Question

For a certain study, the population proportion is known to be 0.8. If we draw a sample of 1000 units from that population what would be the 1 standard deviation?

A> 4%
B> 0.4%
C> 1.264%
D> None of the above

Answer 1

Let $X$ be a binomial random variable based on $n$ trials with success probability $p.$  

If the binomial probability histogram is not too skewed, $X$ has approximately a normal distribution with $\mu=np$ and $\sigma=\sqrt{p(1-p)/n}.$

In practice, the approximation is adequate provided that both $np\geq10$ and $n(1-p)\geq10,$ since there is then enough symmetry in the underlying binomial distribution.

Check normal approximation to binomial

$n=1000, p=0.8$

$np=1000(0.8)=800\geq10,$

$n(1-p)=1000(1-0.8)=200\geq10$

Hence we can use normal approximation to binomial. Then

$$\sigma=\sqrt{p(1-p)/n}=\sqrt{0.8(1-0.8)/1000}\approx0.01265$$

C> 1.264%