Question

Consider random samples of size 40 from a population with proportion 0.15.

Answer 1

Answer:

a) The mean is 0.15 and the standard error is 0.056.

b)  1. Yes

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For proportions p, in samples of size n, the mean is $\mu = p$ and the standard error is $s = \sqrt{\frac{p(1-p)}{n}}$. The Central Limit Theorem applies is np > 5 and np(1-p)>5.

In this question:

$n = 40, p = 0.15$

So

(a) Find the mean and the standard error of the distribution of sample proportions.

$\mu = 0.15, s = \sqrt{\frac{0.15*0.85}{40}} = 0.056$

So the mean is 0.15 and the standard error is 0.056.

(b) Is the sample size large enough for the Central Limit Theorem to apply?

np = 40*0.15 = 6 > 5

np(1-p) = 40*0.15*0.85 = 5.1>5

So yes