Water pH

Module B+F · freq 7

$\bar{x}\ \pm\ \underbrace{t_{n-1,\;\frac{1+\gamma}{2}}}_{\large L}\cdot\frac{s}{\sqrt{n}}$

Signals: You are given a string of raw data (10 to 25 numbers) · a) asks for the average / mean · b) asks for the sample standard deviation · c) asks for a confidence interval for the true mean at some confidence level · d) The width is written as

2\times L\times\frac{s}{\sqrt{n}}

, asking what

L

is and what it is called

Knowledge point:Descriptive statistics →Confidence intervals & proportions →

Generating question…

Method

a) np.mean(x)
b) np.std(x, ddof=1) — ddof=1 gives the sample version dividing by $n-1$
c) Variance unknown → t distribution: tc = stats.t.ppf((1+conf)/2, n-1), CI = m ± tc*s/√n
d) $L$ is just that t quantile; its name = "the $(1+\gamma)/2$ quantile of $t_{n-1}$ distribution"

Pitfalls where marks are lost

Forgetting ddof=1 → dividing by n instead of n−1, all wrong (MATLAB's std/var default to n−1, so they are actually safe)
Using norm.ppf instead of t.ppf — with σ unknown you must use t
Writing the degrees of freedom as n — it is n−1
Writing the quantile as γ — it is (1+γ)/2 (98% → 0.99, not 0.98)
b) asks for the standard deviation — this and the variance question are exact opposites, don't answer the wrong one