The number of equipment breakdowns in a manufacturing plant is closely monitored

Question

The number of equipment breakdowns in a manufacturing plant is closely monitored by the supervisor of operations, because it is critical to the production process. The number averages five per week with a standard deviation of 0.8 per week.

a Find an interval that includes at least 90% of the weekly figures for the number of breakdowns.
b The supervisor promises that the number of breakdowns will rarely exceed 8 in a 1-week period. Is the director safe in making this claim? Why?

I know that the answer for a is:

I am also aware that Chebychev's Theorem must be used to solve the problem.

Please answer both a and b, showing your work!! Thank you.

Explanation / Answer

a)

By Chebyshev's Theorem, at least

1-1/k^2

lie within k standard deviations. Hence,

1 - 1/k^2 = 0.9

1/k^2 = 0.1

k = 3.16227766

Hence,

u - k*sigma = 5 - 3.16227766*0.8 = 2.470177872
u + k*sigma = 5 + 3.16227766*0.8 = 7.529822128

Hence, the widest integer interval between these two numbers is (3, 7). [ANSWER]

*************************

b)

Yes, because 8 is outside the interval in part a.

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