Quartz 4

Sampling Distribution of a Proportion

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Definition

It is a specific type of Sampling Distribution

Let 𝑋 be a binomial distribution, then the proportion of successes is:

̂𝑝=𝑋𝑛

Based on this we know that 𝑝∈[0,1],

𝜇̂𝑝=𝐸(𝑋𝑛)=1𝑛𝐸(𝑋)=𝑛̂𝑝𝑛=̂𝑝𝜎2̂𝑝=Var(𝑋𝑛)=1𝑛2Var(𝑋)=𝑛̂𝑝̂𝑞𝑛2=̂𝑝̂𝑞𝑛

Binomial Form

𝑃(̂𝑝≤𝑝𝑜)=𝑃(𝑋𝑛≤𝑝𝑜)=𝑃(𝑋≤𝑛𝑝𝑜)=∑⌊𝑛𝑝𝑜⌋𝑥=0(𝑛𝑥)̂𝑝𝑥̂𝑞1−𝑥

Normal Approximation

Independence

For a sufficiently large 𝑛, we can apply the Central Limit Theorem

𝑝=𝑁(̂𝑝,̂𝑝(1−̂𝑝)𝑛)→𝑍=𝑝−̂𝑝√̂𝑝(1−̂𝑝)/𝑛

This applies when:

  • 𝑛≥30
  • 𝑋 has a Normal Distribution

Solution Methods

Normal Approximation

𝑝=𝑁(̂𝑝,̂𝑝(1−̂𝑝)𝑛)→𝑍=𝑝−̂𝑝√̂𝑝(1−̂𝑝)/𝑛

In R:

sigma_p <- sqrt( pi*(1-pi)/n )
pnorm( p_valor, pi, sigma_p )

Graph View

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