Confidence Intervals (Part IV)

Adjusted Sample Size

• In the Agresti-Coull Method, the sample size is slightly increased to improve accuracy for small samples

• Adds two imaginary successes and two imaginary failures,

• stabilizing the estimate of the proportion and improving the confidence interval’s coverage.

ñ

Symbol that represents the adjusted sample size

ñ = n + 4

Formula for Adjusted Sample Size

Adjusted Sample Proportion

• The modified estimate of the population proportion used in the Agresti-Coull method

• Where X is the number of observed successes

• This adjustment prevents the interval from being too narrow and keeps it more accurate for small and moderate samples.

p̃

Symbol that represents Adjusted Sample Proportion

p̃ = (X + 2)/ñ

Formula for Adjusted Sample Proportion

p̃ ~ N(p, (p(1-p))/n))

For large sample, the sampling proportion approximately follows a normal distribution

Assuming both np ≥ 10 and n(1 - p) ≥ 10

The two assumptions that shows how Sampling Distribution of Sample Proportions follows from the Central Limit Theorem

Agresti-Coull Correction

A small sample correction that improves Confidence Interval accuracy by adjusting both the number of trials and successes

p̃ ± (1.96)((p̃(1 - p̃))/ñ)

Formula for 95% Confidence interval for p

Because it compensates for the bias in the traditional Confidence Interval when replacing p with p̃. Adding 2 successes and 2 failures stabilizes the estimate, giving better coverage even for moderate or small n.

Why does the Agresti-Coull works better?

One-Sided Confidence Intervals for p

Used when only an upper or lower confidence bound is required. For level 100(1 - α)%

p̃ - (z_α)(((p̃(1 - p̃))/ñ)^1/2

Lower bound confidence Intervals formula for p

p̃ + (z_α)(((p̃(1 - p̃))/ñ)^1/2

Upper bound confidence Intervals formula for p

W = (z_(α/2))(((p̃(1 - p̃))/(n + 4))^1/2

Formula for the half width W

p̃ = 0.5

What should be the value of p̃ in the half-width if it is not provided, in order to ensure a conservative (larger) sample size,

To yield the widest possible interval

And to guarantee that the actual interval will be no wider than desired.