4. The circumferences of battery posts in a nickel-cadmium battery are Weibull-d

Question

4. The circumferences of battery posts in a nickel-cadmium battery are Weibull-distributed with = 3.25 centimeters, = 0.005 centimeters, and = 1/3.

(a) Find the probability that a battery post chosen at random will have a circumference larger than 3.40 centimeters.

(b) If battery posts are larger than 3.50 centimeters, they will not go through the hole provided; if they are smaller than 3.30 centimeters, the clamp will not tighten sufficiently. What proportion of posts will have to be scrapped for one of these reasons?

Explanation / Answer

Weibull Distribution
X ~ WEB(, ) to denote that the X has a Weibull
distribution with parameters and , X ~ WEB(0.005 , 0.33 )
a.
P(X >= 3.4) = e^-(3.4/0.33)^0.005
= e^-(10.303^0.005
= e^-(1.012)
= 0.364 [ Value of e = 2.718 ]
=0.364
b.
LESS THAN
P(X <= 3.3) = 1 - e^-(3.3/0.33)^0.005
= 1 - e^-(10)^0.005
= 1 - e^-(1.012)
= 1 - 0.364 [ Value of e = 2.718 ]
=0.636
GREATER THAN
P(X >= 3.5) = e^-(3.5/0.33)^0.005
= e^-(10.606^0.005
= e^-(1.012)
= 0.364 [ Value of e = 2.718 ]
=0.364
the proportion of posts will have to be scrapped for one of these reasons = P(X <= 3.3) + P(X >= 3.5)
= 0.636318 + 0.363573
= 0.999891

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