Lesson 3.8: Applying the Recipe: CIs for Mean and Variance

We now apply the general 'pivotal method' to construct the two most common confidence intervals. We'll see why the t-distribution is the correct tool for the job when estimating a mean, and why the Chi-squared distribution is necessary when estimating variance.

Part 1: Confidence Interval for a Population Mean (μ)

The Real-World Problem: σ is Unknown

In our general derivation, we used the Z-statistic as our pivot. But this pivot, $(\bar{X} - \mu) / (\sigma/\sqrt{n})$ , has a fatal flaw: we almost never know the true population standard deviation, $\sigma$ .

The only practical solution is to replace $\sigma$ with our sample estimate, $s$ . As we learned in Module 2, this substitution changes the distribution of our pivot.

Choosing the Right Tool: The t-distribution

When we use the sample standard deviation $s$ in our pivot, we introduce extra uncertainty into our calculation. The t-distribution, with its "fatter tails," is the distribution designed to account for precisely this extra uncertainty.

The correct pivot for the mean when $\sigma$ is unknown is the **t-statistic**:

t = \frac{\bar{X} - \mu}{s / \sqrt{n}} \sim t_{n-1}

Deriving the t-Interval

We follow the exact same algebraic inversion from the previous lesson, but we use the t-pivot and its critical values, $\pm t_{\alpha/2, n-1}$ , which are slightly wider than the Z-values.

Starting with $P(-t_{\alpha/2} \le \frac{\bar{X} - \mu}{s / \sqrt{n}} \le t_{\alpha/2}) = 1-\alpha$ and isolating $\mu$ leads directly to the final formula.

The t-Confidence Interval for the Population Mean μ

This is the most widely used confidence interval in all of science and industry.

\text{C.I.} = \left[ \bar{X} - t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}, \quad \bar{X} + t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}} \right]

Part 2: Confidence Interval for a Population Variance (σ²)

To build an interval for variance, we need a different pivot—one that relates our sample variance ( $s^2$ ) to the true population variance ( $\sigma^2$ ).

Choosing the Right Tool: The Chi-Squared (χ²) Distribution

As we learned in Module 2, the distribution that governs the behavior of sample variance (under normality) is the Chi-Squared distribution.

The correct pivot for the variance is:

\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}

A key feature of the $\chi^2$ distribution is that it is **not symmetric**. This makes the derivation a bit trickier.

Derivation: The Chi-Squared Interval for σ²

Step 1: Define the probability region. Because the distribution is skewed, we need two different critical values to chop off $\alpha/2$ from each tail.

Lower Critical Value: $\chi^2_{1-\alpha/2, n-1}$ (the value with $\alpha/2$ area to its left).
Upper Critical Value: $\chi^2_{\alpha/2, n-1}$ (the value with $\alpha/2$ area to its right).

P\left( \chi^2_{1-\alpha/2, n-1} \le \frac{(n-1)s^2}{\sigma^2} \le \chi^2_{\alpha/2, n-1} \right) = 1-\alpha

Step 2: Isolate $\sigma^2$ . Since $\sigma^2$ is in the denominator, we must invert all parts of the inequality, which **reverses the direction** of the inequalities.

P\left( \frac{1}{\chi^2_{1-\alpha/2, n-1}} \ge \frac{\sigma^2}{(n-1)s^2} \ge \frac{1}{\chi^2_{\alpha/2, n-1}} \right) = 1-\alpha

Step 3: Solve for $\sigma^2$ and reorder. Multiply by $(n-1)s^2$ and then flip the expression to the standard format (lower bound on the left).

The Confidence Interval for the Population Variance σ²

\text{C.I.} = \left[ \frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}}, \quad \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}} \right]

Important: Notice how the *upper* Chi-squared critical value appears in the *lower* bound of the interval, and the "lower" $\chi^2$ value forms the *upper* bound. This is a direct result of the inversion in Step 2.

The Payoff: Quantifying Uncertainty in Practice

CIs for Regression Coefficients: Every OLS regression output shows a 95% CI for each coefficient $\hat{\beta}_j$ . That interval is calculated using this lesson's t-interval formula: $\hat{\beta}_j \pm t_{crit} \cdot \text{se}(\hat{\beta}_j)$ . It tells you the range of plausible values for the true effect of that variable.
CIs for Financial Volatility ( $\sigma$ ): A risk manager needs to know the plausible range for an asset's true volatility. They use the Chi-squared method to find the CI for the variance ( $\sigma^2$ ), and then simply **take the square root of both ends of the interval** to get the CI for volatility ( $\sigma$ ). This gives a "best case" and "worst case" for risk.

What's Next? From Ranges to Decisions

Confidence intervals are a powerful tool for quantifying our uncertainty about an estimate. They give us a range of plausible values for the truth.

But often, we need to make a firm, binary decision. Is this new drug effective, yes or no? Is this factor's beta equal to zero, yes or no? This requires a more formal decision-making framework.

In the next lesson, we will introduce the language and logic of **Hypothesis Testing**.

Up Next: Let's Make a Decision: Hypothesis Testing