Chapter 8: Estimating with Confidence

AP Statistics | 2024-25

What is a point estimator?

a statistic that provides an estimate of a population parameter

What is a point estimate?

the value of a statistic from a sample

What is a confidence interval?

the estimate of a parameter given as an interval of values

What does confidence level C give?

the overall success rate of the method used to calculate the confidence interval; in C% of all possible samples, the interval computed from the sample data will capture the true parameter value

How do you interpret a C% confidence interval for an unknown parameter?

“We are C% confidence that the interval from ____ to ____ captures the [parameter in context]”

What does margin of error describe?

how far, at most, we expect the estimate to vary from the true population

What is the distance between the point estimate and true parameter value in a C% confidence interval in relation to the margin of error?

it will be less than the margin of error in C% of all samples

How do you interpret a confidence level?

“If we were to select many random samples of the same size from a population and construct a C% confidence interval using each sample, about C% of the intervals would capture the [parameter in context]”

What are the 2 ways to reduce the margin of error of a confidence interval?

decrease the confidence level and increasing sample size n

What must you be willing to accept if you obtain a smaller margin of error from data?

less confidence

What is the general formula for a confidence interval?

statistic (plus or minus) (critical value) x (standard deviation of statistic)

What is a critical value?

a multiplier that makes the interval wide enough to have the stated captures rate

What is the formula for a C% confidence interval if the three conditions are met?

What are the three conditions for constructing a confidence interval for a proportion?

• Random

• Large Counts

• 10%

What is the standard error of the statistic?

the result when the standard deviation of a statistic is estimated from data

How do you interpret the standard error of a sample proportion?

“In repeated SRSs of size [n], the sample proportion of […..] typically varies from the population proportion by about [SE]”

What does standard error describe?

how close the sample proportion (p-hat) will typically be from the true population proportion p in repeated samples of the same size

What does critical value z* measure?

how many standard errors we need to extend above/below to reach out desired confidence level

What are the four steps of constructing confidence intervals?

1. State

2. Plan

3. Do

4. Conclude

What is the State step of constructing confidence intervals?

state the parameter you want to estimate and the confidence level

“C% confidence for p = the true proportion of […]”

What is the Plan step of constructing confidence intervals?

identify the appropriate inference method and check conditions

“One-sample z interval for p

• Random: […]

• 10%: […]

• Large Counts: […]”

What is the Do step of constructing confidence intervals?

if the conditions are met, perform calculations

(use confidence interval formula)

What is the Conclude step of constructing confidence intervals?

interpret your interval in the context of the problem

“We are C% confident that the interval from ____ to ____ captures p = the true proportion of […]”

What is the formula for margin of error?

How do you choose a value for p̂ when solving for the sample size?

use a guess for p̂ based on a pilot study, or use p̂ = 0.5

What is the Random condition for constructing a confidence interval about a difference in proportions?

the data come from two independent random samples or from two groups in a randomized experiment

What is the 10% condition for constructing a confidence interval about a difference in proportions?

when sampling without replacement, n₁ < 0.10N₁ and n₂ < 0.10N₂

What is the Large Counts condition for constructing a confidence interval about a difference in proportions?

the counts of “successes” and “failures” in each group are all at least 10

What is the formula for a C% confidence interval for p₁-p₂?