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Sample Proportion

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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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4.1 Concepts of Population and Random Samples

Population and Samples

  • The set of all eligible members of a group which we intend to study is called a population.

For example, if we are interested in the IQ scores of a secondary school’s students, then all students in this school is considered as a population; if we want to know the average IQ scores of all students in Australia instead, then this becomes the population.

  • There are difficulties in dealing with a population:

i) The population might be too large – e.g. there are too many students in Australia.

ii) The population may be hard to access – e.g. the blue whales in the Pacific Ocean.

iii) The data collection process might be destructive – testing all the batteries’ durability would mean to use up all the batteries to test, which is a waste.

  • To solve these problems, we use a subset of the population – called a sample – instead, and hope that it can represents the true population.
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