Question

A different species of cockroach has weights that are approximately Normally distributed with a mean of 50 grams. After measuring the weights of many of these cockroaches, a lab assistant reports that 14% of the cockroaches weigh more than 55 grams. Based on this report, what is the approximate standard deviation of weight for this species of cockroaches?

A) Cannot determine without more information.
B) 4.6
C) 6.2
D) 5
E) 14

Answer 1

To calculate the standard deviation of the cockroach species' weight, we use the information that 14% of cockroaches weigh more than 55 grams and the mean weight is 50 grams. Applying the values to the standard normal distribution, we find that the standard deviation is approximately 4.63 grams, close to option B (4.6).

Step-by-step explanation:

The problem states that 14% of the cockroaches weigh more than 55 grams, and we know that the mean weight is 50 grams. Since the weights are normally distributed, we can use the standard normal distribution to find the standard deviation.

The value corresponding to the upper 14% in a standard normal distribution is approximately 1.08 (using a Z-table or calculator). This means that a weight of 55 grams is 1.08 standard deviations above the mean. Therefore, we can set up the equation:

Mean + (Z * Standard Deviation) = X

50 + (1.08 * Standard Deviation) = 55

Solving for the standard deviation, we get:

Standard Deviation = (55 - 50) / 1.08

Standard Deviation ≈ 4.63

Thus, the approximate standard deviation of weight for this species of cockroaches is 4.63 grams, which is closest to choice B (4.6).