## Areas Under the Normal Curve

Examples in this book have used one version of the z-table to find the area under the normal curve. This table provides the cumulative probabilities (or the area under the entire curve to the left of the value).

### Probability Example

Assume that the annual earnings per share (EPS) for a large sample of firms is normally distributed with a mean of \$5.00 and a standard deviation of \$ 1.50. What is the approximate probability of an observed EPS value falling between \$3.00 and \$7.25?

If EPS = x = \$7.25, then z = (x - jx)/o = (\$7.25 - \$5.00)/\$ 1.50 = +1.50

If EPS = x = \$3.00, then z = (x - y,)lc = (\$3.00 - \$5.00)/\$1.50 = -1.33

### Solving Using the Cumulative Z-Table

Forz-value of 1.50: Use the row headed 1.5 and the column headed 0 to find the value 0.9332. This represents the area under the curve to the left of the critical value 1.50.

For z-value of-1.33: Use the row headed 1.3 and the column headed 3 to find the value 0.9082. This represents the area under the curve to the left of the critical value +1.33. The area to the left of-1.33 is 1 - 0.9082 = 0.0918.

The area between these critical values is 0.9332 - 0.0918 = 0.8414, or 84.14%. Hypothesis Testing - One-Tailed Test Example

A sample of a stocks returns on 36 non-consecutive days results in a mean return of 2.0%. Assume the population standard deviation is 20.0%. Can we say with 95% confidence that the mean return is greater than 0%?

H0; ¡-i < 0.0%, Ha: jj, > 0.0%. The test statistic = z-statistic ■= -1=

0 0