Question

The manager of a cultured pearl farm has received a special order for two pearls between 7 millimeters and 9 millimeters in diameter. from past experience, the manager knows that the pearls found in his oyster bed have diameters that are normally distributed with a mean of 8 millimeters and a standard deviation of 2 millimeters. assume that every oyster contains one pearl. the manager wants to know how many oysters he should expect to open to find two pearls of the appropriate size for this special order. complete the following parts to design a simulation to answer the manager’s question.

Answer 1

To calculate the number of oysters the manager should expect to open to find two pearls of the appropriate size, we use the concept of the normal distribution with the mean diameter of 8 millimeters and standard deviation of 2 millimeters.

Step-by-step explanation:

To calculate the number of oysters that the manager should expect to open to find two pearls of the appropriate size, we need to use the concept of the normal distribution. The mean diameter of pearls in the oyster bed is 8 millimeters and the standard deviation is 2 millimeters. We can use the properties of the normal distribution to determine the probability of finding a pearl within the specified range, which is between 7 and 9 millimeters.

First, we calculate the z-scores for the lower and upper limits of the range. The z-score is calculated with the formula: z = (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation.

So for the lower limit, z = (7 - 8) / 2 = -0.5, and for the upper limit, z = (9 - 8) / 2 = 0.5.

Next, we use a z-table or calculator to find the probabilities associated with these z-scores. The probability of finding a pearl within the specified range is then the difference between these two probabilities.

Finally, to find the expected number of oysters to open to find two pearls, we divide the total number of oysters by the probability of finding a pearl within the specified range.