Question

Please make sure to display your thought process? It is imperative to be able to follow how the answer was deduced. Please be as thorough as possible. Please address all parts of this question.

We consider a normal population with known variance, and we want to obtain a confidence interval (CI) of half-length E on the mean value. How is the required sample size affected if all other factors are kept constant, but:

A) The significance level a of the test is increased?

B) We learn that the variance is smaller than the earlier specified value Standard Deviation?

C) We want to decrease E?

Hint: write down the CI formula and then express the required sample size n as a function of alpha level, Standard Deviation, and E.

Answer 1

(A) Required sample size n = ((z*sd/E)^2

where z is critical value

so, when z is increased(significance level), the sample size n will also gets increased due to the direct relationship

(B)  Required sample size n = ((z*sd/E)^2

here sd is the standard deviation

In order to reduce the variance than the earlier speficied value standard deviation, then this will reduce the required sample size because variance and sample size are directly proportional to each.

(C) Required sample size n = ((z*sd/E)^2

here E is the margin of error

In order to reduce the margin of error, we need a larger sample size because sample size and margin of error are inversely proportional to each other.