Area under the curve, Part II. What percent of a standard

Problem 3.2

Area under the curve, Part II. What percent of a standard normal distribution N( = 0, = 1) is found in each region? Be sure to draw a graph.

(a) Z > 1.13 (b) Z < 0.18 (c) Z > 8 (d) |Z| < 0.5

                                                            Step by Step Solution

Step 1 of 4

(a)

If a normal distribution has mean ? and standard deviation , we may write the distribution as .

Given standard normal distribution is

The Z-score of an observation is defined as the number of standard deviations it falls above or below the mean. A normal probability table, which lists Z-scores and corresponding percentiles, can be used to identify a percentile based on the Z-score (and vice versa). A normal probability table is given in Appendix B.1. We use this table to identify the percentile corresponding to any particular Z-score.

Normal distribution of region in Z < 1.13 is 0.8708 (From normal probability table, Appendix B.1)

Z > 1.13 represents the region above Z = 1.13. That is one minus the area of  Z < 1.13, because the total area under the normal curve is always equal to 1. Therefore,

Z > 1.13 = 1 - 0.8708 = 0.1292 = 12. 92%

Corresponding region is shown in the graph below,