Estimation of Population Proportion + Comparing Population Means

population proportion

the percentage of people who agree/disagree

p̂

p hat, the sample proportion, a point estimate of the actual proportion p

p̂ point estimation formula

# of yes/no / n where n is the size of the sample

p̂ interval estimation formula

[ p̂ - Z √ (p̂ (1 - p̂)) / n, p̂ + Z √ (p̂ (1 - p̂)) / n ]

( n mist be sufficiently large )

Prove n is sufficiently large for population proportion

(n * p̂ >= 15) and ( n * (1 - p̂) >= 15)

What is the half length of a population proportion at a 95% confidence level?

Z √ (p̂ (1 - p̂)) / n

Steps to evaluate population proportion

• Formula

• Plug In

• Interperet

• Check Assumptions

Find the sample size of a population proportion given a half length of .05, a confidence interval of 99%, assume p = .5. Is the sample size liberal or conservative?

Z² * .5(.5) / .05²

liberal

Find the sample size of a population proportion given a half length of .1, a confidence interval of 90%, and assume no information on p. Is the sample size liberal or conservative?

Z² * .25 / .1²

conservative

Why is .25 a conservative estimate of p̂?

The maximum value of p(1 - p) is .25

Make a 92% confidence interval estimating the distance of µ₁ - µ₂

[ (ā - ē) ± t_{.96, n+m-2} * S_p √(1/n + 1/m) ]

S_p formula

( (n - 1) * S_x² + (m - 1) * S_y² ) / (n + m - 2)

Interval estimation of the distance between of µ₁ - µ₂ assumptions

• Normal Data for BOTH n and m

• Equal Approximate Variance

• Independent Samples