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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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2.2 Expected Value (Mean)

Expected Value (Mean)

  • You already know what an average is. Expected value is similar in this sense (after all, it is known as mean too).
  • Since random variables have a list or range of possible outcomes with probabilities associated, we can determine the mean of the random variable, known as its expected value.
  • The formula for expected value is given by

E(X)=\sum_{x} x \cdot \operatorname{Pr}(X=x)=\sum_{x} x \cdot p(x)

where E(X) represents the expected value of X.

  • E(X) can also be denoted as the Greek letter \mu (mu), which means the mean of X.
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