Question

If X denotes the number of heads in n tosses of a coin, what is the standard deviation of the fraction of heads, which is X/n? Does this standard deviation get larger or smaller as n gets larger?

Answer 1

Explanation:

Suppose p is the probability of getting head;

The standard deviation can be computed as:

`S.D = \sqrt{(p*(1-p))/(n)`

Thus, the standard deviation gets smaller as n gets larger

For example;

Let assigned some values to the parameters

Assume the standard deviation is ??

sample proportion p = 0.5

and the sample size n = 20

`S.D = \sqrt{(0.5*(1-0.5))/(20)`

S.D = 0.1118

If the sample size increases to 40

`S.D = \sqrt{(0.5*(1-0.5))/(40)`

S.D = 0.0790

Therefore, we can see that as the standard deviation gets smaller as n gets larger