Question

Use Table 4-1 to state what fraction of a Gaussian population lies within the following intervals:

(a)$\mu \pm \sigma$

(b)$\mu \pm 2\sigma$

(c)$\mu$to$+\sigma$

(d)$\mu$to$+0.5\sigma$

(e)$-\sigma$to$-0.5\sigma$

Short answer

a) 0.6826

b) 0.9546

c) 0.3413

d) 0.1915

e) 0.1498

Step-by-step solution

Statement of Gaussian population.

A sample mean from an infinite population is approximately normal, or Gaussian, with a mean equal to the underlying population and variance equal to the population variance divided by the sample size, according to the Gaussian population theory.

Step  2: Find what fraction of a Gaussian population lies within the interval μ±σ

(a)

The area from z = 0 to $z=1\sigma$is 0.3143 .

However, in order to examine both the left (negative) and right (positive) sections of the curve, this must be multiplied by two.

So, the final answer is 0.6826 .

Find what fraction of a Gaussian population lies within the interval μ±2σ .

(b)

According to the Gaussian curve, the area from $z=0$to $z=2\sigma$is 0.4773 .

So, the final answer is 0.9564 .

Find what fraction of a Gaussian population lies within the interval μ to +σ .

(c)

According to the same table, the answer is 0.3413 .

(Note: Since just interested in the positive side of the curve, there's no need to multiply by two.)

Find what fraction of a Gaussian population lies within the interval μ to +0.5σ .

(d)

The answer is 0.1915 .

Find what fraction of a Gaussian population lies within the interval -σ to -0.5σ .

(e)

The expression can be also shown as $\left[\mu to\sigma \right]-\left[\mu to0.5\sigma \right]$.

Since $\mu$to $\sigma$is 0.3413 and $\mu$to $0.5\sigma$is$0,1915$ , substitute these values to the new expression.

Hence, we get 0.1498 .