Full Question:.
Most penguin species are not sexually dimorphic, which means they lack obvious outward body characteristics which indicate sex. Observation of behavior or a blood test can determine Penguin sex. A penguin researcher is interested in estimating the proportion of females in a large penguin population. She takes a random sample of n = 20 penguins and determines the sex of each one using a blood test. She finds 12 males and 8 females. Let π be the proportion of females in the population.
(a) Find a point estimate of π.
(b) Find the estimated standard deviation of your estimate.
(c) Is it reasonable to compute a 95% confidence interval for π using the normal approximation in this case? If it is possible, explain why, and make the interval. If it is not reasonable, explain why.
Answer:
a. The female penguins are 0.4 in proportion to the male.
b. Standard Error = 0.110
c. Yes, it is.
See Explanation
Step-by-step explanation:
Given
n = 20 penguins
m = Male = 12
f = Female = 8
π = the proportion of females in the population...
a. Calculating a point estimate of π.
This is calculated by dividing number of females by total penguins
π = f/n
π = 8/20
π = 0.4
So, the female penguins are 0.4 in proportion to the male.
b. Calculating Standard Error
Standard Error is calculated by √(pq/n)
Where p = 0.4 --- calculated above
q = 1 - p = 1 - 0.4
q = 0.6
n = 20
Standard Error = √(0.4*0.6/20).
Standard Error = 0.109544511501033
Standard Error = 0.110 -; Approximated
c. Checking Possibilities
Expected Value of Male = 20(0.6) = 12
Expected Value of Female = 20(0.4) = 8
20(0.4) and 20(0.6) are both greater than 5, so we can approximate the distribution as normal. The C sill be (0.18, 0.62).
