Question

The basal diameter of a sea anemone is an indicator of its age, and in a certain population of anemones, the distribution of basal diameters is approximately normal with a mean of 5.3 cm and a standard deviation of 1.8 cm. suppose you randomly select five anemones from this population.

a. what is the probability that all five anemones have a basal diameter more than 5.5 cm? (2pt)

Answer 1

Explanation:

Let X be the basal diameter of a sea anemone.

Given that X is N(5.3, 1.8)

Sample size =5

We have to find the prob that all five anemones have a basal diameter more than 5.5 cm

Prob (for one anemone >5.5 cm) =
`P(Z>(5.5-5.3)/(1.8/√(5) ) =0.2485`

Since each anemone selected is independent of the other anemone, the selected anemones follow Binomial with n =5 and p = 0.2485 and q=0.7515

P(X=5) = 1-P(X=0) = 1-0.7515^5

=0.7603

Note that 5 elements selected cannot be taken as normal since sample size is small.

No of anemones selected having a basal diameter more than 5.5 cm is

binomial with n=5 and p = 0.2585

Answer 2

Given:
μ = 5.3 cm, population mean
σ = 1.8 cm, population sandard deviation
n = 5, sample size

The random variable is x = .5 cm.
The z-score is

`z = (x-\mu)/(\sigma / √(n) ) = (5.5-5.3)/(1.8/ √(5) ) =0.2485`

From standard table, obtain
P(x>5.5) = 1 - 0.598 = 0.402

Answer: 0.402