9 The strength of a thread is a random variable with mean 05

9. The strength of a thread is a random variable with mean 0.5 pound and standard deviation 0.2 pound. Assume the strength of a rope is the sum of the strengths of threads in the rope. a) Find the probability that a rope consisting of 100 threads will hold 45 pounds. B) How many threads are needed for a rope that will hold 50 pounds with 99% assurance?

Solution

A)

This is like asking the mean to be at least 45/100 = 0.45.

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    0.45      
u = mean =    0.5      
n = sample size =    100      
s = standard deviation =    0.2      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -2.5      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -2.5   ) =    0.993790335 [ANSWER]

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b)

For a 99% right tailed area, the corresponding z score is, by table/technology,

z = -2.326347874

As

z = (x-u)*sqrt(n)/sigma

As

x = 50/n

Then

-2.326347874 = (50/n - 0.5)*sqrt(n)/0.2

Solving for n,

n = 109.7 --> 110 [ANSWER]