Question

Te weight of an adult bottlenose dolphin was found to follow a normal distribution with a mean of 550 pounds and a standard deviation of 50 pounds.

a. What percentage of adult bottlenose dolphins weigh from 400 to 600 pounds?
b. If X represents the mean weight of a random sample of 9 adult bottlenose dolphins, what is P X (500 580 < < ) ?
c. In a random sample of 9 adult bottlenose dolphins, what is the probability that 5 of them are heavier than 560 pounds?

Answer 1

Mean =
`\mu = 550`

Standard deviation =
`\sigma = 50`

a. What percentage of adult bottlenose dolphins weigh from 400 to 600 pounds?

P(400<x<600)

Formula :
`Z=(x-\mu)/(\sigma)`

at x = 400

`Z=(400-550)/(50)`

`Z=-3`

Refer the z table for p value

p value = 0.0013

at x = 600

`Z=(600-550)/(50)`

`Z=1`

Refer the z table for p value

p value = 0.8413

P(400<x<600)=P(x<600)-P(x<400)=0.8413-0.0013=0.84

So,84% of adult bottlenose dolphins weigh from 400 to 600 pounds

b)If X represents the mean weight of a random sample of 9 adult bottlenose dolphins, what is P (500<x < 580) ?

Formula :
`Z=(x-\mu)/(\sigma)`

at x = 500

`Z=(500-550)/(50)`

`Z=-1`

Refer the z table for p value

p value = 0.1587

at x = 580

`Z=(580-550)/(50)`

`Z=0.6`

Refer the z table for p value

p value = 0.7257

P(500<x<580)=P(x<580)-P(x<500)=0.7257-0.1587=0.84

c). In a random sample of 9 adult bottlenose dolphins, what is the probability that 5 of them are heavier than 560 pounds?

at x = 560

`Z=(560-550)/(50)`

`Z=0.2`

Refer the z table for p value

p value = 0.5793

P(x>560)=1-P(x<560)=1-0.5793=0.4207

Now to find the the probability that 5 of them are heavier than 560 pounds we will use binomial distribution

`P(X=r)=^nC_r p^r q^(n-r)`

p is the probability of success that is 0.4207

q = 1-p = probability of failure

n = 9

r = 5

`P(X=5)=^9C_5 (0.4207)^5 (1-0.4207)^(9-5)`

`P(X=5)=(9!)/(5!(9-5)!)(0.4207)^5 (1-0.4207)^(9-5)`

`P(X=5)=0.187`

Hence In a random sample of 9 adult bottlenose dolphins, the probability that 5 of them are heavier than 560 pounds is 0.187