Question

# The length, X, of a fish from a particular mountain lake in Idaho is normally distributed...

The length, X, of a fish from a particular mountain lake in Idaho is normally distributed with μ= 8.9 and σ=1.6 inches.

(a) Write the event ''a fish chosen has a length of less than 5.9 inches'' in terms of X: ____

(b) Find the probability of this event: ____

(c) Find the probability that the length of a chosen fish was greater than 11.4 inches: ____

(d) Find the probability that the length of a chosen fish was between 5.9 and 11.4 inches:  _____

a)

Event: a fish chosen has a length of less than 5.9 inches

X < 5.9

b)

Given, = 8.9 , = 1.6

We convert this to standard normal as

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

So,

P( X < 5.9) = P( Z < 5.9 - 8.9 / 1.6)

= P( Z < -1.875)

= 0.0304

c)

P( X > 11.4) = P( Z > 11.4 - 8.9 / 1.6)

= P( Z > 1.5625)

= 0.0591

d)

P( 5.9 < X < 11.4 ) = P( X < 11.4) - P( X < 5.9)

= P( Z < 11.4 - 8.9 / 1.6) - P( Z < 5.9 - 8.9 / 1.6)

= P( Z < 1.5625) - P( Z < -1.875)

= 0.9409 - 0.0304

= 0.9105

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