7.6 The Normal Distribution

1. 7.6 The Normal Distribution • Normal Curve • Normally Distributed Outcomes • Properties of Normal Curve • Standard Normal Curve • The Normal Distribution • Percentile • Probability for General Normal Distribution

2. Normal Curve • The bell-shaped curve, as shown below, is call a normal curve.

3. Normally Distributed Outcomes • Examples of experiments that have normally distributed outcomes: • 1. Choose an individual at random and observe his/her IQ. • 2. Choose a 1-day-old infant and observe his/her weight. • 3. Choose a leaf at random from a particular tree and observe its length.

4. Properties of Normal Curve

5. Example Properties of Normal Curve • A certain experiment has normally distributed outcomes with mean equal to 1. Shade the region corresponding to the probability that the outcome • (a) lies between 1 and 3; • (b) lies between 0 and 2; • (c) is less than .5; • (d) is greater than 2.

6. Example Properties of Normal Curve (2)

7. Standard Normal Curve • The equation of the normal curve is The standard normal curve has

8. The Normal Distribution A(z) is the area under the standard normal curve to the left of a normally distributed random variable z.

9. Example The Normal Distribution • Use the normal distribution table to determine the area corresponding to • (a) z< -.5; • (b) 1< z < 2; • (c) z> 1.5.

10. Example The Normal Distribution (2) • (a) A(-.5) = .3085 • (b) A(2) - A(1) = .9772 - 8413 • = .1359 • (c) 1 - A(1.5) = 1 - .9332 • = .0668

11. Percentile • If a score S is the pth percentile of a normal distribution, then p% of all scores fall below S, and (100 - p)% of all scores fall above S. The pth percentile is written as zp.

12. Example Percentile • What is the 95th percentile of the standard normal distribution? • In the normal distribution, find the value of z such that A(z) = .95. • A(1.65) = .9506 • Therefore, z95 = 1.65.

13. Probability for General Normal Distribution • If X is a random variable having a normal distribution with mean and standard deviation then • where Z has the standard normal distribution and A(z) is the area under that distribution to the left of z.

14. Example Probability Normal Distribution • Find the 95th percentile of infant birth weights if infant birth weights are normally distributed with = 7.75 and = 1.25 pounds. • The value for the standard normal random variable is z95 = 1.65. • Then x95 = 7.75 + (1.65)(1.25) = 9.81 pounds.

15. Summary Section 7.6 - Part 1 • A normal curve is identified by its mean ( ) and its standard deviation ( ). The standard normal curve has = 0 and = 1. Areas of the region under the standard normal curve can be obtained with the aid of a table or graphing calculator.

16. Summary Section 7.6 - Part 2 • A random variable is said to be normally distributed if the probability that an outcome lies between a and b is the area of the region under a normal curve from x = a to x = b. After the numbers a and b are converted to standard deviations from the mean, the sought-after probability can be obtained as an area under the standard normal curve.