Question

Kyle goes out fishing for Ahi Tuna. Suppose that Ahi Tuna have weights that are normally distributed with mean 210 pounds and standard deviation 30 pounds.

If Kyle catches an Ahi, what is the probability it weighs between 177 and 219 pounds?
If Kyle catches an Ahi, what is the probability it weighs more than 231 pounds?
Any Ahi that Kyle catches will have 60% chance of weighing more than pounds.
The 25th percentile of the distribution of Ahi weight is pounds.
The 75th percentile of the distribution of Ahi weight is pounds.
If Kyle catches 5 Ahi, the chance that they all weigh more than 210 pounds is.

Answer 1

Kyle goes out fishing for Ahi Tuna. Suppose that Ahi Tuna have weights that are normally distributed with mean 210 pounds and standard deviation 30 pounds.\

Therefore

z-score =

So first we will convert the 'x' to 'z-scores' and then using the normal probability (if 'x' is given) an normal percentage (if given 'p' ) tables to get the required values

(1)If Kyle catches an Ahi, what is the probability it weighs between 177 and 219 pounds?

P(177 < X < 219) = P(X < 219) - P(X < 177)

= P(Z < 0.3) - P(Z < -1.1)

= P(Z < 0.3) - [1 - P(Z <1.1)]

= 0.61791 - (1 - 0.8643)

(2)If Kyle catches an Ahi, what is the probability it weighs more than 231 pounds?

P( X >231) = 1 - P(X <231)

= 1 - P(Z < 0.7)

= 1 - 0.75804

(3)Any Ahi that Kyle catches will have 60% chance of weighing more than pounds.

Here we have been given the probability that more than some weight 'x' has 60% probability. So we are goingto use normal percentage tables

P( X > x)= 0.60

Substraction from 1

P(X > -x) = 0.40 .................WE convert to '>' and '> 0.5' because the tables have greater than and for > 50% values

P(Z > ) = 0.40

= 0.2534

(4)The 25th percentile of the distribution of Ahi weight is pounds.

Percentile is that value below which the 'that percentile % of the data lies. 25% percentile means a value of 'x' belowe which 25% of the data lies.

P(X <x) = 0.25

P(X > -x) = 0.25

P(X > ) = 0.25

= 0.6745

(5)The 75th percentile of the distribution of Ahi weight is pounds.

P(X <x) = 0.75

P(X > x) = 0.25

P(X > ) = 0.25

= 0.6745

(6)If Kyle catches 5 Ahi, the chance that they all weigh more than 210 pounds is

P(X > 210) = 1 - P(X < 210)

= 1 - P(Z < 0)

= 1 - 0.50

= 0.50

We know that probabiltiy of 1 Ahi weighing more than 210 pounds = 0.50

Assuming the weights are independent.

So all 5 weighing more than 210 =

=

We multiply the probabiltiy because catching the Ahi is an independent and simultaneous event.