Quantitative Chemical Analysis
Quantitative Chemical Analysis
9th Edition
ISBN: 9781464135385
Author: Daniel C. Harris
Publisher: W. H. Freeman
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Question
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Chapter 4, Problem 4.2P

(a)

Interpretation Introduction

Interpretation:

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval has to be calculated.

Concept Introduction:

Gaussian curve:

The Gaussian curve is given by the formula:

y=1σe(xμ)2/2

Where,

μ is approximated by x¯

σ is approximated by s

e is the base of the natural logarithm (e=2.71828)

1/σ is normalization factor.

The deviations from mean value is expressed in multiples, z, of the standard deviation and which is given as follows:

z=xμσxx¯s

The area under the whole curve from z= to z=+ must be unity.

To Calculate: Using Table 4-1, the fraction of a Gaussian population that lies within the given interval

(a)

Expert Solution
Check Mark

Answer to Problem 4.2P

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval is 0.6826

Explanation of Solution

Given Interval:

μ±σ

Calculation of Gaussian Population:

The given interval μ±σ corresponds to z=1 to z=+1

From the table 4-1, we know that the area from z=0 to z=+1 is 0.3413

Similarly, the area from z=1 to z=0 is also 0.3413

Therefore, the total area from z=1 to z=+1 is calculated as below:

Total area=(z=0 to z=+1)+(z=1 to z=0)     =0.3413 + 0.3413     =0.6826

Hence, the fraction of a Gaussian population is 0.6826

Conclusion

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval is calculated as 0.6826

(b)

Interpretation Introduction

Interpretation:

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval has to be calculated.

Concept Introduction:

Gaussian curve:

The Gaussian curve is given by the formula:

y=1σe(xμ)2/2

Where,

μ is approximated by x¯

σ is approximated by s

e is the base of the natural logarithm (e=2.71828)

1/σ is normalization factor.

The deviations from mean value is expressed in multiples, z, of the standard deviation and which is given as follows:

z=xμσxx¯s

The area under the whole curve from z= to z=+ must be unity.

To Calculate: Using Table 4-1, the fraction of a Gaussian population that lies within the given interval

(b)

Expert Solution
Check Mark

Answer to Problem 4.2P

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval is 0.9546

Explanation of Solution

Given Interval:

μ±2σ

Calculation of Gaussian Population:

The given interval μ±2σ corresponds to z=2 to z=+2

From the table 4-1, we know that the area from z=0 to z=+2 is 0.4773

Similarly, the area from z=2 to z=0 is also 0.4773

Therefore, the total area from z=2 to z=+2 is calculated as below:

Total area=(z=0 to z=+2)+(z=2 to z=0)     =0.4773 + 0.4773     =0.9546

Hence, the fraction of a Gaussian population is 0.9546

Conclusion

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval is calculated as 0.9546

(c)

Interpretation Introduction

Interpretation:

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval has to be calculated.

Concept Introduction:

Gaussian curve:

The Gaussian curve is given by the formula:

y=1σe(xμ)2/2

Where,

μ is approximated by x¯

σ is approximated by s

e is the base of the natural logarithm (e=2.71828)

1/σ is normalization factor.

The deviations from mean value is expressed in multiples, z, of the standard deviation and which is given as follows:

z=xμσxx¯s

The area under the whole curve from z= to z=+ must be unity.

To Calculate: Using Table 4-1, the fraction of a Gaussian population that lies within the given interval

(c)

Expert Solution
Check Mark

Answer to Problem 4.2P

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval is 0.3413

Explanation of Solution

Given Interval:

μ to +σ

Calculation of Gaussian Population:

The given interval μ to +σ corresponds to z=0 to z=+1

From the table 4-1, we know that the area from z=0 to z=+1 is 0.3413

Therefore, the total area from z=0 to z=+1 is 0.3413

Hence, the fraction of a Gaussian population is 0.3413

Conclusion

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval is calculated as 0.3413

(d)

Interpretation Introduction

Interpretation:

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval has to be calculated.

Concept Introduction:

Gaussian curve:

The Gaussian curve is given by the formula:

y=1σe(xμ)2/2

Where,

μ is approximated by x¯

σ is approximated by s

e is the base of the natural logarithm (e=2.71828)

1/σ is normalization factor.

The deviations from mean value is expressed in multiples, z, of the standard deviation and which is given as follows:

z=xμσxx¯s

The area under the whole curve from z= to z=+ must be unity.

To Calculate: Using Table 4-1, the fraction of a Gaussian population that lies within the given interval

(d)

Expert Solution
Check Mark

Answer to Problem 4.2P

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval is 0.1915

Explanation of Solution

Given Interval:

μ to +0.5σ

Calculation of Gaussian Population:

The given interval μ to +0.5σ corresponds to z=0 to z=0.5

From the table 4-1, we know that the area from z=0 to z=0.5 is 0.1915

Therefore, the total area from z=0 to z=0.5 is 0.1915

Hence, the fraction of a Gaussian population is 0.1915

Conclusion

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval is calculated as 0.1915

(e)

Interpretation Introduction

Interpretation:

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval has to be calculated.

Concept Introduction:

Gaussian curve:

The Gaussian curve is given by the formula:

y=1σe(xμ)2/2

Where,

μ is approximated by x¯

σ is approximated by s

e is the base of the natural logarithm (e=2.71828)

1/σ is normalization factor.

The deviations from mean value is expressed in multiples, z, of the standard deviation and which is given as follows:

z=xμσxx¯s

The area under the whole curve from z= to z=+ must be unity.

To Calculate: Using Table 4-1, the fraction of a Gaussian population that lies within the given interval

(e)

Expert Solution
Check Mark

Answer to Problem 4.2P

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval is 0.1498

Explanation of Solution

Given Interval:

σ to  0.5σ

Calculation of Gaussian Population:

The given interval σ to  0.5σ corresponds to z=1 to z=0.5

From the table 4-1, we know that the area from z=1 to z=0 is 0.3413

The area from z=0.5 to z=0 is 0.1915

Therefore, the total area from z=1 to z=0.5 is calculated as below:

Total area=(z=1 to z=0)(z=0.5 to z=0)     =0.3413  0.1915     =0.1498

Hence, the fraction of a Gaussian population is 0.1498

Conclusion

Using Table 4-1, the fraction of a Gaussian population that lies within the given interval is calculated as 0.1498

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