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4.6 Precision and Margin of Error

Precision and Margin of Error

  • 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,

M=1.96 \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

  • Therefore, from above, we can see that as n increases, M decreases. This also means that as the sample size increases, the precision of the estimate increase.
  • Intuitively, if our random sample is larger, then the random sample should more accurately reflect the population. This also prevents the unintentional bias of selecting samples from a certain subgroup.
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4.5 Point Estimates and Confidence Interval

Point Estimates and 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.
  • 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.
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4.4 The Central Limit Theorem

The Central Limit Theorem (Not Required)

  • 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).
  • We can find the mean and variance of X as below:
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4.3 Approximating Sample Proportion Distribution

Approximating Sample Proportion Distribution

  • 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:
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