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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?

User Gauri
by
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1 Answer

4 votes

Answer:

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

User Rick Pack
by
7.3k points
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