Question

Fruit flies. The common fruit fly Drosophila melano-gaster is the most studied organism in genetic research because it is small, is easy to grow, and reproduces rapidly. The length of the thorax (where the wings and legs attach) in a population of male fruit flies is approximately Normal with mean 0.800 millimeters (mm) and standard deviation 0.078 mm.

(a)What proportion of flies have thorax length less than 0.6 mm?

(b)What proportion have thorax length greater than 0.9 mm?

(c)What proportion have thorax length between 0.6 mm and 0.9 mm?

Answer 1

Let x be a random variable representing the thorax length of a male fruit fly, then
a.) P(x < 0.6) = P(z < (0.6 - 0.8)/0.078) = P(z < -2.564) = 1 - P(z < 2.564) = 1 - 0.99483 = 0.00517 = 0.517%

b.) P(x > 0.9) = 1 - P(z < (0.9 - 0.8)/0.078) = 1 - P(z < 1.282) = 1 - 0.90009 = 0.09991 = 9.991%

c.) P(0.6 < x < 0.9) = 100% - 0.517% - 9.991% = 89.492%

Answer 2

Probability is the likeliness of an event to occur.

• The proportion that have length less than 0.6 mm is 0.0052

• The proportion that have length greater than 0.9 mm is 0.0999

• The proportion that have length between 0.6 mm and 0.9 mm is 0.8949

Given that:

$\mu = 0.800$

$\sigma = 0.078$

(a) Proportion that have length less than 0.6 mm

First, we calculate the z value.

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

$z = \frac{0.6 - 0.800}{0.078}$

$z = \frac{-0.2}{0.078}$

$z = -2.564$

The probability is then represented as:

$P(x < 0.6) = P(z < -2.564)$

From z table, we have:

$P(x < 0.6) = 0.0052$

The proportion that have length less than 0.6 mm is 0.0052

(b) Proportion that have length greater than 0.9 mm

Calculate the z value.

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

$z = \frac{0.9 - 0.800}{0.078}$

$z = \frac{0.1}{0.078}$

$z = 1.282$

The probability is then represented as:

$P(x > 0.9) = P(z > 1.282)$

From z table, we have:

$P(x > 0.9) = 0.0999$

The proportion that have length greater than 0.9 mm is 0.0999

(c) Proportion that have length between 0.6 mm and 0.9 mm

This is represented as:

$P(0.6< x<0.9)$

So, we have:

$P(0.6< x<0.9) = P(x<0.9) - P(x<0.6)$

$P(x<0.9) = 1 - P(x>0.9)$

So, we have:

$P(0.6< x<0.9) = 1 - P(x>0.9) - P(x<0.6)$

$P(0.6< x<0.9) = 1 - 0.0999 - 0.0052$

$P(0.6< x<0.9) = 0.8949$

The proportion that have length between 0.6 mm and 0.9 mm is 0.8949

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