91Ó°ÊÓ

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.

(a)

The area from z = 0 to $z=1\mathrm{σ}$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 .

(b)

According to the Gaussian curve, the area from $z=0$to $z=2\mathrm{σ}$is 0.4773 .

So, the final answer is 0.9564 .

(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.)

(d)

The answer is 0.1915 .

(e)

The expression can be also shown as $\left[\mathrm{μ}to\mathrm{σ}\right]-\left[\mathrm{μ}to0.5\mathrm{σ}\right]$.

Since $\mathrm{μ}$to $\mathrm{σ}$is 0.3413 and $\mathrm{μ}$to $0.5\mathrm{σ}$is$0,1915$ , substitute these values to the new expression.

Hence, we get 0.1498 .