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

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