Question

Fruit fly thorax lengths (2.2) Fruit flies are used frequently in genetic research because of their quick reproductive cycle. The length of the thorax (in millimeters) for male fruit flies is approximately Normally distributed with a mean of 0.80 mm and a standard deviation of 0.08 mm.21

a. What proportion of male fruit flies have a thorax length greater than 1 mm?

b. What is the 30th percentile for male fruit fly thorax lengths?

Short answer

Part (a) The probability is$0.0062$

Part (b) The percentile is$0.758$

Step-by-step solution

Part (a) Step 1: Given information

Population mean $\left(\mu \right)=0.80$

Standard deviation $\left(\sigma \right)=0.08$

Part (a) Step 2: Concept

The formula used:$z=\frac{x-\mu }{\sigma }$

Part (a) Step 3: Calculation

The proportion of male fruit flies that are having thorax length above $1mm$can be calculated as:

$$\begin{gathered} P(x>1)=P(\frac{x-\mu }{\sigma }>\frac{1-\mu }{\sigma }) \\ =P(Z>1-\frac{0.80}{0.08}) \\ =P(Z>2.50) \\ =0.0062 \end{gathered}$$

Therefore, the required proportion is$0.0062$

Part (b) Step 1: Explanation

From standard normal table, the $z-$value at $0.30$ probability is $-0.5244$

The $30th$percentile can be calculated as:

$$\begin{gathered} x=\mu +z\times \sigma \\ =0.80+(-0.5244)\times 0.08 \\ =0.758 \end{gathered}$$

Thus, the required percentile is $0.758$