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:

$y=\frac{\text{1}}{\text{σ}\sqrt{\text{2π}}}{e}^{-{(x-\text{μ)}}^{\text{2}}/{\text{2σ}}^{\text{2}}}$

Where,

$\text{μ}$ is approximated by $\overline{x}$

$\text{σ}$ is approximated by s

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

$1/\text{σ}\sqrt{\text{2π}}$ is normalization factor.

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

$z=\frac{x-\text{μ}}{\text{σ}}\approx \frac{x-\overline{x}}{s}$

The area under the whole curve from $z=-\infty$ to $z=+\infty$ must be unity.

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

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:

$\text{μ±σ}$

Calculation of Gaussian Population:

The given interval $\text{μ±σ}$ corresponds to $z=-1\text{ to }z=+1$

From the table 4-1, we know that the area from $z=0\text{ to }z=+1$ is $0.3413$

Similarly, the area from $z=-1\text{ to }z=0$ is also $0.3413$

Therefore, the total area from $z=-1\text{ to }z=+1$ is calculated as below:

$\begin{array}{l}\text{Total area}=(z=0\text{ to }z=+1)+(z=-1\text{ to }z=0)\\ \text{ =0}\text{.3413 + 0}\text{.3413}\\ \text{ =0}\text{.6826}\end{array}$

Hence, the fraction of a Gaussian population is $\text{0}\text{.6826}$

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:

$y=\frac{\text{1}}{\text{σ}\sqrt{\text{2π}}}{e}^{-{(x-\text{μ)}}^{\text{2}}/{\text{2σ}}^{\text{2}}}$

Where,

$\text{μ}$ is approximated by $\overline{x}$

$\text{σ}$ is approximated by s

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

$1/\text{σ}\sqrt{\text{2π}}$ is normalization factor.

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

$z=\frac{x-\text{μ}}{\text{σ}}\approx \frac{x-\overline{x}}{s}$

The area under the whole curve from $z=-\infty$ to $z=+\infty$ must be unity.

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

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:

$\text{μ±2σ}$

Calculation of Gaussian Population:

The given interval $\text{μ±2σ}$ corresponds to $z=-2\text{ to }z=+2$

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

Similarly, the area from $z=-2\text{ to }z=0$ is also $0.4773$

Therefore, the total area from $z=-2\text{ to }z=+2$ is calculated as below:

$\begin{array}{l}\text{Total area}=(z=0\text{ to }z=+2)+(z=-2\text{ to }z=0)\\ \text{ =0}\text{.4773 + 0}\text{.4773}\\ \text{ =0}\text{.9546}\end{array}$

Hence, the fraction of a Gaussian population is $\text{0}\text{.9546}$

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:

$y=\frac{\text{1}}{\text{σ}\sqrt{\text{2π}}}{e}^{-{(x-\text{μ)}}^{\text{2}}/{\text{2σ}}^{\text{2}}}$

Where,

$\text{μ}$ is approximated by $\overline{x}$

$\text{σ}$ is approximated by s

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

$1/\text{σ}\sqrt{\text{2π}}$ is normalization factor.

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

$z=\frac{x-\text{μ}}{\text{σ}}\approx \frac{x-\overline{x}}{s}$

The area under the whole curve from $z=-\infty$ to $z=+\infty$ must be unity.

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

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:

$\text{μ to +σ}$

Calculation of Gaussian Population:

The given interval $\text{μ to +σ}$ corresponds to $z=0\text{ to }z=+1$

From the table 4-1, we know that the area from $z=0\text{ to }z=+1$ is $0.3413$

Therefore, the total area from $z=0\text{ to }z=+1$ 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:

$y=\frac{\text{1}}{\text{σ}\sqrt{\text{2π}}}{e}^{-{(x-\text{μ)}}^{\text{2}}/{\text{2σ}}^{\text{2}}}$

Where,

$\text{μ}$ is approximated by $\overline{x}$

$\text{σ}$ is approximated by s

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

$1/\text{σ}\sqrt{\text{2π}}$ is normalization factor.

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

$z=\frac{x-\text{μ}}{\text{σ}}\approx \frac{x-\overline{x}}{s}$

The area under the whole curve from $z=-\infty$ to $z=+\infty$ must be unity.

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

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:

$\text{μ to +0}\text{.5σ}$

Calculation of Gaussian Population:

The given interval $\text{μ to +0}\text{.5σ}$ corresponds to $z=0\text{ to }z=0.5$

From the table 4-1, we know that the area from $z=0\text{ to }z=0.5$ is $0.1915$

Therefore, the total area from $z=0\text{ to }z=0.5$ 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:

$y=\frac{\text{1}}{\text{σ}\sqrt{\text{2π}}}{e}^{-{(x-\text{μ)}}^{\text{2}}/{\text{2σ}}^{\text{2}}}$

Where,

$\text{μ}$ is approximated by $\overline{x}$

$\text{σ}$ is approximated by s

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

$1/\text{σ}\sqrt{\text{2π}}$ is normalization factor.

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

$z=\frac{x-\text{μ}}{\text{σ}}\approx \frac{x-\overline{x}}{s}$

The area under the whole curve from $z=-\infty$ to $z=+\infty$ must be unity.

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

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:

$-\text{σ to }-\text{ 0}\text{.5σ}$

Calculation of Gaussian Population:

The given interval $-\text{σ to }-\text{ 0}\text{.5σ}$ corresponds to $z=-1\text{ to }z=-0.5$

From the table 4-1, we know that the area from $z=-1\text{ to }z=0$ is $0.3413$

The area from $z=-0.5\text{ to }z=0$ is $0.1915$

Therefore, the total area from $z=-1\text{ to }z=-0.5$ is calculated as below:

$\begin{array}{l}\text{Total area}=(z=-1\text{ to }z=0)-(z=-0.5\text{ to }z=0)\\ \text{ =0}\text{.3413 }-\text{ 0}\text{.1915}\\ \text{ =0}\text{.1498}\end{array}$

Hence, the fraction of a Gaussian population is $\text{0}\text{.1498}$

Conclusion

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