Session 8.1.1 - Confidence Interval for μ

Why a confidence interval?

We rarely know the population mean $ \mu $. A small sample gives a point estimate $ \bar x $, but we need a range showing plausible values for $ \mu $. A $100(1-\alpha)\%$ confidence interval (CI) is that range: if we repeated this process many times, about $100(1-\alpha)\%$ of such intervals would capture the true $ \mu $.

Interpretation one-liner: “We are 95% confident that $ \mu $ lies between the two endpoints.” (It’s about the method’s long-run success rate, not a probability statement about this fixed $ \mu $.)

What is α (alpha)?

Alpha is the miss rate. For a two-sided $100(1-\alpha)\%$ CI, the area outside totals $ \alpha $, split into $\alpha/2$ in each tail. The critical value is $ z_{\alpha/2}=\Phi^{-1}(1-\alpha/2) $.

Confidence Pick 90% (α=0.10), 95% (α=0.05), or 99% (α=0.01).

Confidence

95%

$ \alpha $

0.05

$ \alpha/2 $

0.025

$ z_{\alpha/2} $

1.960

Right-tail Excel

=NORM.S.INV(1-0.05/2)

Left-tail Excel

=NORM.S.INV(0.05/2)

Sketch: shaded tails total $ \alpha $; cutpoints at $ \pm z_{\alpha/2} $.

How it works (σ known)

\[ Z \;=\; \frac{\bar X - \mu}{\sigma/\sqrt{n}} \sim \mathcal{N}(0,1) \;\Rightarrow\; \bar x \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}. \]

If $ \sigma $ is unknown (typical), use the t-interval page.

What to expect

Higher confidence ⇒ wider interval.
Larger $n$ or smaller $ \sigma $ ⇒ narrower interval.
Width $= 2\,z_{\alpha/2}\cdot \sigma/\sqrt{n}$.

How accurate?

Margin of error $ \text{ME}=z_{\alpha/2}\sigma/\sqrt{n} $. To hit a target ME*, use $ n=\big(\frac{z_{\alpha/2}\sigma}{\text{ME}^{*}}\big)^2 $.

Formula recap & how to use it

1) The confidence interval formula

\[ \text{CI for }\mu\ (\sigma\text{ known}):\quad \boxed{\ \bar x \pm z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}\ } \] \[ \text{Lower}=\bar x - z_{\alpha/2}\frac{\sigma}{\sqrt{n}},\quad \text{Upper}=\bar x + z_{\alpha/2}\frac{\sigma}{\sqrt{n}}. \]

2) Precision pieces

Standard error: $ \mathrm{SE}=\sigma/\sqrt{n} $
Margin of error: $ \boxed{\mathrm{ME}=z_{\alpha/2}\,\mathrm{SE}=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}} $
Width: $ \text{Upper}-\text{Lower} = \boxed{2\cdot \mathrm{ME}} $
Plan sample size for target accuracy $ \mathrm{ME}^{*} $: $ \boxed{n=\left(\dfrac{z_{\alpha/2}\sigma}{\mathrm{ME}^{*}}\right)^2} $ (round up)

3) Quick workflow

Choose confidence $1-\alpha$ → get $z_{\alpha/2}$ (see the alpha box above).
Compute $ \mathrm{SE}=\sigma/\sqrt{n} $.
Compute $ \mathrm{ME}=z_{\alpha/2}\cdot \mathrm{SE} $.
Report CI: $ [\,\bar x-\mathrm{ME},\ \bar x+\mathrm{ME}\,] $.
Interpret: “We are $100(1-\alpha)\%$ confident that $ \mu $ lies in this interval.”

Worked example (calculator + mini graph)

Type your numbers or pick a preset.

Preset:

Sample mean $ \bar x $

Known $ \sigma $

Sample size $ n $

Confidence

z-CI:

—

Margin of error

—

SE $=\sigma/\sqrt{n}$

—

z for confidence

—

Enter values and click Compute z-CI to see the step-by-step math here.

Graph: dot = $ \bar x $; line = CI; tick marks = endpoints.

Excel — quick steps

Cell map

B2 = $ \bar x $ (sample mean)
B3 = $ \sigma $ (known SD)
B4 = $ n $ (sample size)
B5 = confidence as decimal (e.g., 0.95)
B6 = $ z_{\alpha/2} $ → =NORM.S.INV(1-(1-$B$5)/2)
B7 = SE → =$B$3/SQRT($B$4)
B8 = ME → =$B$6*$B$7
B9 = Lower bound → =$B$2-$B$8
B10 = Upper bound → =$B$2+$B$8
B11 = Target ME* (you type this)
B12 = Required n → =ROUNDUP(($B$6*$B$3/$B$11)^2,0)

All-in-one formulas

ME in one cell: =NORM.S.INV(1-(1-$B$5)/2)*$B$3/SQRT($B$4)
n for target ME*: =ROUNDUP((NORM.S.INV(1-(1-$B$5)/2)*$B$3/$B$11)^2,0)

Tip: To plan sample size, enter your target ME* in B11 and read the result in B12. Then set B4 = B12 to compute the CI. If $ \sigma $ is unknown, use the t-interval page.

Common Student Q&A — quick explanations + tiny math

1) Does higher confidence mean more accuracy?

No—higher confidence → wider interval (less precise).

Example (commute): $ \sigma=4,\ n=25\Rightarrow \mathrm{SE}=4/\sqrt{25}=0.8$.
95%: $z_{\alpha/2}=1.96\Rightarrow \mathrm{ME}=1.96\times0.8=1.568$.
99%: $z_{\alpha/2}=2.576\Rightarrow \mathrm{ME}=2.576\times0.8=2.0608$.
99% is “safer” but less precise.

2) If I collect more data, how much narrower does the CI get?

Width shrinks like $1/\sqrt{n}$. Quadrupling $n$ halves the ME.

With 95%, $ \sigma=4 $.
$n=25\Rightarrow \mathrm{SE}=0.8,\ \mathrm{ME}=1.96\times0.8=1.568$.
$n=100\Rightarrow \mathrm{SE}=0.4,\ \mathrm{ME}=1.96\times0.4=0.784$ (half as wide).

3) Is the CI centered at $ \mu $ or at $ \bar x $?

Always centered at the sample mean $ \bar x $. That’s why your line is symmetric around the dot ( $ \bar x $ ).

Example: $ \bar x=11.7,\ \mathrm{ME}=2.48 \Rightarrow [\,9.22,\ 14.18\,] $ is symmetric around 11.7.

4) What is “width” vs “margin of error”?

Width = Upper − Lower = $2\times \mathrm{ME}$.

If $ \mathrm{ME}=1.568 \Rightarrow \text{Width}=2\times1.568=3.136$.

5) How many students do I need to be within ±1 minute at 95%?

Use $ n=\left(\dfrac{z_{\alpha/2}\sigma}{\mathrm{ME}^{*}}\right)^2 $ and round up.

With $ \sigma\approx4,\ \mathrm{ME}^{*}=1 $:
$ n=(1.96\times4/1)^2=(7.84)^2\approx 61.47 \Rightarrow \boxed{62}$.

6) σ is unknown—can I still use this page?

For real data, use the t-interval page. z is mainly for when $ \sigma $ is known or planning.

Tiny math: if $n=10$ and you estimate $s=4$:
$ \mathrm{SE}=4/\sqrt{10}=1.265.$ 95%:
z: $1.96\times1.265=2.48$ vs t$_{9,0.025}=2.262$: $2.262\times1.265=2.86$.
t makes ME a bit larger at small n.

7) One-sided bound example?

For a 95% lower bound (only want a minimum), use $ \bar x - z_{\alpha}\dfrac{\sigma}{\sqrt{n}} $ with $z_{\alpha}=1.645$.

Example: $ \bar x=11.7,\ \sigma=4,\ n=10.\ \mathrm{SE}=1.265$.
Lower 95% bound: $11.7-1.645\times1.265\approx \boxed{9.62}$.

8) Student-life example: “average study hours within ±0.5 hr at 95%”

Suppose prior $ \sigma\approx 2 $ hours.

$ n=(1.96\times2/0.5)^2=(3.92/0.5)^2=(7.84)^2\approx 61.47 \Rightarrow \boxed{62}$.

9) Engineering example: “mean shaft diameter within ±0.02 mm at 95%”

Assume historical $ \sigma\approx 0.08 $ mm.

$ n=(1.96\times0.08/0.02)^2=(0.1568/0.02)^2=(7.84)^2\approx 61.47 \Rightarrow \boxed{62}$.

10) Does a mean CI say 95% of individual items are inside?

No. That’s a CI for the mean, not for individual observations.

For coverage of individuals (e.g., protein bar weights), use a tolerance interval or a prediction interval.

11) Is a hypothesized mean plausible?

If $ \mu_0 $ falls inside your CI, it’s consistent with the data at that confidence.

Example: CI $[9.22,14.18]$. Is $ \mu_0=11 $ plausible? Yes (inside).

12) Units & rounding?

Keep units consistent (minutes with minutes, mm with mm). Report ME and endpoints to a sensible number of decimals (usually 2–3).

Quick Excel map: B6 = $z_{\alpha/2}$, B7 = SE, B8 = ME, B9/B10 = lower/upper; B11 = target ME*, B12 = required $n$.