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EXAMPLE 1

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

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EXAMPLE 1

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  1. A normal distribution has mean xand standard deviation σ. For a randomly selected x-value from the distribution, find P(x – 2σ ≤ x ≤ x). x x x x The probability that a randomly selected x-value lies between –2σ and is the shaded area under the normal curve shown. P(–2σ ≤ x ≤ ) EXAMPLE 1 Find a normal probability SOLUTION = 0.135 + 0.34 = 0.475

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

  3. 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 SOLUTION

  4. 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%. EXAMPLE 2 Interpret normally distribute data

  5. A normal distribution has mean and standard deviation σ. Find the indicated probability for a randomly selected x-value from the distribution. x x P(≤ ) x 1. ANSWER 0.5 for Examples 1 and 2 GUIDED PRACTICE

  6. P(> ) x 2. x ANSWER 0.5 for Examples 1 and 2 GUIDED PRACTICE

  7. P(<< + 2σ ) x 3. x x ANSWER 0.475 for Examples 1 and 2 GUIDED PRACTICE

  8. P( – σ<x<) x x ANSWER 0.34 for Examples 1 and 2 GUIDED PRACTICE 4.

  9. P(x ≤ – 3σ) 5. x ANSWER 0.0015 for Examples 1 and 2 GUIDED PRACTICE

  10. P(x > + σ) 6. x ANSWER 0.16 for Examples 1 and 2 GUIDED PRACTICE

  11. WHAT IF?In Example 2, what percent of the women have readings between 172 and 200? 7. ANSWER 47.5% for Examples 1 and 2 GUIDED PRACTICE

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