Question

Suppose that the body weights of lemmings are normally distributed with a mean of 63.8g and a standard seviation of 12.2g.

(a) What proportions of lemmings weigh more than 90g ?

(b)What proportion of lemmings weigh between 40 and 90 g?

(c)We can say that 99% of lemmings weigh more than ____ g.

(d)In a random sample of 20 lemmings what is the probability that exactly 3 weigh more than 90g ?

Answer 1

Solution:

Given: the body weights of lemmings are normally distributed with a mean of 63.8g and a standard deviation of 12.2g.

thus and

Part a) What proportions of lemmings weigh more than 90g ?

That is find:
P( X > 90) =....?

Find z score for x = 90

thus we get:

P( X > 90) = P( Z > 2.15)

P( X > 90) = 1 - P( Z < 2.15)

Look in z table for z = 2.1 and 0.05 and find area.

From z table we have: P( Z< 2.15) = 0.9842

Thus

P( X > 90) = 1 - P( Z < 2.15)

P( X > 90) = 1 - 0.9842

P( X > 90) = 0.0158

Part b) What proportion of lemmings weigh between 40 and 90 g?

P( 40 < X < 90 ) =........?

P( 40 < X < 90 ) =P( X < 90)- P( X < 40)

Find z score for x = 90 and for x = 40

P( 40 < X < 90 ) =P( X < 90)- P( X < 40)

P( 40 < X < 90 ) =P( Z < 2.15)- P( Z < -1.95)

Look in z table for z = -1.9 and 0.05 and find area.

P( Z< -1.95) = 0.0256

Thus

P( 40 < X < 90 ) =P( Z < 2.15)- P( Z < -1.95)

P( 40 < X < 90 ) = 0.9842 - 0.0256

P( 40 < X < 90 ) = 0.9586

Part c) We can say that 99% of lemmings weigh more than ____ g.

That is find x value such that:

P( X > x ) = 0.99

Thus find z such that:

P(Z > z ) =0.99

that is find z such that:

P( Z < z ) = 1 - P( Z> z )

P( Z < z ) = 1 - 0.99

P( Z < z ) = 0.01

look in z table for area = 0.0100 or its closest area and find z value.

Area 0.0099 is closest to 0.0100 and it corresponds to -2.3 and 0.03

thus z critical value = -2.33

Use following formula to find x:

thus We can say that 99% of lemmings weigh more than 35.4 g.

Part d) In a random sample of 20 lemmings what is the probability that exactly 3 weigh more than 90g ?

We have: P( X > 90) = 0.0158

thus p = 0.0158

n = sample size =20

We have to find:

P(X = 3) =......?

Here X = number of lemmings weigh more than 90g follows a binomial distribution,

thus using Binomial distribution we get:

Where

q = 1 - p = 1 - 0.0158 = 0.9842

thus