Question

The weight of fish in Lake Paradise follows a normal distribution with mean of 7.6 lbs and standard deviation of 2 lbs.

a. What proportion of fish are between 9 lbs and 12 lbs? Give your answer to 3 decimal places.

b. Alex boasts that he once caught a fish that was just big enough to be in the top 3.5% of of the fish population. How much did his fish weigh? Give your answer to 2 decimal places. (lbs)

c. If one catches a fish from the bottom 20% of the population, the fish must be returned to the lake. What is the weight of the smallest fish that one can keep? Give your answer to 2 decimal places.

lbs

Answer 1

Solution :

Given that ,

mean = = 7.6

standard deviation = = 2

(a)

P(9 < x < 12) = P((9 - 7.6)/ 2) < (x - ) / < (12 - 7.6) / 2) )

= P(0.7 < z < 2.2)

= P(z < 2.2) - P(z < 0.7)

= 0.9861 - 0.758

= 0.2281

Proportion = 0.228

(b)

P(Z > z) = 3.5%

1 - P(Z < z) = 0.035

P(Z < z) = 1 - 0.035 = 0.965

P(Z < 1.81) = 0.965

z = 1.81

Using z-score formula,

x = z * +

x = 1.81 * 2 + 7.6 = 11.22

Fish weight = 11.22 lbs

(c)

P(Z < z) = 20%

P(Z < -0.84) = 0.20

z = -0.84

Using z-score formula,

x = z * +

x = -0.84 * 2 + 7.6 = 5.92

Weight of smallest fish = 5.92 lbs