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Recall that:

• If p is the true proportion of "successes" in a population of N members, then the number of successes X in a sample of size n from the population will follow a Binomial distribution, as long as n is small relative to the size of the population N.
• If in addition, n itself is reasonably large, then both X as well as  are approximately Normal.

We test

H₀: p=p₀ vs. H_a: p ¹ p₀ OR p>p₀ OR p<p₀

CASE 1: Large sample Tests (valid as long as both np₀ and n(1-p₀) ³ 10)

Test Statistic is z =

H_a: p>p₀Þ Reject if z³z_a
H_a: p<p₀ Þ Reject if z£-z_a
H_a: p¹ p₀ Þ Reject if z³z_a/2 OR z£-z_a/2, i.e. |z|³z_a/2

CASE 2: Small-Sample Tests  (i.e., either one or both of np₀ and n(1-p₀) < 5)

Can’t use Normal approximation here - tests are based on the exact Binomial probabilities.

E.g. Suppose H₀: p=p₀and H_a: p<p₀. A sample of size n yields X successes and we reject H₀ if X£c.
Therefore a =P{X£c when X~Binomial(n,p₀)} = B(c-1; n,p₀)
Solve for c.

In general, the Test Statistic is X
 

H_a: p>p₀ - Reject if X³c, where c is the smallest integer for which 1-B(c-1;n,p₀) £a

H_a: p<p₀ - Reject if X£c, where c is the largest integer for which B(c;n,p₀) £a

H_a: p=p₀ - Reject if X³c₁ or X£c₂, where c₁ and c₂ are integers such that c₁>c₂ and 1-B(c₁-1;n,p₀) + B(c₂;n,p₀) £a

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