Statistics for Engineers and Scientists
Statistics for Engineers and Scientists
4th Edition
ISBN: 9780073401331
Author: William Navidi Prof.
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 7, Problem 1SE

The Beer–Lambert law relates the absorbance A of a solution to the concentration C of a species in solution by A = MLC, where L is the path length and M is the molar absorption coefficient. Assume that L = 1 cm. Measurements of A are made at various concentrations. The data are presented in the following table.

Chapter 7, Problem 1SE, The BeerLambert law relates the absorbance A of a solution to the concentration C of a species in

  1. a. Let A = β ^ 0 + β ^ 1 C be the equation of the least-squares line for predicting absorbance (A) from concentration (C). Compute the values of β ^ 0 and β ^ 1 .
  2. b. What value does the Beer–Lambert law assign to β0?
  3. c. What physical quantity does β ^ 1 estimate?
  4. d. Test the hypothesis H0 : β0 = 0. Is the result consistent with the Beer–Lambert law?

a.

Expert Solution
Check Mark
To determine

Find the values of β^0 and β^1.

Answer to Problem 1SE

The values of β^0=0.0390 and β^1=1.0169.

Explanation of Solution

Calculation:

The given information is that the Beer–Lambert law relates the absorbance A of a solution to the concentration C of a species in solution by,

A=MLC

Where L is the path length and M is the molar absorption coefficient.

Least-squares line:

Software Procedure:

Step-by-step procedure to obtain the least-squares line using the MINITAB software is given below:

  • Choose Stat > Regression > Regression > Fit Regression Model.
  • In Responses, enter “Absorbance”.
  • In Continuous predictors, enter “Concentration”.
  • Check Results.
  • In Display of results, choose Simple tables.
  • Click OK.

Output using the MINITAB software is given below:

Statistics for Engineers and Scientists, Chapter 7, Problem 1SE

From the MINITAB output, β^0=0.0390 and β^1=1.0169.

b.

Expert Solution
Check Mark
To determine

Find the value does the Beer-Lambert law assign to β0.

Explanation of Solution

Calculation:

Here, the variable C is independent, the variable A is dependent, the length L is constant and M is the molar absorption coefficient.

The equation of the Beer-Lambert law is,

A=MLC (1)

The equation of the least-squares line for predicting absorbance (A) from concentration (C) is,

A=β^0+β^1C (2)

On comparing equation (1) and (2),

MLC=β^0+β^1CMC=β^0+β^1C(Since, L=1)

Thus, the coefficient β^1 estimates the molar absorption coefficient M. Therefore, the Beer-Lambert law assign to β0 is 0.

c.

Expert Solution
Check Mark
To determine

Find the physical quantity does β^1 estimate.

Explanation of Solution

Justification:

From part b., comparing equation (1) and (2) is, MC=β^0+β^1C.

Therefore, the coefficient β^1 estimates the molar absorption coefficient M.

d.

Expert Solution
Check Mark
To determine

Test the hypothesis H0:β0=0.

Check whether the result is consistent with the Beer-Lambert law.

Answer to Problem 1SE

Yes, the result is consistent with the Beer-Lambert law.

Explanation of Solution

Calculation:

State the null and alternative hypotheses.

Null hypothesis:

H0:β0=0

Alternative hypothesis:

H1:β00

From the MINITAB obtained in part a., the test statistic for slope is –0.43 and the P- value is 0.697.

Conclusion:

Here, the P-value is not small.

That is, the P-value is greater than the level of significance, 0.05.

Therefore, the null hypothesis is not rejected.

Hence, it can be concluded that β0=0.

Thus, the result is consistent with the Beer-Lambert law.

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Chapter 7 Solutions

Statistics for Engineers and Scientists

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