In the example, we saw the 95% confidence interval is (0.499,\ 0.901), which is quite big, and therefore is not very useful. There are measures to describe situations like these too.

The distance between the sample estimate and the endpoints of the confidence interval is called the margin of error (M). For a 95% confidence interval,

All this while, we assume that we know some of the population proportion p. However, in reality we do not know the population proportion, which is why we conduct experiments to try to find it.

However, conducting experiments is not feasible (refer to 4.1 Concepts of Population and Random Samples). This means that p is unknown, since we cannot conduct experiments to find it out.

Therefore, we use random sample(s) instead, which gives the sample proportion(s) \hat{p}. Value(s) of \hat{p} gives some indication of the value of p, and can be used when we have no other information.

The value of \hat{p} (from a single experiment) can be used to estimate p. Since this is a single-valued estimate, it is called a point estimate of p.

The observation in 4.3 Approximating Sample Proportion Distribution is a well-known effect. Statisticians has investigated such phenomenon and came up with a theorem for it, known as the Central Limit Theorem.

The Central Limit Theorem (CLT) states that, in many situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.

To explain in math, suppose that X=\frac{1}{n}\left(X_{1}+X_{2}+X_{3}+\ldots+X_{n}\right), then we can assume X \sim \color{red}N\color{black}\left(\mu, \frac{\sigma^{2}}{n}\right). The most important part here is N. as for the mean of variance of X, it can be calculated without the usage of any approximation/assumption (shown below).

Previously in 4.2 Distribution of Sample Proportion, we have seen that we can assume the distribution of a sample proportion. With that said, since each random sample we use will be different for each experiment, it is almost certain that the sample proportion will be different each time.

For instance, suppose there are 55% male, 45% female in Australia. However, we do not know this, and we want to test for the proportion of males to females in Australia. A random sample of 100 people is chosen and apparently 58 of them is male, hence \hat{p}=0.58. Repeat the process again, this time we have 44 males, therefore \hat{p}=0.44. Repeat the experiment 200 times, and we would have something like this: