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Introduction

Confidence intervals provide a range of plausible values for a population parameter. Rather than giving a single point estimate, they acknowledge the uncertainty inherent in sampling.

What is a Confidence Interval?

A confidence interval is calculated from sample data and has a specified probability (confidence level) of containing the true population parameter.

CI Structure

Point Estimate ± Margin of Error

Lower Bound

Estimate - Margin of Error

Upper Bound

Estimate + Margin of Error

Correct Interpretation

Correct

"If we repeated this sampling process many times, 95% of the intervals constructed would contain the true population parameter."

Incorrect

"There is a 95% probability that the true parameter is in this interval." (The parameter is fixed; it either is or isn't in the interval.)

Confidence Interval Formula

For a Mean (σ known)

CI = x̄ ± z × (σ / √n)

For a Mean (σ unknown)

CI = x̄ ± t × (s / √n)

Use the t-distribution when population standard deviation is unknown (most real situations).

Factors Affecting Width

Sample Size (n)

Larger sample = Narrower interval (more precise)

Confidence Level

Higher confidence = Wider interval (more certain but less precise)

Standard Deviation

More variability = Wider interval

Common Confidence Levels

Confidence Level	Z-Score	Common Use
90%	1.645	Exploratory research
95%	1.96	Standard in most fields
99%	2.576	High-stakes decisions

Common Misconceptions

The parameter has a 95% chance of being in the interval

Wrong! The parameter is fixed. The interval either contains it or doesn't.

A wider interval is always better

Wrong! Wider intervals are less informative. We want narrow AND reliable.

Summary

Key Takeaways

1.CI = Point Estimate ± Margin of Error
2.The confidence level refers to the method, not a specific interval.
3.Larger samples and lower confidence levels give narrower intervals.
4.95% confidence is standard in most fields.