EXAMPLE 30Case 3: Find the area between two (checking the accuracy of the Empirical Rule)

Recall that the Empirical Rule (page 135 of Chapter 3) states that about 68% of the area under the curve lies within 1 standard deviation of the mean, that is, between and . Check this result for the standard normal distribution by using the table.

Solution

For the standard normal random variable and , so that and . Thus, using Case 3, we have and .

  • Step 1 Draw the standard normal curve. Label the and .
  • Step 2 Shade the area between − 1 and 1, as shown in Figure 32a.
  • Step 3 Find the area to the left of and the area to the left of . The table gives these areas as follows: area to the left of is 0.1587, and area to the left of is 0.8413. We subtract the smaller area from the larger to give us the area between − 1 and 1, as shown in Figures 32a–32c.
    image
    Figure 6.34: FIGURE 32a To get the area we are looking for…FIGURE 32b Find the area to the left of 1…FIGURE 32c And subtract the area to the left of −1.

NOW YOU CAN DO

Exercises 49–58.