Confidence Intervals

Student's t-test originated when a Guinness brewer (who used the alias “Joe Student”) wanted to compare one beer batch to another without associating his name with the test.
The test formalized into the concept of confidence intervals, which quantify how likely the true value μ is to lie near the observed sample statistic.

Confidence interval answers: Is the true value likely to be within a certain distance from the measured value?
Everyday analogy: If you’re told you’re about 5'8" tall, that statement is made with some degree of confidence, not absolute certainty.
Infinite trial thought experiment: If you kept measuring forever, you’d refine your estimate of μ, but 100% confidence is unattainable in practice.
Common practice: 95% confidence interval is a standard rule of thumb in many scientific contexts.
100% confidence is impossible in finite samples; with more data you can approach it, but never reach it.

μ (mu): the true population value (the parameter of interest).
x̄ (x-bar): the sample mean, i.e., the average of the measured values.
s: the sample standard deviation.
n: the number of trials/measurements in the sample.
t: the value from the t-distribution table corresponding to the desired confidence level and degrees of freedom.
df (degrees of freedom) = n − 1 for the one-sample t-interval.
Confidence interval formula (for μ):
ar{x} \,\pm\; t_{\mathrm{df},\,1-\alpha/2}\; \frac{s}{\sqrt{n}}
The t-value you look up depends on df = n − 1 and the chosen confidence level (e.g., 90%, 95%, 99%).
The standard error is the term \frac{s}{\sqrt{n}} which shrinks with more data or with smaller variability.

The t-table provides critical values t_{df, 1−α/2} for different dfs and confidence levels.
Example: For n = 5 measurements, df = 4.
The t-value for a 90% CI with df = 4 is approximately t_{4,0.95} = 2.132 ).
For a 95% CI with df = 4, t_{4,0.975} ≈ 2.776.
For a 99% CI with df = 4, t_{4,0.995} ≈ 4.601.

Given: ar{x} = 0.674,\ s = 0.012,\ n = 5,\ df = n-1 = 4.
Standard error: SE = \frac{s}{\sqrt{n}} = \frac{0.012}{\sqrt{5}} \approx 0.012 / 2.236 \approx 0.005367.
90% confidence interval:
- t_{df,0.95} = 2.132
- Margin of error: ME = t \times SE = 2.132 \times 0.005367 \approx 0.0114.
- CI: \mu \in [\bar{x} - ME, \bar{x} + ME] = [0.674 - 0.0114, 0.674 + 0.0114] \approx [0.6626, 0.6854].
The notes in the transcript report the same approach: 90% CI gives a margin around 0.011 (i.e., ±0.011).
95% confidence interval (for df = 4):
- t_{4,0.975} ≈ 2.776
- ME ≈ 2.776 × 0.005367 ≈ 0.0149
- CI ≈ [0.674 − 0.0149, 0.674 + 0.0149] ≈ [0.6591, 0.6890].
99% confidence interval (for df = 4):
- t_{4,0.995} ≈ 4.601
- ME ≈ 4.601 × 0.005367 ≈ 0.0247
- CI ≈ [0.674 − 0.0247, 0.674 + 0.0247] ≈ [0.6493, 0.6987].

Higher confidence level → wider interval (larger margin of error).
Lower confidence level → narrower interval (smaller margin of error).
If you want 50% confidence, the interval would be smaller than for 90% CI; at 100% confidence, the interval would theoretically extend from −∞ to +∞.
The general practice in many applied fields is to use 95% as the “gold standard.”

Center of the interval is the sample mean, 0.674 in this example.
The interval expands/contracts symmetrically around 0.674 as the confidence level changes:
- 50% around 0.674 would be relatively narrow.
- 95% around 0.674 would be wider than 90% but not as wide as 99%.
- 99% would be even wider, capturing more potential μ values.
A 100% confidence interval would require extending to infinity on one or both sides, which is not possible with finite data.

They quantify uncertainty and make it possible to communicate how close the sample statistic is to the true value with a stated level of confidence.
The phrase “the gold standard is 95%” reflects a balance between precision and reliability.
In real-world settings, we can never claim absolute certainty about μ from a finite sample; CI communicates the degree of certainty.

Using confidence intervals to compare a new experiment to a known value:
- If the known value lies within the CI, the result is not contradicted at that confidence level.
Comparing two datasets:
- If two independent samples each have CIs, overlap between CIs can give a rough sense of agreement, though formal tests (e.g., two-sample t-test) are more precise.
Comparing two methods (e.g., gravimetric analysis vs. ion chromatography):
- Use CIs to assess whether the two methods agree within a stated confidence level.
The transcript emphasizes: confidence intervals help remove subjective claims by providing an objective, probabilistic interval around the measured mean.
Foundational connections:
- T-test and confidence intervals build on the idea of sampling distributions and estimation under uncertainty.
- They link experimental results to known standards and to alternatives, enabling quantitative comparisons.

Degrees of freedom (df) for the t-interval are df = n − 1 because the sample standard deviation s is estimated from the data.
As n increases, SE = s/√n decreases, narrowing the interval for a fixed confidence level.
A smaller s (less variability) also tightens the interval for a fixed n and confidence level.
The t-distribution is used instead of the normal distribution when σ (the population standard deviation) is unknown and the sample size is relatively small.
The 95% confidence interval is a conventional standard in many scientific contexts; it reflects an acceptable balance between precision and confidence.

The t-test framework provides a way to express uncertainty in the population mean via confidence intervals:
- CI for μ: \bar{x} \pm t_{\mathrm{df},1-\alpha/2} \frac{s}{\sqrt{n}} \quad (df = n-1)
Example with n = 5, x̄ = 0.674, s = 0.012:
- 90% CI: [0.6626, 0.6854] (approx., using $t_{4,0.95}=2.132$)
- 95% CI: [0.6591, 0.6890] (approx., using $t_{4,0.975}=2.776$)
- 99% CI: [0.6493, 0.6987] (approx., using $t_{4,0.995}=4.601$)
Increasing confidence level widens the interval; decreasing it narrows the interval.
The “gold standard” 95% CI is widely used to remove subjectivity and standardize interpretation.
Confidence intervals enable practical comparisons across experiments, datasets, and measurement methods.
Always report the sample size, sample mean, sample standard deviation, and the appropriate t-value (with df = n − 1) when presenting CIs.