Statistics for Engineers and Scientists
Statistics for Engineers and Scientists
4th Edition
ISBN: 9780073401331
Author: William Navidi Prof.
Publisher: McGraw-Hill Education
Question
Book Icon
Chapter 3.2, Problem 19E

a.

To determine

Find the uncertainty in the average of ten independent measurements of length of a component.

a.

Expert Solution
Check Mark

Answer to Problem 19E

The uncertainty in the average of ten independent measurements of length of a component is σX¯=0.016 mm_.

Explanation of Solution

Given info:

Ten independent measurements are made on the length of a component using an instrument. The uncertainty in the instrument is σX=0.05mm.

Justification:

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 linear combinations of measurements it is known that,

  • If X1,X2,...,Xn are independent measurements with mean μ and uncertainty σ then the mean and uncertainty of sample mean measurement X¯ is are μX¯=μ and σX¯=σn.

Here, the number of measurements made on the length of a component using the instrument is n=10 and uncertainty in the instrument is σX=0.05mm.

The uncertainty in the average of ten measurements of length of a component is,

σX¯=σXn=0.0510=0.016

Hence, the uncertainty in the average of ten measurements of length of a component is σX¯=0.016 mm_.

b.

To determine

Find the uncertainty in the average of five independent measurements of length of a component using the new measuring device.

b.

Expert Solution
Check Mark

Answer to Problem 19E

The uncertainty in the average of five independent measurements of length of a component using the new measuring device is σY¯=0.0089 mm_.

Explanation of Solution

Given info:

Five independent measurements are made on the length of a component using a new measuring device. The uncertainty in the new measuring device is σY=0.02mm.

Justification:

From the properties of linear combinations of measurements it is known that,

  • If X1,X2,...,Xn are independent measurements with mean μ and uncertainty σ then the mean and uncertainty of sample mean measurement X¯ is are μX¯=μ and σX¯=σn.

Here, the number of measurements made on the length of a component using the new measuring device is n=5 and uncertainty in the new measuring device is σY=0.02mm.

The uncertainty in the average of five measurements of length of a component is,

σY¯=σYn=0.025=0.0089

Hence, the uncertainty in the average of five measurements of length of a component is σY¯=0.0089 mm_.

c.

To determine

Find the uncertainty in X¯+Y¯2.

Find the uncertainty in (1015)X¯+(515)Y¯.

c.

Expert Solution
Check Mark

Answer to Problem 19E

The uncertainty in X¯+Y¯2 is σX¯+Y¯2=0.0091 mm_.

The uncertainty in (1015)X¯+(515)Y¯ is σ(1015)X¯+(515)Y¯=0.0111 mm_.

Explanation of Solution

Justification:

Here, the number of measurements made on the length of a component using the instrument is n=10 and uncertainty in the instrument is σX=0.05mm.

The uncertainty in the average of ten independent measurements of length of a component is σX¯=0.016 mm.

Here, the number of measurements made on the length of a component using the new measuring device is n=5 and uncertainty in the new measuring device is σY=0.02mm.

The uncertainty in the average of five independent measurements of length of a component using the new measuring device is σY¯=0.0089 mm.

From the properties of linear combinations of measurements it is known that,

  • If X1,X2,...,Xn are independent measurements and c1,c2,...,cn are constants then the uncertainty of the random variable c1X1+c2X2+...+cnXn is σc1X1+c2X2+...+cnXn=c12σX12+c22σX22+...+cn2σXn2.

Uncertainty inX¯+Y¯2:

The uncertainty of the random variable X¯+Y¯2 is,

σX¯+Y¯2=(12)2×σX¯2+(12)2×σY¯2=14×0.0162+14××0.00892=0.000825=0.0091

Thus, the standard deviation of random variable X¯+Y¯2 is 0.0091.

Hence, the uncertainty of the random variable X¯+Y¯2 is σX¯+Y¯2=0.0091 mm_.

Uncertainty in (1015)X¯+(515)Y¯:

The uncertainty of the random variable (1015)X¯+(515)Y¯ is,

σ(1015)X¯+(515)Y¯=(1015)2×σX¯2+(515)2×σY¯2=49×0.0162+19××0.00892=0.00012258=0.0111

Thus, the standard deviation of random variable (1015)X¯+(515)Y¯ is 0.0111.

Hence, the uncertainty of the random variable (1015)X¯+(515)Y¯ is σ(1015)X¯+(515)Y¯=0.0111 mm_.

d.

To determine

Find the value of c that minimizes the uncertainty in the weighted average cX¯+(1c)Y¯.

Find the uncertainty the weighted average cX¯+(1c)Y¯.

d.

Expert Solution
Check Mark

Answer to Problem 19E

The uncertainty in the weighted average cX¯+(1c)Y¯ is minimum for c=0.24_.

The uncertainty the weighted average cX¯+(1c)Y¯. is σcX¯+(1c)Y¯=0.007778 mm_.

Explanation of Solution

Justification:

The value of c that minimizes the weighted average of X¯andY¯:

From the properties of linear combinations of measurements it is known that,

  • If XandY are independent measurements of the same quantity with uncertainties σX and σY then the weighted average of the independent measurements XandY with minimum uncertainty is cX+(1c)Y. Here, the value of c is c=σY2σX2+σY2 and 1c=σX2σX2+σY2.

Here, the uncertainty in the average of ten independent measurements of length of a component is σX¯=0.016 mm and the uncertainty in the average of five independent measurements of length of a component using the new measuring device is σY¯=0.0089 mm.

Here, X¯andY¯ are two independent measurements made on same quantity.

Therefore, the weighted average of X¯andY¯ is cX¯+(1c)Y¯.

The value of c that minimizes the weighted average cX¯+(1c)Y¯ is,

c=σY2σX2+σY2=0.008920.00892+0.0162=0.23630.24

Thus, the uncertainty in the weighted average cX¯+(1c)Y¯ is minimum for c=0.24_.

Uncertainty in the weighted average cX¯+(1c)Y¯:

From the properties of linear combinations of measurements it is known that,

  • If X1,X2,...,Xn are independent measurements and c1,c2,...,cn are constants then the uncertainty of the random variable c1X1+c2X2+...+cnXn is σc1X1+c2X2+...+cnXn=c12σX12+c22σX22+...+cn2σXn2.

The uncertainty in the weighted average cX¯+(1c)Y¯ is,

σcX¯+(1c)Y¯=(0.24)2×σX¯2+(10.24)2×σY¯2=(0.24)2×0.0162+(0.76)2××0.00892=0.007778

Thus, the standard deviation of random variable cX¯+(1c)Y¯ is 0.007778.

Hence, the uncertainty the weighted average cX¯+(1c)Y¯. is σcX¯+(1c)Y¯=0.007778 mm_.

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!
Students have asked these similar questions
Using the accompanying Home Market Value data and associated regression​ line, Market ValueMarket Valueequals=​$28,416+​$37.066×Square ​Feet, compute the errors associated with each observation using the formula e Subscript ieiequals=Upper Y Subscript iYiminus−ModifyingAbove Upper Y with caret Subscript iYi and construct a frequency distribution and histogram.   LOADING... Click the icon to view the Home Market Value data.       Question content area bottom Part 1 Construct a frequency distribution of the​ errors, e Subscript iei.   ​(Type whole​ numbers.) Error Frequency minus−15 comma 00015,000less than< e Subscript iei less than or equals≤minus−10 comma 00010,000 0   minus−10 comma 00010,000less than< e Subscript iei less than or equals≤minus−50005000 5   minus−50005000less than< e Subscript iei less than or equals≤0 21   0less than< e Subscript iei less than or equals≤50005000 9…
The managing director of a consulting group has the accompanying monthly data on total overhead costs and professional labor hours to bill to clients. Complete parts a through c   Overhead Costs    Billable Hours345000    3000385000    4000410000    5000462000    6000530000    7000545000    8000
Using the accompanying Home Market Value data and associated regression​ line, Market ValueMarket Valueequals=​$28,416plus+​$37.066×Square ​Feet, compute the errors associated with each observation using the formula e Subscript ieiequals=Upper Y Subscript iYiminus−ModifyingAbove Upper Y with caret Subscript iYi and construct a frequency distribution and histogram. Square Feet    Market Value1813    911001916    1043001842    934001814    909001836    1020002030    1085001731    877001852    960001793    893001665    884001852    1009001619    967001690    876002370    1139002373    1131001666    875002122    1161001619    946001729    863001667    871001522    833001484    798001589    814001600    871001484    825001483    787001522    877001703    942001485    820001468    881001519    882001518    885001483    765001522    844001668    909001587    810001782    912001483    812001519    1007001522    872001684    966001581    86200

Chapter 3 Solutions

Statistics for Engineers and Scientists

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
Background pattern image
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,
Text book image
Glencoe Algebra 1, Student Edition, 9780079039897...
Algebra
ISBN:9780079039897
Author:Carter
Publisher:McGraw Hill
Text book image
College Algebra
Algebra
ISBN:9781938168383
Author:Jay Abramson
Publisher:OpenStax
Text book image
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
Text book image
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage