4. Estimation I: Confidence Intervals for the Population Mean :math:`\mu` (Known :math:`\sigma` and Unknown :math:`\sigma`)
======================================================================
4.0 Notation Table
------------------
.. list-table::
:header-rows: 1
:widths: 22 78
* - Symbol
- Meaning
* - :math:`X_1,\dots,X_n`
- independent sample observations
* - :math:`n`
- sample size
* - :math:`\mu`
- population mean
* - :math:`\sigma,\ \sigma^2`
- population standard deviation, variance
* - :math:`\bar{X}`
- sample mean
* - :math:`S,\ S^2`
- sample standard deviation, variance
* - :math:`\alpha`
- CI error rate (:math:`1-\alpha` confidence)
* - :math:`z_{\alpha/2}`
- Normal critical value
* - :math:`t_{\alpha/2,\ n-1}`
- t critical value (df :math:`n-1`)
* - :math:`E`
- margin of error (half-width)
* - :math:`\text{s.e.}(\bar{X})`
- standard error of :math:`\bar{X}`
4.1 Introduction
----------------
In the previous module, the sampling distribution of :math:`\bar{X}` was used to quantify how sample means vary across repeated samples. That variability is the source of uncertainty statements in estimation.
This module uses that idea to estimate the population mean :math:`\mu` in a disciplined way. We begin with point estimation and then construct confidence intervals (CIs) that communicate precision under explicit sampling conditions.
4.2 Learning Outcomes
----------------------
After this session, you should be able to:
- Distinguish point estimation from interval estimation for :math:`\mu`
- State and interpret a :math:`100(1-\alpha)\%` CI correctly, using repeated-sampling language
- Construct a CI for :math:`\mu` when :math:`\sigma` is known (z-based) and when :math:`\sigma` is unknown (t-based)
- Explain how :math:`n` affects the standard error and CI width
- Plan a sample size :math:`n` to achieve a target margin of error :math:`E` (under stated assumptions)
- Avoid common interpretation and condition-checking errors
4.3 Main Concepts
------------------
4.3.1 Point estimation for :math:`\mu`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
A point estimate is a single-number estimate of a parameter. For the population mean :math:`\mu`, the standard estimator is the sample mean :math:`\bar{X}`.
.. math::
\bar{X}=\frac{1}{n}\sum_{i=1}^{n} X_i
This estimator summarizes the center of the observed data and targets :math:`\mu` under random sampling. Its variability decreases as :math:`n` increases, so larger samples usually produce more stable estimates.
The uncertainty of :math:`\bar{X}` is summarized by the standard error. When the population standard deviation :math:`\sigma` is treated as known, the standard error is
.. math::
\text{s.e.}(\bar{X})=\frac{\sigma}{\sqrt{n}}
When :math:`\sigma` is unknown, it is estimated by :math:`S`, and :math:`S/\sqrt{n}` is used as the estimated standard error. This substitution changes the reference distribution used for interval estimation when :math:`n` is small.
4.3.2 Confidence intervals: meaning and interpretation discipline
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
A confidence interval is a random interval computed from sample data. The method is designed so that, across repeated samples taken under the same conditions, the interval covers the fixed parameter :math:`\mu` with a pre-specified frequency.
For a two-sided :math:`100(1-\alpha)\%` CI, we construct random endpoints :math:`L` and :math:`U` such that
.. math::
P(L \le \mu \le U)=1-\alpha
This probability statement describes the long-run performance of the procedure before data are observed. After the sample is collected, :math:`L` and :math:`U` become fixed numbers, and :math:`\mu` is either inside the interval or not.
A correct CI interpretation must include all of the following elements:
- the parameter (:math:`\mu`)
- the confidence level (:math:`100(1-\alpha)\%`)
- the repeated-sampling meaning (long-run coverage under the same procedure and conditions)
- the conditions under which the method is intended to work
4.3.3 CI for :math:`\mu` when :math:`\sigma` is known (z-based)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Assume :math:`X_1,\dots,X_n` are independent and represent a random sample from a stable process. If the population is Normal, then :math:`\bar{X}` is Normal; if the population is not Normal but :math:`n` is sufficiently large and not extremely skewed, :math:`\bar{X}` is approximately Normal by the CLT. In this section, :math:`\sigma` is treated as known from reliable historical process knowledge or a validated measurement system.
A two-sided :math:`100(1-\alpha)\%` CI for :math:`\mu` is
.. math::
\bar{x} \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}
This can also be written as :math:`[\bar{x}-E,\ \bar{x}+E]`, where the margin of error (half-width) is
.. math::
E=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}
The margin of error decreases at the rate :math:`1/\sqrt{n}`, so increased sample size improves precision but with diminishing returns. The confidence level affects the critical value :math:`z_{\alpha/2}`, so higher confidence implies a larger :math:`E` when :math:`n` and :math:`\sigma` are fixed.
Figure 4.1 — Repeated confidence intervals and long-run coverage
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This figure uses simulated data from a Normal population because it provides a clean baseline for understanding CI logic without additional distribution complications. “Repetition” means drawing many independent samples of the same size :math:`n` from the same population and constructing a CI from each sample.
To read the plot, first locate the vertical reference line for the true mean :math:`\mu`. Each horizontal segment is one confidence interval computed from one sample mean, and its position shows the interval endpoints on the mean scale. The intervals are empirical outcomes from simulation, while the confidence level is a theoretical design target for the interval procedure.
The statistical message is that a :math:`100(1-\alpha)\%` CI method is calibrated for long-run coverage, not guaranteed success in any one repetition. When the confidence level increases (for fixed :math:`n` and :math:`\sigma`), intervals become wider because the critical value increases, and coverage tends to be higher across repetitions. This tradeoff is central to operational decision-making because confidence and precision cannot usually be maximized at the same time.
.. raw:: html
Example 4.1
^^^^^^^^^^^
A call center monitors customer handling time (minutes) under a stable staffing plan and routing rule. From long-run monitoring and process control records, management treats the standard deviation as known: :math:`\sigma=4.0` minutes.
**Question:** A random sample of :math:`n=64` calls yields :math:`\bar{x}=18.6`. Construct a 95% CI for :math:`\mu` and interpret it correctly.
Because :math:`\sigma` is treated as known and :math:`n` is moderately large, a z-based CI is appropriate under Normal sampling or CLT approximation. For 95% confidence, :math:`z_{\alpha/2}\approx 1.96`, and the standard error is :math:`\sigma/\sqrt{n}=4/\sqrt{64}=0.5`.
The margin of error is :math:`E=1.96(0.5)=0.98`, so the CI is :math:`18.6\pm 0.98`, which gives endpoints 17.62 and 19.58. The parameter is the process mean handling time, not the mean of the observed sample.
**Answer:** A 95% CI for :math:`\mu` is approximately :math:`[17.62,\ 19.58]` minutes. Interpreted correctly, if this sampling-and-interval procedure were repeated many times with :math:`n=64` under the same stable process, about 95% of the resulting intervals would contain the true mean handling time.
4.3.4 CI for :math:`\mu` when :math:`\sigma` is unknown (t-based)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
In many operations studies, :math:`\sigma` is not known and must be estimated from the same sample used to estimate :math:`\mu`. When the population distribution is approximately Normal (or when the sample does not show severe skewness or extreme outliers), the standardized statistic
.. math::
T=\frac{\bar{X}-\mu}{S/\sqrt{n}}
follows a t distribution with :math:`n-1` degrees of freedom. This requires :math:`n\ge 2` and relies on independent sampling.
A two-sided :math:`100(1-\alpha)\%` CI for :math:`\mu` is
.. math::
\bar{x} \pm t_{\alpha/2,\ n-1}\frac{s}{\sqrt{n}}
Compared with the z-based interval, the t critical value is larger for small :math:`n`, so the interval is wider. As :math:`n` increases, the t distribution approaches the standard Normal distribution, and the difference between the two methods becomes small.
Figure 4.3 — How t critical values approach z as df increases
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This figure uses theoretical critical values rather than simulated samples because the goal is to isolate the effect of degrees of freedom. Here :math:`n` enters through :math:`\text{df}=n-1`, and there is no repeated sampling because we are not generating data-driven intervals.
To read the plot, first identify the horizontal line for the z critical value at the selected confidence level. Then track the t critical value curve as :math:`\text{df}` increases; the vertical gap shows the extra conservatism from using t when :math:`\sigma` is unknown. The curve is theoretical, while an applied CI is empirical because it uses the sample value :math:`s` and the observed :math:`\bar{x}`.
The statistical message is that small-sample estimation is less precise because :math:`\sigma` is replaced by :math:`s`, and the reference distribution must account for that added uncertainty. As :math:`n` grows, :math:`s` stabilizes and the t reference becomes close to Normal, so CI width is increasingly dominated by :math:`1/\sqrt{n}` rather than by the choice of z versus t.
.. raw:: html
Example 4.2
^^^^^^^^^^^
A warehouse tests a new picking method and records picking time per order (minutes). The method is new, so there is no validated historical value for :math:`\sigma`, and variability must be estimated from the sample.
**Question:** A random sample of :math:`n=12` orders yields :math:`\bar{x}=31.4` and :math:`s=6.0`. Construct a 95% CI for :math:`\mu` and state the conditions that support this method.
Because :math:`\sigma` is unknown, the t-based interval is used with :math:`\text{df}=11`. For 95% confidence, :math:`t_{\alpha/2,\ 11}\approx 2.201`, and the estimated standard error is :math:`s/\sqrt{n}=6/\sqrt{12}\approx 1.732`.
The margin of error is :math:`E\approx 2.201(1.732)\approx 3.81`, so the interval is :math:`31.4\pm 3.81`, giving endpoints 27.59 and 35.21. The interval is wider than a z-based interval because the variability is estimated rather than known.
**Answer:** A 95% CI for :math:`\mu` is approximately :math:`[27.59,\ 35.21]` minutes. This relies on random, independent sampling and an approximately Normal population distribution (or, at minimum, no severe skewness or extreme outliers at :math:`n=12`).
4.3.5 Planning precision: choosing :math:`n` for a target :math:`E`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Operational decisions often require that an estimate be sufficiently precise. For a z-based CI with known :math:`\sigma`, the margin of error is :math:`E=z_{\alpha/2}\sigma/\sqrt{n}`, so solving for :math:`n` yields
.. math::
n=\left(\frac{z_{\alpha/2}\sigma}{E}\right)^2
This formula is a planning tool, so the input :math:`\sigma` should represent expected operating conditions, typically from historical data or a pilot study. The computed :math:`n` must be rounded up to ensure that the target margin of error is not overstated.
Figure 4.2 — Margin of error versus sample size (diminishing returns)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This figure plots the margin of error formula using theoretical values because the relationship between :math:`E` and :math:`n` is deterministic once :math:`\sigma` and the confidence level are specified. Here :math:`n` represents the planned number of independent observations that will be averaged to produce :math:`\bar{x}`, and there is no repeated sampling because the plot is not based on simulated intervals.
To read the plot, select a confidence level from the control at the top-right, and then follow the curve as :math:`n` increases. The curve shows the computed margin of error :math:`E(n)`, while reference lines indicate a target half-width and the implied minimum :math:`n` to meet that target. The curve is theoretical, not a fitted model from observed data.
The statistical message is diminishing returns: to reduce :math:`E` by a factor of 2, the sample size must increase by a factor of 4. Higher confidence levels produce larger critical values, shifting the curve upward, so achieving the same :math:`E` requires a larger :math:`n`. This tradeoff matters directly in operations because sampling effort, time, and cost increase with :math:`n`.
.. raw:: html
Example 4.3
^^^^^^^^^^^
A logistics team wants a precise estimate of mean delivery delay (minutes) for a stable route during standard operating hours. A pilot study suggests the standard deviation is approximately :math:`\sigma\approx 2.5` minutes for that route.
**Question:** For a 95% z-based planning target, what sample size :math:`n` is needed to achieve margin of error :math:`E=0.5` minutes?
For 95% confidence, :math:`z_{\alpha/2}\approx 1.96`. Using the planning formula,
.. math::
n=\left(\frac{1.96(2.5)}{0.5}\right)^2
Compute :math:`1.96(2.5)=4.9`, then :math:`4.9/0.5=9.8`, and :math:`9.8^2=96.04`. Because :math:`n` must be an integer and planning should not understate uncertainty, the value is rounded up.
**Answer:** Plan for :math:`n=97` deliveries. This plan assumes independent observations and that :math:`\sigma\approx 2.5` remains valid under the same route and operating conditions.
4.4 Discussion and Common Errors
--------------------------------
A frequent error is to say “there is a 95% probability that :math:`\mu` is in the interval.” The confidence level describes the long-run success rate of the method under repetition, not a probability distribution for :math:`\mu` after the data are observed.
Another error is to substitute :math:`s` for :math:`\sigma` while still using :math:`z_{\alpha/2}` in small samples. When :math:`\sigma` is unknown and :math:`n` is small, the intended coverage relies on the t reference distribution under the stated conditions.
It is also common to ignore distribution shape when :math:`n` is small. The t-based method depends on approximately Normal sampling behavior, so strong skewness or extreme outliers can invalidate the long-run coverage claim.
Students sometimes confuse a CI for :math:`\mu` with a statement about individual future observations. A CI for the mean can be narrow even when individual outcomes are highly variable, so it should not be used as a prediction interval.
Finally, sample-size planning often fails due to rounding down or using an unrealistic variability input. Planning must round up :math:`n` and should use a defensible value of :math:`\sigma` based on stable historical data or a carefully designed pilot study.
4.5 Summary
------------
A point estimate for :math:`\mu` is typically :math:`\bar{x}`, and its uncertainty is measured by the standard error. A confidence interval reports a plausible range for :math:`\mu` with a method calibrated to cover :math:`\mu` in a fixed proportion of repeated samples.
When :math:`\sigma` is treated as known, a z-based CI uses :math:`\bar{x}\pm z_{\alpha/2}\sigma/\sqrt{n}`. When :math:`\sigma` is unknown, a t-based CI uses :math:`\bar{x}\pm t_{\alpha/2,\ n-1}s/\sqrt{n}`, which is wider for small :math:`n` and approaches the z-based result as :math:`n` increases.
Precision planning uses the margin of error relationship to choose :math:`n` for a target half-width. The dependence :math:`E\propto 1/\sqrt{n}` explains diminishing returns and supports realistic operational budgeting.