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# Specifications for an aircraft bolt require that the

ISBN: 9780073401331 38

## Solution for problem 12E Chapter 4.5

Statistics for Engineers and Scientists | 4th Edition

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Problem 12E

Specifications for an aircraft bolt require that the ultimate tensile strength be at least 18 kN. It is known that 10% of the bolts have strengths less than 18.3 kN and that 5% of the bolts have strengths greater than 19.76 kN. It is also known that the strengths of these bolts are normally distributed.

a. Find the mean and standard deviation of the strengths.

b. What proportion of the bolts meet the strength specification?

Step-by-Step Solution:

Answer:

Step 1 of 2:

(a)

In this question, we are asked to find the mean and standard deviation of the strengths.

Specifications for an aircraft bolt require that the ultimate tensile strength be at least 18.

Given 10% of the bolts have strengths less than 18.3and that 5% of the bolts have strengths greater than 19.76

Strengths of these bolts are normally distributed.

Let  represent the strength of these bolts.

we are given,

and

Since  is normally distributed, we can write,

Now we will use the formula to calculate the mean and standard deviation, since our probability (area) value is given.

Case 1:

From the  table, the closest area to 0.10 is 0.1003, and corresponding   is .

……………(1)

Case 2:

From the  table, the closest area to 0.95 is 0.9495, and corresponding   is .

……………(2)

On solving equation (1) and (2), our mean and standard deviation .

Hence mean and standard deviation .

Step 2 of 2

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Specifications for an aircraft bolt require that the