Question

The length of an adult oceanic manta ray is normally distributed with a mean of 14.9 ft and a standard deviation of 3.65 ft. Forty-nine adult oceanic manta ray is selected at random, and we are interested in the mean length.

a) Define the random variable of interest.
b) State the distribution of the RV of interest.
c) What is the probability that the mean length is between 12 and 16 ft?
d) What is the probability that the mean length is at least 15.5 ft?
e) What is the probability that the mean length is 14.9 ft?

Answer 1

a) The variable of interest is the mean length of 49 adult oceanic manta rays.

b) Given, = 14.9 ft

= 3.65 ft

n = 49

According to Central Limit Theorem, the sampling distribution of mean will be normally distributed with mean, = = 14.9 ft

and standard error, =

=

= 0.5214

P( < A) = P(Z < (A - )/)

c. P(12 < < 16) = P( < 16) - P( < 12)

= P(Z < (16 - 14.9)/0.5214) - P(Z < (12 - 14.9)/0.5214)

= P(Z < 2.11) - P(Z < -5.56)

= 0.9826 - 0

= 0.9826

d) P(at least 15.5 ft) = P( 15.5)

= 1 - P( < 15.5)

= 1 - P(Z < (15.5 - 14.9)/0.5214)

= 1 - P(Z < 1.15)

= 1 - 0.8749

= 0.1251

e) P( = 14.9) = 0

For normal distribution which is a continuous distribution, the probability at any particular point will be 0.