Solution Found!

Statistics / Statistics for Engineers and Scientists 4 / Chapter 4.5 / Problem 12E

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Change Question

Specifications for an aircraft bolt require that the

Chapter 4, Problem 12E

(choose chapter or problem)

Get Unlimited Answers

QUESTION:

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?

Questions & Answers

QUESTION:

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?

ANSWER:

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 $kN$ .

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

Strengths of these bolts are normally distributed.

Let $X$ represent the strength of these bolts.

we are given,

$P(X<18.3)\ =\ 0.1$ and $P(X>19.76)\ =\ 0.05$

Since $X$ is normally distributed, we can write,

$X\sim{}N(\mu{},\ {\sigma{}}^{2})$

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

Case 1: $P(X<18.3)\ =\ 0.1$

From the $Z$ table, the closest area to 0.10 is 0.1003, and corresponding $Z-score$ is $-1.28$ .

$z\ =\ \left({\frac{X\ -\ {\mu{}}_{\ }}{{\sigma{}}_{\ }}}\right)$

$-1.28\ =\ \left({\frac{18.3\ -\ {\mu{}}_{\ }}{{\sigma{}}_{\ }}}\right)$

$18.3\ -\ {\mu{}}_{\ }\ =\ -1.28{\sigma{}}_{\ }$

$\ {\mu{}}_{\ }-1.28{\sigma{}}_{\ }\ =\ 18.3$ ……………(1)

Case 2: $P(X>19.76)\ =\ 0.05$

$P(X<19.76)\ =1\ \ -\ \ 0.05\ =\ 0.95$

From the $Z$ table, the closest area to 0.95 is 0.9495, and corresponding $Z-score$ is $1.64$ .

$z\ =\ \left({\frac{X\ -\ {\mu{}}_{\ }}{{\sigma{}}_{\ }}}\right)$

$1.64\ =\ \left({\frac{19.76\ -\ {\mu{}}_{\ }}{{\sigma{}}_{\ }}}\right)$

$19.76\ -\ {\mu{}}_{\ }\ =\ 1.64{\sigma{}}_{\ }$

$\ {\mu{}}_{\ }+1.64{\sigma{}}_{\ }\ =\ 19.76$ ……………(2)

On solving equation (1) and (2), our mean $\mu{}\ =\ 18.94\ kN$ and standard deviation $\sigma{}\ =\ 0.5\ kN$ .

Hence mean $\mu{}\ =\ 18.94\ kN$ and standard deviation $\sigma{}\ =\ 0.5\ kN$ .

Study Tools You Might Need

Chapter: 2 Problem: 33E

Conceptual Physics 12
Problem 33E Consider the normal force on a book at rest on a tabletop. If the table is tilted so that the surface forms an inclined plane, will the magnitude of the normal force change? If so, how?

Chapter: 12 Problem: 12.157

Chemistry: Atoms First 1
Problem 157AP Which of the following compounds is most likely to exist as a liquid at room temperature: ethane (C2H6), hydrazine (N2H4), fluoromethane (CH3F)? Explain

Chapter: 4 Problem: 22E

Elementary Statistics: Picturing the World 6
Problem 22E Finding Probabilities Use the probability distribution you made in Exercise to find the probability of randomly selecting an employee whose overtime is (a) two or three hours, (b) three hours or less, and (c) between two and five hours, inclusive. Exercise Overtime Hours The number of o

Chapter: 1 Problem: 42P

Chemistry 11
How many seconds are there in a solar year (365.24 days)?

Chapter: 24 Problem: 18

Anatomy & Physiology 1
If an amino acid is not used to create new proteins, it can be ________. a. converted to acetyl CoA b. converted to glucose or ketones c. converted to nitrogen d. stored to be used later

Chapter: 50 Problem: 5

Campbell Biology 10
The transduction of sound waves into action potentials occurs a. within the tectorial membrane as it is stimulated by the hair cells. b. when hair cells are bent against the tectorial membrane, causing them to depolarize and release neurotransmitter that stimulates sensory neurons. c. as the basilar

Chapter: 9 Problem: 57

Applied Calculus for the Life and Social Sciences 1
In Exercises, find all second partial derivatives. \(f(x, y)=3 x^{2}-x y+2 y^{3}\)

Chapter: 36 Problem: 5

Principles of Economics 9
Advocates for setting monetary policy by rule rather than discretion often argue that a. central bankers with discretion are tempted to renege on their announced commitments to low inflation. b. central bankers following a rule will be more responsive to the needs of the political process. c. fiscal

Not The Solution You Need? Search for Your Answer Here:

×

Login

Login or Sign up for access to all of our study tools and educational content!

×

Register

Sign up for access to all content on our site!

Or login if you already have an account