23-Ind-B1 Reliability and Maintainability · December 2018

Question 5 of 9: Simple Linear Regression — Age vs. BMI

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — December 2018 — 17-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 (20 marks); Section B: do 2 of 3 (40 marks); Section C: do 1 of 2 (20 marks) — a 9-question, printed-total-80-mark paper as actually structured (the front page's own summary line states “Exam: 5 Questions. Total marks: 100”, but 20+40+20=80 by the section instructions and per-question mark values printed beside each question — the front page's 100 is internally inconsistent with its own section table; the 80-mark reading is used throughout, consistent with the section-by-section split printed on pages 2, 4 and 6). All nine questions across the three sections are solved below for completeness.

Reference texts: Montgomery & Runger, Applied Statistics and Probability for Engineers (7th ed., Wiley) — normal distribution and sums of normals (ch. 4–5), point/interval estimation (ch. 8), two-sample hypothesis testing (ch. 9–10), simple linear regression (ch. 11), single-factor ANOVA and multiple comparisons (ch. 13). Montgomery, Peck & Vining, Introduction to Linear Regression Analysis (6th ed., Wiley) — multiple regression by matrices, confidence/prediction intervals (ch. 2–3). Montgomery, Design and Analysis of Experiments (9th ed., Wiley) — Bartlett's test, $2^k$ factorial designs (ch. 3, 6).

Question 5 (Section B.1): Simple Linear Regression — Age vs. BMI (20 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. $n=9$ patients; predictor $x=$ Age, response $y=$ BMI (per the stated model form $y=ax+b$, i.e. BMI regressed on age).

Patient	1	2	3	4	5	6	7	8	9
Age ($x$)	45	34	40	32	28	30	52	33	47
BMI ($y$)	20	21	30	20	26	22	24	18	36

Find. (a) Least-squares line $\hat y=ax+b$; (b) is the regression significant overall ($F$-test)? (c) is the slope significant ($t$-test)? (d) $\hat y$ at age 42, and is $(42,28)$ an anomaly?

Approach. Compute $S_{xx}$, $S_{xy}$, $S_{yy}$ from the raw data, form the least-squares slope and intercept, then run the $F$-test (regression significance) and $t$-test (slope significance) from the ANOVA decomposition of $S_{yy}$.

(a) Sums of squares and the fitted line. $$\bar x=37.89,\ \bar y=24.11,\qquad S_{xx}=\sum(x_i-\bar x)^2=570.89,\quad S_{xy}=\sum(x_i-\bar x)(y_i-\bar y)=154.11,\quad S_{yy}=264.89$$ $$a=\frac{S_{xy}}{S_{xx}}=\frac{154.11}{570.89}=0.2699,\qquad b=\bar y-a\bar x=24.11-0.2699(37.89)=13.883$$ $$\boxed{\hat y=0.2699x+13.883}$$
(b) $F$-test for overall regression significance, $\alpha=0.05$. $SS_R=aS_{xy}=41.60$, $SS_E=S_{yy}-SS_R=223.29$, $df_E=n-2=7$. $$MS_R=41.60,\qquad MS_E=\frac{223.29}{7}=31.90,\qquad F=\frac{MS_R}{MS_E}=\boxed{1.304}$$ Critical value $F_{0.05,1,7}=5.591$ ($p=0.291$). Since $F=1.304<5.591$, fail to reject $H_0:\beta_1=0$ — the regression is not statistically significant at the 5% level.
(c) $t$-test on the slope, $\alpha=0.05$. $$SE(a)=\sqrt{\frac{MS_E}{S_{xx}}}=\sqrt{\frac{31.90}{570.89}}=0.2364,\qquad t=\frac{a}{SE(a)}=\frac{0.2699}{0.2364}=\boxed{1.142}$$ Critical value $t_{0.025,7}=2.365$ ($p=0.291$, and indeed $t^2=1.304=F$, the standard identity for simple regression). Since $|t|=1.142<2.365$, the slope is not significant — consistent with (b). Only $r^2=SS_R/S_{yy}=15.7\%$ of the BMI variation is explained by age in this sample ($r=0.396$).
(d) Predict at age 42, and assess $(42,28)$. $$\hat y(42)=0.2699(42)+13.883=\boxed{25.22}$$ 95% confidence interval for the mean BMI at age 42 ($t_{0.025,7}=2.365$): $$\hat y\pm t_{0.025,7}\sqrt{MS_E\left(\frac1n+\frac{(42-\bar x)^2}{S_{xx}}\right)}=25.22\pm5.01=[20.21,\ 30.23]$$ 95% prediction interval for a single new patient's BMI at age 42: $$\hat y\pm t_{0.025,7}\sqrt{MS_E\left(1+\frac1n+\frac{(42-\bar x)^2}{S_{xx}}\right)}=25.22\pm14.26=[10.96,\ 39.48]$$ A BMI of 28 at age 42 falls comfortably inside the 95% prediction interval $[10.96,39.48]$, so it is not a statistical anomaly.

Final Results
Quantity	Value
Fitted line	$\hat y=0.270x+13.883$
Regression $F$ (vs. crit.)	$1.304$ (vs. $5.591$) — not significant
Slope $t$ (vs. crit.)	$1.142$ (vs. $2.365$) — not significant
$\hat y(42)$, 95% PI	25.22, $[10.96,39.48]$
Age 42, BMI 28	Not an anomaly (within PI)

Check The question's own phrase “age (x) versus BMI (y)” together with the stated model form $y=ax+b$ is read as: $x=$ Age (predictor), $y=$ BMI (response) — i.e. the model predicts BMI from age, which is also the only reading consistent with part (d) asking for “the expected BMI for a 42 year-old.” Separately, because the overall regression is not significant (part b), the point prediction and intervals in part (d) describe a very weak linear relationship (only 15.7% of BMI variance explained by age in this small, $n=9$ sample) — the wide prediction interval reflects that weakness honestly rather than a computational error.

Topic: Simple linear regression — least squares, significance testing, confidence vs. prediction intervals.

Key relations: $a=S_{xy}/S_{xx}$, $b=\bar y-a\bar x$; $F=MS_R/MS_E\sim F_{1,n-2}$; $t=a/SE(a)\sim t_{n-2}$, and $t^2=F$ in simple regression; CI half-width $<$ PI half-width always, since the PI adds the individual observation's own variance $\sigma^2$ on top of the estimation uncertainty in the fitted line.

Why this works: The $F$-test and slope $t$-test are algebraically identical tests of the same null hypothesis $\beta_1=0$ in simple linear regression — they must agree, and did here.

Common pitfall: Reporting a point prediction without also reporting (and reading) its interval — a non-significant regression can still produce a plausible-looking point estimate that hides how little confidence the data actually support.

Source: Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th ed., §11.2–11.5 (least squares, hypothesis tests, CI/PI in simple regression).

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