EXAMPLE 1 Find a normal probability

SOLUTION

The probability that a randomly selected x-value lies between – 2σ and is the shaded area under the normal curve shown.

xx

P( – 2σ ≤ x ≤ )x x

A normal distribution has mean x and standard deviation σ. For a randomly selected x-value from the distribution, find P(x – 2σ ≤ x ≤ x).

= 0.135 + 0.34 = 0.475

EXAMPLE 2 Interpret normally distribute data

Health

The blood cholesterol readings for a group of women are normally distributed with a mean of 172 mg/dl and a standard deviation of 14 mg/dl.

a. About what percent of the women have readings between 158 and 186?

Readings higher than 200 are considered undesirable. About what percent of the readings are undesirable?

b.

EXAMPLE 2 Interpret normally distribute data

SOLUTION

a. The readings of 158 and 186 represent one standard deviation on either side of the mean, as shown below. So, 68% of the women have readings between 158 and 186.

EXAMPLE 2 Interpret normally distribute data

b. A reading of 200 is two standard deviations to the right of the mean, as shown. So, the percent of

readings that are undesirable is 2.35% + 0.15%, or 2.5%.

GUIDED PRACTICE for Examples 1 and 2

A normal distribution has mean and standard deviation σ. Find the indicated probability for a randomly selected x-value from the distribution.

x

1. P( ≤ )x x

0.5ANSWER

GUIDED PRACTICE for Examples 1 and 2

2. P( > )x x

0.5ANSWER

GUIDED PRACTICE for Examples 1 and 2

3. P( < < + 2σ ) x x x

0.475ANSWER

GUIDED PRACTICE for Examples 1 and 2

4. P( – σ < x < ) x x

0.34ANSWER

GUIDED PRACTICE for Examples 1 and 2

5. P(x ≤ – 3σ)x

0.0015ANSWER

GUIDED PRACTICE for Examples 1 and 2

6. P(x > + σ)x

0.16ANSWER

GUIDED PRACTICE for Examples 1 and 2

7. WHAT IF? In Example 2, what percent of the women have readings between 172 and 200?

47.5%ANSWER