The minimum force required to break a particular type of cable is normally distributed with mean 12,432 and standard deviation 25. A random sample of 400 cables of this type is selected. Calculate the probability that at least 349 of the selected cables will not break under a force of 12,400.

• 0.62
• 0.67
• 0.92
• 0.97
• 1.00

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

The probability that a randomly selected cable will not break under a force of 12,400 is

[[math]]\operatorname{P}(Y \gt 12, 400) =\gt P[ Z (12, 400 − 12, 432) / 25 =−1.28] =0.9. [[/math]]

The number of cables, [math]N[/math], that will not break has the binomial distribution with [math]n = 400 [/math] and [math]p = 0.9[/math]. This can be approximated by a normal distribution with mean 360 and standard deviation 6. With the continuity correction,

[[math]]\operatorname{P}( N ≥ 349) =≥ P[ Z (348.5 − 360) / 6 = −1.9167] = 0.97. [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.