that 80% of the diamonds fail to qualify, the population proportion (p) is 0.80, and the standard deviation of the sampling distribution of the sample proportion ($\sigma_{\hat{p}}$) can be calculated using the formula $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$, where n is the sample size (94 in this case). To find the probability that the sample proportion ($\hat{p}$) falls between 0.76 and 0.83, we convert these proportions to z-scores and use the standard normal distribution to find the probability between these z-scores. The calculation results in a probability of 0.6013, indicating that there's about a 60.13% chance the proportion of diamonds that fail to qualify as gemstone grade in the sample is between 0.76 and 0.83.

For a Particular Diamond Mine, 80% of the Diamonds Fail