Question

The weight of a certain type of fishes follows a normal distribution with mean 250 grams and standard deviation 20 grams.

(a) Find the probability that the weight of a fish will be less than 260 grams.

(b) Find the probability that the weight of a fish will exceed 245 grams.

(c) Find the probability that the weight of a fish will be from 200 to 270 grams.

(d) Find the weight of the fish that exceeds 90% of the weights.

Answer 1

Let X be weight of the fish

X ~ Normal (250 , 20)

a) P( X < 260) = P( < )

= P( z < 0.5)

= 0.69146

b) P( X > 245) = P( >   )

= P( z > -0.25)

= P( z < 0.25)

= 0.59871

c) P( 200 < X < 270) = P( < >   )

= P( -2.5 < z <1 )

= P( z < 1) - P( z < -2.5)

= P( z < 1) - 1 + P(z < 2.5)

= 0.84134 - 1 + 0.99379

= 0.83513

d) We need to find the 90th percentile

P( Z < z) = 0.9

P( Z < 1.282) =0.9

z = 1.282

= 1.282

   = 1.282

X= 250 + 20*1.282

X= 275.64