Assignment 1

1 Clock-out times of the staff

1 (a) Mean, median, mode

$$\bar{t}=\frac{1}{17}\sum _{i=1}^{17}{t}_i=292.1\text{ min}\approx 4\text{ h}52.1\text{ min}$$

Median:

$$t_{\text{med}}=315\text{ min}=5:15\text{ pm}$$

Mode:

$$t_{\text{mode}}=330\text{ min}=5:30\text{ pm}$$

1 (b) Variance & standard deviation

$${\sigma }^2=\frac{1}{17}\sum _{i=1}^{17}(t_i-\bar{t}{)}^2=3.609\times 10^3{\text{min}}^2$$ $$\sigma =\sqrt{{\sigma }^2}=60.07\text{min}\approx 1.00\text{ h}$$

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2 Probability of drawing a white or green ball

Total balls:

$$4+6+3+8=21$$

Favourable (white + green):

$$4+8=12$$ $$P(\text{white or green})=\frac{12}{21}=\frac{4}{7}\approx 0.5714$$

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3 Distinct sums from two coins

Five coins: $0.01,0.05,0.10,0.25,0.50$ There are $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$ unordered pairs ⇒ 10 unique sums:

$$\{0.06,0.11,0.26,0.51,0.15,0.30,0.55,0.35,0.60,0.75\}\$$$

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4 Milled metal plate

4 (a) Nominal size and combined tolerance

Initial thickness:

$$1.200\pm 0.001\text{in}$$

Material removed:

$$0.200\pm 0.002\text{in}$$

Nominal result:

$$\mu =1.200-0.200=1.000\text{in}$$

Combined (RSS) tolerance:

$${\sigma }_{\text{tot}}=\sqrt{0.001^2+0.002^2}=0.00224\text{in}$$

Therefore:

$$t=1.000\pm 0.0022\text{in}$$

4 (b) Probability thickness > 1.0075 in

$$z=\frac{1.0075-1.000}{0.00224}=3.35$$ $$P(t>1.0075)=1-\Phi (3.35)\approx 4.0\times 10^{-4}(0.04\%)$$

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5 Students completing their program

Binomial: $X\sim \text{Binom}(n=6,p=0.60)$.

5 (a) More than half succeed ($X\ge 4$)

$$P=\sum _{k=4}^6\left(\genfrac{}{}{0ex}{}{6}{k}\right)p^k(1-p{)}^{6-k}=0.5443$$

5 (b) Plot

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6 Fair die, 120 throws

Let $X⟹(120,\frac{1}{6})$.

$$\mu =120\cdot \frac{1}{6}=20,\sigma =\sqrt{120\cdot \frac{1}{6}\cdot \frac{5}{6}}=4.082$$

Continuity-corrected $z$ for $X\le 18$:

$$z=\frac{18.5-20}{4.082}=-0.368$$ $$P(X\le 18)\approx \Phi (-0.368)=0.356$$

The exact binomial sum is

• $P(X\le 18)=0.3657$

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7 Choosing the cheapest machine setting

Specification limits: $9.88\le L\le 10.12\text{ mm}$ (tolerance ± 0.12 mm).
For each variance ${\sigma }^2$ compute

$$P_{\text{def}}=2[1-\Phi (\frac{0.12}{\sigma }\left)\right]$$

${\sigma }^2$ (mm²)	$\sigma$ (mm)	$P_{\text{def}}$	Defects in 10 000	Run Cost ($)	Total Cost ($)
0.0016	0.040	0.00270	27	800	827
0.0025	0.050	0.0164	164	500	664
0.0036	0.060	0.0455	455	50	505

Best choice: ${\sigma }^2=0.0036{\text{ mm}}^2$.