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23-Ind-B1 Reliability and Maintainability · May 2015

Question 3 of 10: Class-Average Sampling Distribution — Normal Probability and a One-Observation z -Test

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Notes on this paper

National Exams — May 2015 — 98-Ind-B1 Applied Probability & Statistics. Three-hour, closed-book exam; one of two permitted calculators (Sharp or Casio), one 8.5″×11.0″ aid sheet (both sides), statistical tables supplied. Format: three sections — Section A: do 2 of 4 questions (10 marks); Section B: do 2 of 3 (20 marks); Section C: do 1 of 3 (20 marks) — a 5-question, 50-mark paper as printed. All ten questions across the three sections are solved below for completeness.

Reference texts: Montgomery & Runger, Applied Statistics and Probability for Engineers (7th ed., Wiley) — joint distributions (ch. 5), binomial/geometric probability (ch. 3), normal distribution and sampling distributions (ch. 4–7), point/interval estimation (ch. 8–9), hypothesis testing incl. sample-size design (ch. 9–10), simple linear regression and ANOVA (ch. 11–13), Bartlett's and Tukey's tests, single-degree-of-freedom contrasts and 23 factorial designs (ch. 13–14). Montgomery, Peck & Vining, Introduction to Linear Regression Analysis (6th ed., Wiley) — multiple regression by matrices, confidence/prediction intervals (ch. 2–3).

Question 3 (Section A.3): Class-Average Sampling Distribution — Normal Probability and a One-Observation z-Test (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $X=$ a 3rd-year class average $\sim N(\mu=72,\ \sigma^2=3)$, so $\sigma=\sqrt{3}=1.732$.

Find. (a) $P(75\lt X\lt 80)$; (b) whether $x=68.3$ (COMP 3311's average) is unusual at $\alpha=0.05$.

Approach. Standardize and read normal-curve areas for (a); for (b), COMP 3311 is itself one of the 3rd-year classes the $N(72,3)$ model describes, so its observed average is a single draw from that distribution — test $H_0:x$ is a typical draw via a $z$-statistic.

  1. (a) Probability the average falls in (75, 80). $z_1=(75-72)/1.732=1.732$, $z_2=(80-72)/1.732=4.619$. $$P(75\lt X\lt 80)=\Phi(4.619)-\Phi(1.732)\approx 1.0000-0.9584=\boxed{0.0416}$$
  2. (b) Is 68.3 unusual? $$z=\frac{68.3-72}{\sqrt{3}}=\frac{-3.7}{1.732}=-2.136$$ At $\alpha=0.05$ two-tailed, the critical values are $\pm 1.96$; since $|-2.136|\gt 1.96$ (two-tailed $p=0.033$), $H_0$ is rejected: $\boxed{\text{yes, 68.3 is an unusual result}}$.
Summary
PartResult
(a) $P(75\lt X\lt 80)$0.0416
(b) $z$, verdict−2.136; unusual (reject $H_0$, $p=0.033$)
Check — reading of "the term consists of 5 courses" This sentence establishes that COMP 3311 is one of several 3rd-year courses running that term, i.e. one legitimate draw from the historical $N(72,3)$ population being tested — it does not change the calculation. If instead the intent were to flag that COMP 3311 was singled out for scrutiny from among 5 courses (a multiple-comparisons concern), a Bonferroni-adjusted two-tailed critical value would be $z_{0.005}=\pm 2.576$; at that stricter threshold $|-2.136|\lt 2.576$ and the result would NOT be flagged as unusual. The primary boxed answer uses the direct single-test reading, which is the standard first read of this question type.