Answer:
(a) Mean = 122.9, σ = 30.071
(b) No. of failed specimens at less than 115k cycles are 27.
(c) μ = 39.07
Explanation:
We are given:
L 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210
f 2 1 3 5 8 12 6 10 8 5 2 3 2 1 0 1
(a) First we need to calculate the mean and standard deviation. The formula for calculating mean is:
Mean = ∑fx/∑f
And for standard deviation we have:
S.D. = √Var
Var = ∑fx²/∑f - (Mean)²
∑fx = (2*60) + (1*70) + (3*80) + (5*90) + (8*100) + (12*110) + (6*120) + (10*130) + (8*140) + (5*150) + (2*160) + (3*170) + (2*180) + (1*190) + (0*200) + (1*210)
= 120 + 70 + 240 + 450 + 800 + 1320 + 720 + 1300 + 1120 + 750 + 320 + 510 + 360 + 190 + 0 + 210
∑fx = 8480
Mean = ∑fx/∑f
= 8480/69
Mean = 122.9
∑fx² = (2*60²) + (1*70²) + (3*80²) + (5*90²) + (8*100²) + (12*110²) + (6*120²) + (10*130²) + (8*140²) + (5*150²) + (2*160²) + (3*170²) + (2*180²) + (1*190²) + (0*200²) + (1*210²)
=7200+4900+19200+40500+80000+145200+86400+169000+156800+112500+51200+86700+64800+36100+0+44100
∑fx² = 1104600
Var = ∑fx²/∑f - (Mean)²
= 1104600/69 - (122.9)²
= 16008.69565 - 15104.41
Var = 904.2856
S.D = √Var
σ = √904.2856
σ = 30.071
(b) Let X be the number of failed specimen.
We will use the z-score to calculate the probability. The formula for z-score is:
z = (X-μ)/σ
P(X<115) = P(z<(115-122.9)/30.071)
= P(z<-0.26)
Using the normal distribution probability table, we can compute the value of P(z<-0.26).
P(X<115) = 0.3974
So, no. of failed specimens at less than 115k cycles are: 0.3974*69 = 27 specimens
(c) σ = 30.071
P(x<115) = 0.99
P(z<(115-μ)/30.071) = 0.99
From the normal distribution table we find that 0.99 lies between the z values 2.52 and 2.33. Hence, we get 2.525 as the z-value at which the probability is 0.99.
z = (x-μ)/σ
2.525 = (115 - μ)/30.071
75.93 = 115 - μ
μ = 115 - 75.93
μ = 39.07
