Question

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

The length, X X , of a fish from a particular mountain lake in Idaho is normally distributed with μ=9.6 μ = 9.6 inches and σ=1.4 σ = 1.4 inches. (a) Is X X a discrete or continuous random variable?

(b) Write the event ''a fish chosen has a length of less than 6.6 inches'' in terms of X X :

(c) Find the probability of this event:

(d) Find the probability that the length of a chosen fish was greater than 12.6 inches: .

(e) Find the probability that the length of a chosen fish was between 6.6 and 12.6 inches: .

a)

X is continuous random variable

b)
X <= 6.6

c)
Here, μ = 9.6, σ = 1.4 and x = 6.6. We need to compute P(X <= 6.6). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (6.6 - 9.6)/1.4 = -2.14

Therefore,
P(X <= 6.6) = P(z <= (6.6 - 9.6)/1.4)
= P(z <= -2.14)
= 0.0162

d)
z = (x - μ)/σ
z = (12.6 - 9.6)/1.4 = 2.14

Therefore,
P(X >= 12.6) = P(z <= (12.6 - 9.6)/1.4)
= P(z >= 2.14)
= 1 - 0.9838
= 0.0162

e)
P(6.6 <= X <= 12.6) = P((12.6 - 9.6)/1.4) <= z <= (12.6 - 9.6)/1.4)
= P(-2.14 <= z <= 2.14) = P(z <= 2.14) - P(z <= -2.14)
= 0.9838 - 0.0162
= 0.9676

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