Module 6 Study Guide – Comprehensive Reference

How to Use This Study Guide

Print this guide for offline studying
Review before taking the quiz or post-assessment
Use as a reference while working on practice problems
Focus on formulas and conditions - these are tested frequently

1. Introduction to Confidence Intervals

Point Estimate: A single number estimating a parameter (x̄ for μ, p̂ for p)

Interval Estimate (CI): A range of plausible values for the parameter

CI = Point Estimate ± Margin of Error

CI = Point Estimate ± (Critical Value) × (Standard Error)

Confidence Level: The percentage of intervals (constructed the same way) that would capture the true parameter

90% CI: 90% of intervals capture the parameter
95% CI: 95% of intervals capture the parameter (most common)
99% CI: 99% of intervals capture the parameter

Correct Interpretation: "We are 95% confident that the true parameter is between ___ and ___."
Incorrect: "There's a 95% chance the parameter is between ___ and ___."
Why: The parameter is fixed (not random). The 95% refers to confidence in the method.

2. Confidence Intervals for Means

CI for Population Mean (μ)

x̄ ± t* × (s / √n)

Where: x̄ = sample mean, t* = critical t-value, s = sample SD, n = sample size

The t-Distribution

Use when σ is unknown (most common)
Similar to normal but with heavier tails
Shape depends on degrees of freedom: df = n - 1
As df increases, t approaches z

Common t* Values (for reference)

df	90% CI	95% CI	99% CI
10	1.812	2.228	3.169
20	1.725	2.086	2.845
30	1.697	2.042	2.750
∞ (z)	1.645	1.960	2.576

When to Use z vs. t

Use t: σ unknown (almost always for means)
Use z: σ known (rare) OR working with proportions

3. Confidence Intervals for Proportions

CI for Population Proportion (p)

p̂ ± z* × √(p̂(1-p̂)/n)

Where: p̂ = sample proportion, z* = critical z-value, n = sample size

Success-Failure Condition

Before using normal approximation, check:

np̂ ≥ 10 AND n(1-p̂) ≥ 10

This ensures enough successes and failures for normal approximation to be valid.

Critical z* Values

Confidence Level	z*
90%	1.645
95%	1.960
99%	2.576

4. Determining Sample Size

Sample Size for Means

n = (z*σ / E)²

Where: E = desired margin of error, σ = population SD (estimate), z* = critical value

Sample Size for Proportions

n = p̂(1-p̂) × (z*/E)²

Conservative (no prior estimate): Use p̂ = 0.5 → n = 0.25 × (z*/E)²

Important Rules:

Always round UP to next whole number
To cut margin of error in half → multiply n by 4
Higher confidence → larger n needed
Use p̂ = 0.5 when no prior estimate (gives maximum n)

5. Quick Formula Reference

Type	Formula	When to Use
CI for Mean	x̄ ± t*(s/√n)	σ unknown, estimating μ
CI for Proportion	p̂ ± z*√(p̂(1-p̂)/n)	Estimating p, check conditions
Sample Size (Mean)	n = (z*σ/E)²	Plan study for means
Sample Size (Prop)	n = p̂(1-p̂)(z*/E)²	Plan study for proportions
Standard Error (Mean)	SE = s/√n	Variability of x̄
Standard Error (Prop)	SE = √(p̂(1-p̂)/n)	Variability of p̂
Degrees of Freedom	df = n - 1	For t-distribution

6. Step-by-Step Procedures

Constructing CI for a Mean:

Check conditions (random sample, n≥30 or population normal)
Calculate x̄ and s from data
Find df = n - 1
Look up t* for desired confidence level and df
Calculate SE = s/√n
Calculate E = t* × SE
Construct interval: x̄ ± E
Interpret in context

Constructing CI for a Proportion:

Check conditions (random sample, np̂≥10 and n(1-p̂)≥10)
Calculate p̂ = x/n
Look up z* for desired confidence level
Calculate SE = √(p̂(1-p̂)/n)
Calculate E = z* × SE
Construct interval: p̂ ± E
Interpret in context