Scenario

The Birganj Sub-Metropolitan Office publishes a citizen charter claiming:

“Mean time to issue a recommendation letter is 8 working days.”

A monitoring cell draws a simple random sample of $n = 36$ completed cases from the last quarter. Processing times (working days):

10, 7, 9, 12, 8, 11, 6, 9, 14, 8,
7, 10, 9, 8, 13, 11, 7, 9, 10, 12,
8, 6, 15, 9, 10, 8, 11, 7, 9, 10,
12, 8, 9, 11, 7, 10

Summary statistics (you may use these):

  • $\bar{x} = 9.47$ days (approx.)
  • $s = 2.15$ days (approx.)
  • Population $\sigma$ is unknown; $n = 36$ is large.

Leadership asks whether the charter claim $\mu = 8$ is still tenable at $\alpha = 0.05$, and wants a one-page memo for the Executive Officer.

Your tasks

  1. Choose the appropriate procedure (name the test; justify briefly).
  2. State hypotheses and assumptions.
  3. Compute the test statistic and decision (or outline every numerical step clearly if calculator-limited).
  4. Optionally note what a 95% CI for $\mu$ would add for managers.
  5. Write a short PA memo using the rubric below.

Memo rubric

Use these five beats (a few sentences each is enough):

Beat What to include
Claim Restate the charter claim and the decision question in plain language.
Evidence Sample size, mean, SD (or SE), test used, statistic, and α.
Decision Reject or fail to reject $H_0$; avoid “prove true/false.”
Practical meaning What the result implies for citizens and for publishing the 8-day figure.
Caveats Sampling frame, working-day definition, seasonality, and that significance ≠ importance.

Hints (optional)

  • Large $n$, unknown $\sigma$, one mean → large-sample z (using $s$) or t with large df; either is acceptable if you state the choice.
  • Two-sided vs one-sided: the charter is a point claim unless the brief says “no more than 8.”
Full solution (click to reveal) ### 1. Method **Large-sample z-test for a single mean** (σ unknown, replace with $s$; $n = 36 \ge 30$). A one-sample **t-test** with $df = 35$ is also defensible; critical values are nearly the same. ### 2. Hypotheses (two-sided) $$H_0: \mu = 8 \qquad H_1: \mu \neq 8$$ Assumptions: random sample; approx. normal sampling distribution of $\bar{x}$ by CLT. ### 3. Calculations $$SE = \frac{s}{\sqrt{n}} = \frac{2.15}{\sqrt{36}} = \frac{2.15}{6} \approx 0.358$$ $$z = \frac{\bar{x} - \mu_0}{SE} = \frac{9.47 - 8}{0.358} \approx 4.11$$ Critical values at $\alpha = 0.05$: $\pm 1.96$. $|z| = 4.11 > 1.96$ → **reject $H_0$**. Approximate 95% CI: $$\bar{x} \pm 1.96\,SE \approx 9.47 \pm 0.70 \Rightarrow (8.77,\ 10.17)$$ The interval lies entirely above 8, consistent with the test. ### 4. Sample memo (model) **To:** Executive Officer **From:** Monitoring cell **Re:** Citizen-charter claim of 8-day mean service time **Claim.** The charter states that mean recommendation-letter time is 8 working days. We asked whether recent completed cases still support that figure. **Evidence.** A random sample of 36 cases had mean 9.47 days (SD 2.15). Using a large-sample test of $H_0: \mu = 8$ at 5% significance, $z \approx 4.11$. **Decision.** We reject the null: the data are not consistent with a population mean of 8 days. **Practical meaning.** Typical recent times appear closer to about 9–10 days (95% CI roughly 8.8–10.2). Publishing “8 days” risks over-promising unless processes improve or the charter is revised. **Caveats.** Results apply to completed cases in the sampled quarter, use working-day definitions, and do not by themselves measure citizen harm. A follow-up should check peak-season loads and incomplete applications excluded from the frame.