Confidence Intervals

What is a Confidence Interval?

A confidence interval is a range of values that likely contains an unknown population parameter. Instead of providing a single point estimate, confidence intervals give us a range with an associated confidence level.

For example, instead of saying “the average height is 170 cm,” we might say “we are 95% confident that the average height is between 168 cm and 172 cm.”

Components of a Confidence Interval

Every confidence interval consists of three key components:

1. Point Estimate

The sample statistic used to estimate the population parameter (e.g., sample mean $\bar{x}$)

2. Margin of Error

The amount we add and subtract from the point estimate to create the interval

3. Confidence Level

The probability that the interval contains the true population parameter (commonly 90%, 95%, or 99%)

Confidence Interval for a Mean

The most common type of confidence interval is for a population mean $\mu$.

When Population Standard Deviation (σ) is Known

Where:

• $\bar{x}$ = sample mean
• $z_{\alpha/2}$ = critical z-value (1.96 for 95% confidence)
• $\sigma$ = population standard deviation
• $n$ = sample size

When Population Standard Deviation (σ) is Unknown

In most real-world situations, we don’t know the population standard deviation. Instead, we use the sample standard deviation $s$ and the t-distribution.

Where:

• $s$ = sample standard deviation
• $t_{\alpha/2, df}$ = critical t-value with degrees of freedom $df = n - 1$

Common Confidence Levels

Confidence Level	z-value	Interpretation
90%	1.645	Less precise, but more certain to capture true parameter
95%	1.96	Standard choice, good balance
99%	2.576	More precise, but wider interval

Factors Affecting Interval Width

The width of a confidence interval depends on three factors:

1. Sample Size (n)

• Larger samples → Narrower intervals
• Standard error decreases as $1/\sqrt{n}$

2. Confidence Level

• Higher confidence → Wider intervals
• More critical z or t value needed

3. Population Variability (σ or s)

• More variability → Wider intervals
• Less certainty about the true parameter

Interpreting Confidence Intervals

✅ Correct Interpretations

1. “We are 95% confident that the true population mean lies between [a] and [b].”
2. “If we repeated this study many times, approximately 95% of the constructed intervals would contain the true mean.”

❌ Incorrect Interpretations

1. “There is a 95% probability that the true mean is in this interval.” (The parameter is fixed, not random!)
2. “95% of the data falls in this interval.” (This describes the data distribution, not the parameter)

Confidence Intervals for Proportions

We can also construct confidence intervals for population proportions $p$.

Where $\hat{p}$ is the sample proportion.

Sample Size Determination

Sometimes we want to determine how large a sample we need to achieve a desired margin of error.

Common Mistakes to Avoid

1. Confusing confidence level with probability: The interval either contains the parameter or it doesn’t; the confidence level refers to the long-run success rate of the method.

2. Using z instead of t: When σ is unknown and sample size is small, always use the t-distribution.

3. Ignoring assumptions: Confidence intervals assume random sampling and approximately normal distributions (or large sample sizes).

4. Misinterpreting width: A wider interval doesn’t mean we’re less certain; it means we’re being more cautious.

Summary

In this lesson, you learned:

• Confidence intervals provide a range of plausible values for population parameters
• The interval is constructed as: point estimate ± margin of error
• Common confidence levels are 90%, 95%, and 99%
• Use z-distribution when σ is known, t-distribution when unknown
• Interval width depends on sample size, confidence level, and variability
• Larger samples produce narrower, more precise intervals

Next Steps

Continue your learning with:

• Hypothesis Testing - Use intervals to make decisions about parameters
• T-Test Calculator - Compute confidence intervals automatically
• Sample Size Determination - Plan your studies effectively