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Sample Proportion Distribution

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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4.2 Distribution of Sample Proportion

Sampling Distribution of a Small Proportion

  • Recall that

\text { Sample proportion, }\ \hat{p}=\frac{\text { number in population with attribute }}{\text { population size }}

and can be treated as a random variable.

  • In a small enough population, we are able to list out all the possible samples, the probability of getting each sample, and therefore the sample proportion.
  • Therefore, we are able to construct the probability distribution of \hat{p}.
  • The distribution of a statistic which is calculated from a sample (such as the sample proportion) is called a sampling distribution.
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