Confidence Intervals

Why This Matters

Confidence intervals are the bridge between sample data and population truth—and in engineering probability, you're being tested on your ability to construct, interpret, and apply them correctly. Every time you estimate a parameter from data (a mean strength, a failure rate, a proportion of defective parts), you need to quantify how uncertain that estimate is. CIs do exactly that, connecting directly to concepts like sampling distributions, the Central Limit Theorem, and hypothesis testing. Expect exam questions that require you to choose the right formula, justify your distributional assumptions, and interpret results in engineering contexts.

Don't just memorize the formulas—know when each interval type applies and why the mechanics work. Can you explain why larger samples shrink your interval? Why you switch from Z to t? Why a 99% CI is wider than a 95% CI? These conceptual questions appear constantly on FRQs, and the formulas alone won't save you. Master the underlying logic: confidence level trade-offs, sample size effects, distributional assumptions, and the duality between CIs and hypothesis tests.

---

Foundational Concepts

Before diving into specific interval formulas, you need rock-solid understanding of what a confidence interval actually means and the components that control its width. The confidence level determines how often the procedure captures the true parameter; the margin of error determines how precise your estimate is.

Definition and Interpretation

• A confidence interval is a range of plausible values for an unknown population parameter, constructed from sample data—it is not a probability statement about where the parameter lies
• The confidence level (e.g., 95%) means that if you repeated the sampling procedure infinitely, 95% of constructed intervals would contain the true parameter
• Width reflects uncertainty—wider intervals indicate less precision, narrower intervals indicate more confidence in the point estimate

Confidence Level and Significance Level

• Confidence level $1 - \alpha$ and significance level $\alpha$ are complementary—a 95% CI corresponds to $\alpha = 0.05$
• Higher confidence levels produce wider intervals because you're demanding more certainty that you've captured the true value
• The trade-off is precision vs. confidence—you can't have both a narrow interval and high confidence without increasing sample size

Margin of Error

• Margin of error (ME) equals half the interval width—it's the "±" portion of your estimate and quantifies maximum expected sampling error
• ME depends on three factors: confidence level (higher → larger ME), sample size (larger → smaller ME), and data variability (more spread → larger ME)
• Engineering applications often specify a required ME first, then solve for the necessary sample size to achieve it

---

Choosing the Right Distribution

The critical decision in constructing any CI is selecting the appropriate sampling distribution. Your choice depends on what you know about the population and how large your sample is.

Z-Score Intervals

• Use the Z-distribution when $\sigma$ (population standard deviation) is known and either the population is normal or $n > 30$ (CLT applies)
• Z critical values are fixed for each confidence level: $Z_{0.95} = 1.96$, $Z_{0.99} = 2.576$, $Z_{0.90} = 1.645$
• Common in quality control where historical process data provides a reliable $\sigma$ value

T-Score Intervals

• Use the t-distribution when $\sigma$ is unknown and you're estimating it with the sample standard deviation $s$
• Degrees of freedom ($df = n - 1$) control the shape—smaller samples have heavier tails, producing wider intervals to account for extra uncertainty
• As $n \to \infty$, the t-distribution approaches the Z-distribution—this is why the "$n > 30$" rule exists, though using t is always valid

Chi-Squared Distribution for Variance

• Variance intervals use the $\chi^2$ distribution because sample variance follows a chi-squared distribution when data is normal
• The interval is asymmetric: $\left(\frac{(n-1)s^2}{\chi^2_{\alpha/2}}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}\right)$—note the "flipped" critical values
• Normality assumption is critical here—variance intervals are more sensitive to non-normality than mean intervals

---

Intervals for Means

These are your workhorse formulas for estimating central tendency. The structure is always: point estimate ± (critical value) × (standard error).

CI for Mean (Known $\sigma$)

• Formula: $\bar{x} \pm Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$ where $\frac{\sigma}{\sqrt{n}}$ is the standard error of the mean
• Requires normality or large sample ($n > 30$) for the sampling distribution of $\bar{x}$ to be approximately normal
• Rare in practice since $\sigma$ is seldom truly known, but common on exams to test your understanding of the Z framework

CI for Mean (Unknown $\sigma$)

• Formula: $\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}$ where $s$ replaces $\sigma$ and introduces additional sampling variability
• The t critical value depends on degrees of freedom—always check your t-table or calculator with $df = n - 1$
• This is the default real-world scenario—when in doubt on an exam and $\sigma$ isn't explicitly given, use t

CI for Difference Between Two Means

• For independent samples with known variances: $(\bar{x}_1 - \bar{x}_2) \pm Z_{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$
• For unknown variances, use pooled or Welch's t-interval: $(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$
• If the interval contains zero, you cannot conclude the means differ at that confidence level—direct connection to two-sample hypothesis tests

---

Intervals for Proportions

Proportion intervals apply when your data is categorical (success/failure, defective/non-defective). The normal approximation requires checking sample size conditions.

CI for Single Proportion

• Formula: $\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ where $\hat{p} = x/n$ is the sample proportion
• Check conditions: $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$ to ensure the normal approximation is valid
• Used extensively in reliability engineering to estimate defect rates, failure probabilities, and compliance percentages

CI for Difference Between Two Proportions

• Formula: $(\hat{p}_1 - \hat{p}_2) \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$
• Both samples must independently satisfy the normal approximation conditions ($np \geq 10$, $n(1-p) \geq 10$)
• Interpretation: if zero is in the interval, you cannot conclude the proportions differ—critical for A/B testing and comparative studies

---

Sample Size and Interval Design

Engineers often work backwards: given a desired precision, how much data do you need? Sample size determination is about controlling the margin of error before collecting data.

Sample Size Effects

• Larger $n$ shrinks the interval because standard error contains $\sqrt{n}$ in the denominator—doubling precision requires quadrupling sample size
• The relationship is square-root: to cut ME in half, you need $4\times$ the sample size; to cut it by $1/3$, you need $9\times$
• Resource constraints matter—engineers must balance statistical precision against time, cost, and feasibility

One-Sided vs. Two-Sided Intervals

• Two-sided intervals (most common) bound the parameter above and below: "we're 95% confident $\mu$ is between A and B"
• One-sided intervals provide only an upper or lower bound: "we're 95% confident $\mu$ is at least A" or "at most B"
• Use one-sided when the research question is directional—e.g., "does the new process reduce defects?" only cares about the lower bound

---

Assumptions and Connections

Valid inference requires meeting assumptions, and CIs connect directly to hypothesis testing. Understanding these relationships separates strong exam performance from formula memorization.

Assumptions for Valid CIs

• Random sampling and independence are non-negotiable—biased samples produce meaningless intervals regardless of formula correctness
• For means: population normality or $n > 30$ (CLT); for proportions: $np \geq 10$ and $n(1-p) \geq 10$
• Variance intervals are most sensitive—the chi-squared method requires strict normality; consider bootstrap methods for non-normal data

CI and Hypothesis Test Duality

• A two-sided $(1-\alpha)$ CI corresponds to a two-tailed test at significance level $\alpha$—if the null value falls outside the CI, reject $H_0$
• This provides a visual decision rule: construct the CI, check if the hypothesized value is inside or outside
• CIs give more information than tests—they show the range of plausible values, not just "reject" or "fail to reject"

Interpreting and Reporting CIs

• Never say "95% probability the parameter is in this interval"—the parameter is fixed; the interval is random
• Correct interpretation: "We are 95% confident that [parameter] lies between [lower] and [upper]"
• Report context: state the confidence level, sample size, assumptions checked, and practical implications for engineering decisions

---

Quick Reference Table

---

Self-Check Questions

1. You're estimating the mean tensile strength of a new alloy. You have $n = 25$ specimens and calculate $s = 12$ MPa. Should you use a Z or t interval, and why does this choice affect your interval width?

2. Compare the CI for a single proportion to the CI for the difference between two proportions. What additional complexity arises in the two-sample case, and what conditions must both samples satisfy?

3. An engineer wants to estimate a defect rate with a margin of error of ±2% at 95% confidence. If a pilot study suggests $\hat{p} \approx 0.10$, approximately what sample size is needed? How would this change if $\hat{p} \approx 0.50$?

4. You construct a 95% CI for $\mu_1 - \mu_2$ and obtain $(-3.2, 1.8)$. What can you conclude about a hypothesis test of $H_0: \mu_1 = \mu_2$ at $\alpha = 0.05$? What if the interval were $(-3.2, -0.4)$?

5. Explain why a 99% confidence interval is wider than a 95% confidence interval for the same data. If you needed to maintain the same width while increasing confidence from 95% to 99%, what would you have to change?