30 Confidence Intervals and Estimation
Estimation is the process of using sample data to infer the value of an unknown population parameter. A point estimate (such as a sample mean) gives a single best guess, but it will almost never exactly equal the true population value. A confidence interval improves on this by providing a range of plausible values, together with a stated level of confidence — combining the point estimate with the uncertainty quantified by the standard error from the previous chapter.
30.1 Point Estimates vs Interval Estimates
- Point Estimate: A single value, such as the sample mean \(\bar{x}\) or sample proportion \(\hat{p}\), used as the best guess for the population parameter.
- Interval Estimate (Confidence Interval): A range of values, calculated from the sample, that is likely to contain the true population parameter at a stated confidence level (commonly 90%, 95%, or 99%).
- Confidence Level: The long-run proportion of intervals, constructed the same way from repeated samples, that would contain the true population parameter. A 95% confidence level means that if the same sampling procedure were repeated many times, about 95% of the resulting intervals would contain the true parameter — it is not the probability that this particular interval contains the true value.
30.2 Confidence Interval for a Mean
When the Population Standard Deviation Is Known
\[ \bar{x} \pm Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \]
When the Population Standard Deviation Is Unknown (the common case)
The sample standard deviation \(s\) is used instead of \(\sigma\), and the \(t\)-distribution replaces the \(Z\)-distribution to account for the extra uncertainty: \[ \bar{x} \pm t_{\alpha/2,\, n-1} \cdot \frac{s}{\sqrt{n}} \] where \(t_{\alpha/2,\, n-1}\) is the critical value from the \(t\)-distribution with \(n-1\) degrees of freedom.
Confidence Interval for a Proportion
\[ \hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
30.2.1 Worked Example
A sample of 25 customers has a mean satisfaction score of 78 with a sample standard deviation of 12. Construct a 95% confidence interval for the true mean satisfaction score.
Since the population standard deviation is unknown and \(n = 25\) is small, the \(t\)-distribution is used with \(df = n - 1 = 24\). The critical value \(t_{0.025,\,24} \approx 2.064\).
\[ 78 \pm 2.064 \times \frac{12}{\sqrt{25}} = 78 \pm 2.064 \times 2.4 = 78 \pm 4.95 \]
\[ \text{95% CI} = (73.05,\ 82.95) \]
The analyst can be 95% confident that the true mean satisfaction score across all customers lies between 73.05 and 82.95.
30.3 Interpreting Confidence Intervals
- A narrower interval indicates a more precise estimate, achieved by increasing the sample size or reducing variability.
- A higher confidence level (e.g. 99% instead of 95%) produces a wider interval, since more certainty requires a larger margin of safety.
- A confidence interval that does not include a hypothesised value (e.g. a target score of 80) is informative for decision-making, and is directly connected to the hypothesis-testing framework covered in the next part of the book.
30.4 Confidence Intervals in R and Python
Transition to Inferential Statistics and Hypothesis Testing
Confidence intervals estimate what a population parameter probably is. The remaining chapters of this module take the next step — formal hypothesis testing — to decide whether an observed difference or relationship in the data is statistically significant, starting with the framework for choosing the right test.
Summary
| Concept | Description |
|---|---|
| Foundations | |
| Point Estimate | A single best-guess value for a population parameter, such as the sample mean |
| Interval Estimate (Confidence Interval) | A range of values, calculated from a sample, likely to contain the true population parameter |
| Confidence Level | The long-run proportion of similarly constructed intervals that would contain the true parameter |
| CI Formulas | |
| CI for a Mean (sigma known) | x-bar plus or minus Z times sigma over the square root of n |
| CI for a Mean (sigma unknown) | x-bar plus or minus the t critical value times s over the square root of n, using n-1 degrees of freedom |
| CI for a Proportion | p-hat plus or minus Z times the square root of p-hat times (1 minus p-hat) over n |
| Margin of Error | The plus-or-minus amount added to and subtracted from the point estimate to form the interval |
| Interpretation | |
| Width vs Precision | A narrower interval reflects a more precise estimate, from a larger sample or less variability |
| Width vs Confidence Level | A higher confidence level produces a wider interval, trading precision for certainty |
| In R and Python | |
| R t.test() | Built-in R function that returns a confidence interval alongside a one-sample t-test |
| Python scipy.stats.t.interval() | SciPy function that computes a confidence interval directly from a distribution's parameters |