Question

The length, ?, of a fish from a particular mountain lake in Idaho is normally distributed with ?=9.8 inches and ?=1.1 inches.

(a) Is ? a discrete or continuous random variable? (Type: DISCRETE or CONTINUOUS) ANSWER:

(b) Write the event ''a fish chosen has a length of less than 7.8 inches'' in terms of ?: .

(c) Find the probability of this event:

(d) Find the probability that the length of a chosen fish was greater than 11.3 inches: .

(e) Find the probability that the length of a chosen fish was between 7.8 and 11.3 inches:

Please show all work so I can understand how you got to the answer. R is not to be used for this question, please complete it by hand!!

Thanks!

Answer 1

a)

X ~ Normal( , ²)

Where = 9.8 , = 1.1

X is normally distributed. Therefore, this is continuous distribution.

b)

Let X be length of fish from particular mountain lake in Idaho.

Event: a fish chosen has length of less than 7.8 inches.

That is

X < 7.8

c)

We have to calculate P( X < 7.8) = ?

Given = 9.8 , = 1.1

We convert this to standard normal as

P( X < x) = P( Z < x - / )

That is

P( X < 7.8) = P( Z < 7.8 - 9.8 / 1.1)

= P (Z < -1.8182)

= 1 - P (Z < 1.8182)

= 1 - 0.9655

= 0.0345

d)

We have to calculate P( X > 11.3) = ?

P( X > 11.3) = P( Z > 11.3 - 9.8 / 1.1)

= P( Z > 1.3636)

= 1 - P( Z < 1.3636)

= 1 - 0.9137

= 0.0863

e)

We have to calculate P( 7.8 < X < 11.3) = ?

P( 7.8 < X < 11.3) = P( X < 11.3) - P( X < 7.8)

= P( Z < 11.3 - 9.8 / 1.1) - P( Z < 7.8 - 9.8 / 1.1)

= P(Z < 1.3636) - P( Z < -1.8182)

= P( Z < 1.3636) - ( 1 - P( Z < 1.8182) )

= 0.9137 - ( 1 - 0.9655)

= 0.8791