Given μ = 30, σ = 4
(a) P(X < 40)
When X = 40, Z = \(\frac { X-\mu }{ \sigma } \)
=\(\frac { 40-30 }{ 4 } =\frac { 10 }{ 4 } \) = 2.5
∴ P(X<40) = P(Z<2.5)

= P(-∞) = 0.5+.4938
P(X<40) = 0.9938
(b) P(X>21)
When X = 21, Z = \(\frac { 21-30 }{ 4 } \quad \)
= \(\frac { -9 }{ 4 } \) = -2.25
∴ P(X>21) = P(Z>-2.25)

= 0.4878 + 0.5
P(X > 21) = 0.9878
(c) P(30 When X = 30, Z₁ =\(\frac { 30-30 }{ 4 } \) = 0
When X = 35, Z₂ = \(\frac { 35-30 }{ 4 } \) = 1.25
P(30 = 0.3944

Given μ = 20, σ = 2
a) P(X<19.5)
When X = 19.5, Z = \(\frac { X-\mu }{ \sigma } \)
= \(\frac { 19.5-20 }{ 20 } \) = -0.25
∴ P(Z<-0.25) = P(-∞ =P(-∞

= 0.5-P(0 = 0.5-0.0987
P(X<19.5) = 0.4013
b) P(between 20 and 22 hours)
= P(20 When X = 20, Z₁ = \(\frac { 20-20 }{ 2 } \) = 0
When X = 22, Z₂ = \(\frac { 22-20 }{ 2 } \) = 1

∴ P (20 < X < 22) - P(0 < Z < 1)
∴ P (20 < X < 22) = 0.3413

Let p be the probability getting his piston rejected
Given p = 12% = \(\frac { 12 }{ 100 } \) = 0.12
∴ q = 1-p = 1-0.12 = 0.88
n = 10
(a) P (not more than 2 rejects)
= P(X≤2) = P(X = 0) + P(X = 1) + P(X = 2)
= 10C₀(0.12)⁰ (0.88)¹⁰ + 10C₁ (0.12)¹ (0.88)⁹ + 10C₂ (0.12)² (0.88)⁸
[∵ P(x) = nC_x p^xq^n-x]

= (0.88)⁸ [(0.88)² + 10(0.12) (0.88) + 45 (0.12)²]
= (0.88)⁸ [0.7744 + 1.056 + 0.648]
= (0.3596) (2.4784) = 0.8913
∴ Probability of not more than 2 rejects = 0.8913
b) P(at least 2 rejects)
= P(X ≥ 2) = 1 - P (X < 2)
= 1 - [P(X = 0) + P (X = 1)]
= 1-[10C₀(0.12)⁰ (0.88)¹⁰ + 10C₁ (0.12)¹ (0.88)⁹ ]
= 1 - [(0.88)¹⁰ + 10 (0.12) (0.88)⁹]
= 1 - (0.88)⁹ [0.88 + 1.2] = 1 - (0.31647) (2.08)
= 1 - 0.6583 = 0.34173
P (atleast 2 rejects) = 0.34173

Given μ = 16.28 seconds and σ = 0.12 second
P(X<16.35)
When X = 16.35, Z = \(\frac { X-\mu }{ \sigma } \)
= \(\frac { 16.35-16.28 }{ 0.12 } \)
= \(\frac { 0.007 }{ 0.12 } \) = 0.583
∴ P(X < 16.35) = (Z < 0.583)
= P(-∞  = 0.5 + 0.2190 = 0.7190

∴ Probability that it will take less than 16.35 sec to develop prints = 0.719.

Given μ = 12 and σ = 4
(1) P( ≤ 20)
When X = 20, Z = \(\frac { X-\mu }{ \sigma } \)
= \(\frac { 20-12 }{ 4 } =\frac { 8 }{ 4 } \) = 2
∴ P(X≤20) = P(Z≤2)
= P(-∞ = 0.5+0.4772
P(X≤20) = 0.9772

(ii) P(0≤X≤12)
When X=0, Z=\(\frac { X-\mu }{ \sigma } \)
\(\frac { 0-12 }{ 4 } =\frac { -12 }{ 4 } \)=-3
When X=12, Z=\(\frac { X-\mu }{ \sigma } \)
=\(\frac { 12-12 }{ 4 } =\frac { 0 }{ 4 } \)=0
∴ P(0≤X≤12) = P(-3≤Z≤0)
= P(0≤Z≤3) (By symmetry)
P(0≤X≤12) = 0.4987.