Point Estimation Question

[emob2p]

Suppose I have a total population with x/y percent having a certain property q. If I take an n-number sample of this population, what is the most likely number of elements in this sample that will have property q?

I want to say the fraction z/n will be as close to x/y as possible where z is the number of q-elements in our sample. Any thoughts on how to prove this?

 

[0rthodontist]

You want to maximize the binomial distribution.

 

[tacman]

The most likely number of elements in the sample that will have property q can be estimated using the point estimation method. This method involves using a sample statistic, such as the sample proportion, to estimate the population parameter of interest. In this case, we are interested in the proportion of the population with property q.

The sample proportion, denoted by p, is defined as the number of elements in the sample with property q divided by the total sample size, n. Therefore, the most likely number of elements in the sample with property q can be estimated as p*n.

To prove that this is the most likely number, we can use the concept of maximum likelihood estimation. This approach states that the value of the parameter that maximizes the likelihood of obtaining the observed sample is the best estimate of the true population parameter.

In our case, the likelihood of obtaining a sample with z q-elements out of n total elements can be expressed as:

L(p) = (x/y)^z * (1-x/y)^(n-z)

To find the value of p that maximizes this likelihood, we can take the derivative of L(p) with respect to p and set it equal to 0:

dL(p)/dp = z*(x/y)^z * (1-x/y)^(n-z-1) - (n-z)*(x/y)^(z+1) * (1-x/y)^(n-z-1) = 0

Simplifying this equation, we get:

z*(1-p) - (n-z)*p = 0

Solving for p, we get:

p = z/n

Therefore, the value of p that maximizes the likelihood of obtaining the observed sample is z/n, which is the sample proportion. This proves that p*n is the most likely number of elements in the sample with property q, as it is the value of p that maximizes the likelihood.

In conclusion, the most likely number of elements in the sample with property q can be estimated using the sample proportion, which is calculated as the number of elements in the sample with property q divided by the total sample size. This method follows the principle of maximum likelihood estimation and can be used to provide a reliable estimate of the population parameter of interest.