Statistics for Engineers and Scientists - With Access
Statistics for Engineers and Scientists - With Access
4th Edition
ISBN: 9781259275975
Author: Navidi
Publisher: MCG
Question
Chapter 3, Problem 1SE

a.

To determine

Find the uncertainty in U=XY+Z.

a.

Expert Solution
Check Mark

Answer to Problem 1SE

The uncertainty in U=XY+Z is σU=9_.

Explanation of Solution

Given info:

The form of the measurements of the variables X, Y and Z are as follows: X=25±1, Y=5.0±0.3 and Z=3.5±0.2.

Calculation:

The form of the measurements of a process is,

Measuredvalue(μ)±Standard deviation(σ).

Here, for a random sample the measured value will be sample mean and the population standard deviation will be sample standard deviation.

The form of the measurements of the variable X is X=25±1.

Here, the measured value or mean of the variable X is X=25 and the uncertainty in the variable X is σX=1.

The form of the measurements of the variable Y is Y=5.0±0.3.

Here, the measured value or mean of the variable Y is Y=5.0 and the uncertainty in the variable Y is σY=0.3.

The form of the measurements of the variable Z is Z=3.5±0.2.

Here, the measured value or mean of the variable Z is Z=3.5 and the uncertainty in the variable Z is σZ=0.2.

Uncertainty:

The uncertainty of a process is determined by the standard deviation of the measurements. In other words it can be said that, measure of variability of a process is known as uncertainty of the process.

Therefore, it can be said that uncertainty is simply (σ).

Standard deviation:

The standard deviation is based on how much each observation deviates from a central point represented by the mean. In general, the greater the distances between the individual observations and the mean, the greater the variability of the data set.

The general formula for standard deviation is,

s=xi2(xi)2nn1.

From the properties of uncertainties for functions of one measurement it is known that,

  • If X1,X2,...,Xn are independent measurements with uncertainties σX1,σX2,...,σXn and if U=U(X1,X2,..,Xn) is a function of X1,X2,...,Xn then the uncertainty in the variable U is σU=(UX1)2σX12+(UX2)2σX22+....+(UXn)2σXn2.

Uncertainty in U:

The uncertainty of the random variable U=XY+Z is,

σU=XY+Z=(UX)2σX2+(UY)2σY2+(UZ)2σZ2=((XY+Z)X)2σX2+((XY+Z)Y)2σY2+((XY+Z)Z)2σZ2=(Y)2σX2+(X)2σY2+σZ2=(5)2×(1)2+(25)2×(0.3)2+(1)2×(0.2)2

            =9

Thus, the uncertainty of the random variable U=XY+Z is σU=9_.

b.

To determine

Find the uncertainty in U=ZX+Y.

b.

Expert Solution
Check Mark

Answer to Problem 1SE

The uncertainty in U=ZX+Y is σU=0.0078_.

Explanation of Solution

Calculation:

From part (a), X=25, σX=1, Y=5.0, σY=0.3 and Z=3.5, σZ=0.2.

Uncertainty in U:

The uncertainty of the random variable U=ZX+Y is,

σU=ZX+Y=(UX)2σX2+(UY)2σY2+(UZ)2σZ2=((ZX+Y)X)2σX2+((ZX+Y)Y)2σY2+((ZX+Y)Z)2σZ2=(Z(X+Y)2)2σX2+(Z(X+Y)2)2σY2+(1X+Y)2σZ2=(3.5(25+5)2)2×(1)2+(3.5(25+5)2)2×(0.3)2+(125+5)2×(0.2)2

=0.0078

Thus, the uncertainty of the random variable U=ZX+Y is σU=0.0078_.

c.

To determine

Find the uncertainty in U=X(lnY+Z).

c.

Expert Solution
Check Mark

Answer to Problem 1SE

The uncertainty in U=X(lnY+Z) is σU=0.32_.

Explanation of Solution

Calculation:

From part (a), X=25, σX=1, Y=5.0, σY=0.3 and Z=3.5, σZ=0.2.

Uncertainty in U:

The uncertainty of the random variable U=X(lnY+Z) is,

σU=X(lnY+Z)=(UX)2σX2+(UY)2σY2+(UZ)2σZ2=((XlnY+XZ)X)2σX2+((XlnY+XZ)Y)2σY2+((XlnY+XZ)Z)2σZ2=(lnY+Z2X(lnY+Z))2σX2+(X2YX(lnY+Z))2σY2+(X2X(lnY+Z))2σZ2=((ln(5)+3.5225(ln(5)+3.5))2×(1)2+(252×525(ln(5)+3.5))2×(0.3)2+(25225(ln(5)+3.5))2×(0.2)2)

=0.32

Thus, the uncertainty of the random variable U=X(lnY+Z) is σU=0.32_.

d.

To determine

Find the uncertainty in U=XeZ22Y.

d.

Expert Solution
Check Mark

Answer to Problem 1SE

The uncertainty in U=XeZ22Y is σU=361.41_.

Explanation of Solution

Calculation:

From part (a), X=25, σX=1, Y=5.0, σY=0.3 and Z=3.5, σZ=0.2.

Uncertainty in U:

The uncertainty of the random variable U=XeZ22Y is,

σU=XeZ22Y=(UX)2σX2+(UY)2σY2+(UZ)2σZ2=((XeZ22Y)X)2σX2+((XeZ22Y)Y)2σY2+((XeZ22Y)Z)2σZ2=(eZ22Y)2σX2+(2X×eZ22Y)2σY2+(2XZ×eZ22Y)2σZ2=((e(3.5)22×5)2×(1)2+(2×25×e(3.5)22×5)2×(0.3)2+(2×25×3.5×e(3.5)22×5)2×(0.2)2)

                   =361.41

Thus, the uncertainty of the random variable U=XeZ22Y is σU=361.41_.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Chapter 3 Solutions

Statistics for Engineers and Scientists - With Access

Ch. 3.1 - The length of a rod was measured eight times. The...Ch. 3.2 - Assume that X and Y are independent measurements...Ch. 3.2 - The length of a rod is to be measured by a process...Ch. 3.2 - The volume of a cone is given by V = r2h/3, where...Ch. 3.2 - In the article The Worlds Longest Continued Series...Ch. 3.2 - A cylindrical hole is bored through a steel block,...Ch. 3.2 - A force of F = 2.2 0.1 N is applied to a block...Ch. 3.2 - The period T of a simple pendulum is given by...Ch. 3.2 - The specific gravity of a substance is given by G...Ch. 3.2 - Prob. 10ECh. 3.2 - According to Newtons law of cooling, the...Ch. 3.2 - Prob. 12ECh. 3.2 - Nine independent measurements are made of the...Ch. 3.2 - A certain scale has an uncertainty of 3 g and a...Ch. 3.2 - The volume of a rock is measured by placing the...Ch. 3.2 - A student measures the spring constant k of a...Ch. 3.2 - A certain chemical process is run 10 times at a...Ch. 3.2 - An object is weighed four times, and the results,...Ch. 3.2 - Prob. 19ECh. 3.2 - Prob. 20ECh. 3.3 - Find the uncertainty in Y, given that X = 2.0 0.3...Ch. 3.3 - Given that X and Y are related by the given...Ch. 3.3 - The volume of a cone is given by V = r2h/3, where...Ch. 3.3 - Prob. 4ECh. 3.3 - The period T of a simple pendulum is given by...Ch. 3.3 - The change in temperature of an iron bar brought...Ch. 3.3 - The friction velocity F of water flowing through a...Ch. 3.3 - The refractive index n of a piece of glass is...Ch. 3.3 - The density of a rock will be measured by placing...Ch. 3.3 - The conversion of ammonium cyanide to urea is a...Ch. 3.3 - Prob. 11ECh. 3.3 - Prob. 12ECh. 3.3 - The acceleration g due to gravity is estimated by...Ch. 3.3 - Refer to Exercise 4. Assume that T = 298.4 0.2 K....Ch. 3.3 - Refer to Exercise 5. a. Assume g = 9.80 m/s2...Ch. 3.3 - Refer to Exercise 6. Assume that c = 448 J/kgC and...Ch. 3.3 - Prob. 17ECh. 3.3 - Refer to Exercise 8. Assume the critical angle is...Ch. 3.3 - Refer to Exercise 9. Assume that the mass of the...Ch. 3.3 - Prob. 20ECh. 3.4 - Find the uncertainty in U, assuming that X = 10.0 ...Ch. 3.4 - The volume of a cone is given by V = r2h/3, where...Ch. 3.4 - From a fixed point on the ground, the distance to...Ch. 3.4 - Refer to Exercise 10 in Section 3.2. Assume that ...Ch. 3.4 - When air enters a compressor at pressure P1 and...Ch. 3.4 - One way to measure the water content of a soil is...Ch. 3.4 - Prob. 7ECh. 3.4 - Prob. 8ECh. 3.4 - The Beer-Lambert law relates the absorbance A of a...Ch. 3.4 - In the article Temperature-Dependent Optical...Ch. 3.4 - Refer to Exercise 12 in Section 3.2. Assume that 0...Ch. 3.4 - Prob. 12ECh. 3.4 - Archaeologists studying meat storage methods...Ch. 3.4 - Prob. 14ECh. 3.4 - A cylindrical wire of radius R elongates when...Ch. 3.4 - Prob. 16ECh. 3.4 - Refer to Exercise 16. In an experiment to...Ch. 3.4 - The vertical displacement v of a cracked slurry...Ch. 3.4 - The shape of a bacterium can be approximated by a...Ch. 3.4 - Prob. 20ECh. 3.4 - Refer to Exercise 10 in Section 3.2. Assume that ...Ch. 3.4 - Refer to Exercise 5. Assume that P1 = 15.3 0.2...Ch. 3.4 - Refer to Exercise 7. Assume that p = 4.3 0.1 cm...Ch. 3.4 - Prob. 24ECh. 3.4 - Refer to Exercise 12. Estimate n, and find the...Ch. 3.4 - Refer to Exercise 14. Assume that l = 10.0 cm ...Ch. 3.4 - Prob. 27ECh. 3.4 - Refer to Exercise 16. Assume that T0 = 73.1 0.1F,...Ch. 3.4 - Prob. 29ECh. 3.4 - Prob. 30ECh. 3.4 - Prob. 31ECh. 3 - Prob. 1SECh. 3 - Prob. 2SECh. 3 - Prob. 3SECh. 3 - Prob. 4SECh. 3 - Prob. 5SECh. 3 - Let A and B represent two variants (alleles) of...Ch. 3 - The heating capacity of a calorimeter is known to...Ch. 3 - Sixteen independent measurements were made of the...Ch. 3 - If two gases have molar masses M1 and M2, Grahams...Ch. 3 - A piece of plywood is composed of five layers. The...Ch. 3 - The article Effect of Varying Solids Concentration...Ch. 3 - Prob. 13SECh. 3 - Prob. 14SECh. 3 - Prob. 15SECh. 3 - The mean yield from process A is estimated to be...Ch. 3 - The flow rate of water through a cylindrical pipe...Ch. 3 - Prob. 18SECh. 3 - The decomposition of nitrogen dioxide (NO2) into...Ch. 3 - Prob. 20SECh. 3 - A track has the shape of a square capped on two...Ch. 3 - Prob. 22SECh. 3 - Prob. 23SE
Knowledge Booster
Recommended textbooks for you
  • College Algebra
    Algebra
    ISBN:9781305115545
    Author:James Stewart, Lothar Redlin, Saleem Watson
    Publisher:Cengage Learning
    College Algebra
    Algebra
    ISBN:9781938168383
    Author:Jay Abramson
    Publisher:OpenStax
    Algebra: Structure And Method, Book 1
    Algebra
    ISBN:9780395977224
    Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
    Publisher:McDougal Littell
  • College Algebra
    Algebra
    ISBN:9781305115545
    Author:James Stewart, Lothar Redlin, Saleem Watson
    Publisher:Cengage Learning
    College Algebra
    Algebra
    ISBN:9781938168383
    Author:Jay Abramson
    Publisher:OpenStax
    Algebra: Structure And Method, Book 1
    Algebra
    ISBN:9780395977224
    Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
    Publisher:McDougal Littell