U2: Confidence Intervals

Confidence Interval Definition

● A confidence interval is a range of values obtained from a sample that is likely to contain the parameter

● The basic formula for a confidence interval is: Statistic ± (Critical Value)(Standard Deviation of Statistic)

● The margin of error is defined by the Critical Value (Standard Deviation of Statistic)

● The critical values for each confidence level can be found on Table B of the AP Statistics formula sheets (p. 5 of the 2020 version)

Three Steps

1. Conditions/Assumptions

2. Formula

3. Interpretation

Step 1: Conditions/Assumptions

● Random Sample

○ “The stem of the problem states that [sample] was chosen at random”

○ “The stem of the problem states that [participants] were randomly assigned to the groups”

● Approximately Normal Distribution

○ “The stem of the problem states that the distribution is approximately normal”

○ “Since n = _ ≥ 30, by the Central Limit Theorem, we can assume the distribution is approximately normal”

● For two samples, both n1 and n2 must be ≥ 30

○ “Since np = _ ≥ 10 and nq = _ ≥ 10, we can assume the distribution is approximately normal”

● For two samples, n1p̂1, n1q̂1, n2p̂2, and n2q̂2must all be ≥ 10

○ “Since the [graphical display] shows no outliers or strong skewness, we can assume the distribution is approximately normal”

● You must provide a graphical display (preferably a box plot) if normality cannot be assumed by the other three ways

Step 2: Formula

● List the formula, your substitution, degrees of freedom (if using t*) and your unrounded answer

● One-Sample Means and Two-Sample Means

● Proportions

Step 3: Interpretation

● “Based on these samples, we are _% confident that the true [context of problem] is between [lower value] and [upper value]”

● “Since zero [is/is not] in the _% confidence interval [interval], we [do not/do] have sufficient evidence to suggest there is a difference at the α = [1 - %] level”

● Always put the % as a decimal (e.g. 95% = .95)

● Use this second statement for two-sample confidence intervals only

Z vs t Distribution

● Means

○ Use a Z distribution when you have σ

○ Use a t distribution when you do not have σ (i.e. you have S)

● Proportions

○ ALWAYS use a Z distribution

Degrees of Freedom

● Only applies to t-distributions

○ The t-distribution varies with degrees of freedom

● df = n - 1

● For a Z-distribution, df = ∞

Finding Critical Values

● AP Statistics formula sheet table B

● TI-84 Plus

○ 2nd → Vars

■ 3 = Z-distribution

■ 4 = t-distribution

■ Area = 1/2 * (1 - %)

● Always put the % as a decimal (e.g. 95% = .95)

Calculator (TI-84 Plus)

● Stat → Test

○ Means

■ 7 = 1-sample Z-Interval

■ 8 = 1-sample t-Interval

■ 9 = 2-sample Z-Interval

■ 10 = 2-sample t-Interval

○ Proportions

■ A = 1-Sample

■ B = 2-Sample