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?

Answer 1

Answer:

The standard deviation of weight for this species of cockroaches is 4.62.

Step-by-step explanation:

Given : 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.

To find : What is the approximate standard deviation of weight for this species of cockroaches?

Solution :

We have given,

Mean $\mu=50$

The sample mean x=55

A lab assistant reports that 14% of the cockroaches weigh more than 55 grams.

i.e. P(X>55)=14%=0.14

The total probability needs to sum up to 1,

$P(X\leq 55)=1-P(X>55)$

$P(X\leq 55)=1-0.14$

$P(X\leq 55)=0.86$

The z-score value of 0.86 using z-score table is z=1.08.

Applying z-score formula,

$z=\frac{x-\mu}{\sigma}$

Where, $\sigma$ is standard deviation

Substitute the values,

$z=\frac{x-\mu}{\sigma}$

$1.08=\frac{55-50}{\sigma}$

$1.08=\frac{5}{\sigma}$

$\sigma=\frac{5}{1.08}$

$\sigma=4.62$

The standard deviation of weight for this species of cockroaches is 4.62.