Question

Table 1: Cumulative distribution function of the standard Normal distribution
z: 0 1 2 3 Probability to the left of z: .5000 .84134 .97725 .99865 Probability to the right of z: .5000 .15866 .02275 .00135 Probability between z and z: .6827 .9544 .99730
Table 2: Inverse of the cumulative distribution function of the standard Normal distribution
Probability to the left of z: . 5000 .92 .95 .975 .9990 z: 0.00 1.405 1.645 1.960 3.09
1 Normal Distributions
1. What proportion of a Normal distribution is within one standard deviation of the mean?
2. What proportion of a Normal distribution is within two standard deviations of the mean ?
3. What proportion of a Normal distribution is within three standard deviations of the mean ?
4. What is the 95th percentile of the standard Normal distribution?
5. What is the area to the right of z = 2 in a standard Normal distribution?

Answer 1

1)

proportion of a Normal distribution is within one standard deviation of the mean =P(-1<Z<1)

=0.84134-0.15866=0.68268

2)

proportion of a Normal distribution is within two standard deviations of the mean=P(-2<Z<2)

=0.97725-0.02275=0.9545

3)

proportion of a Normal distribution is within three standard deviations of the mean=P(-3<Z<3)

=0.99865-0.00135=0.9973

4)

95th percentile of the standard Normal distribution ; z =1.645

5)

area to the right of z = 2 in a standard Normal distribution =0.02275