1. CONTINUOUS RANDOM VARIABLES AND THE NORMAL DISTRIBUTION

2. 2 CONTINUOUS PROBABILITY DISTRIBUTION Two characteristics 1. The probability that x assumes a value in any interval lies in the range 0 to 1 2. The total probability of all the intervals within which x can assume a value of 1.0

3. 3 Area under a curve between two points. x = a x = b x Shaded area is between 0 and 1

4. 4 Total area under a probability distribution curve. Shaded area is 1.0 or 100% x

5. 5 Area under the curve as probability. a b x Shaded area gives the probability P ( a ≤ x ≤ b )

6. 6 The probability of a single value of x is zero.

7. 7 THE NORMAL DISTRIBUTION  Normal Probability Distribution  A normal probability distribution, when plotted, gives a bell-shaped curve such that 1. The total area under the curve is 1.0. 2. The curve is symmetric about the mean. 3. The two tails of the curve extend indefinitely.

8. 8 Normal distribution with mean μ and standard deviation σ. Standard deviation = σ Mean = μ x

9. 9 Total area under a normal curve. The shaded area is 1.0 or 100% μx

10. 10 A normal curve is symmetric about the mean. Each of the two shaded areas is.5 or 50%.5 μx

11. 11 Areas of the normal curve beyond μ ± 3σ. μ – 3σ μ – 3σ μ + 3 σ Each of the two shaded areas is very close to zero μ x

12. 12 Three normal distribution curves with the same mean but different standard deviations. σ = 5 σ = 10 σ = 16 x μ = 50

13. 13 Three normal distribution curves with different means but the same standard deviation. σ = 5 σ = 5 σ = 5 µ = 20 µ = 30 µ = 40 x

14. 14 THE STANDARD NORMAL DISTRIBTUION  Definition  The normal distribution with μ = 0 and σ = 1 is called the standard normal distribution.

15. 15 The standard normal distribution curve. σ = 1 µ = 0 -3 -2 -1 0 1 2 3 z

16. 16 THE STANDARD NORMAL DISTRIBTUION  Definition  The units marked on the horizontal axis of the standard normal curve are denoted by z and are called the z values or z scores. A specific value of z gives the distance between the mean and the point represented by z in terms of the standard deviation.

17. 17 Area under the standard normal curve. -3 -2 -1 0 1 2 3 z. 5 Each of these two areas is.5

18. 18 Examples Using The Standard Normal Table …

19. 19 STANDARDIZING A NORMAL DISTRIBUTION  Converting an x Value to a z Value  For a normal random variable x, a particular value of x can be converted to its corresponding z value by using the formula  where μ and σ are the mean and standard deviation of the normal distribution of x, respectively.

20. 20 Examples … IQ ~ Normal mean = 100 stdev = 15 …

21. 21 DETERMINING THE z AND x VALUES WHEN AN AREA UNDER THE NORMAL DISTRIBUTION CURVE IS KNOWN Examples …

22. 22 Finding an x Value for a Normal Distribution For a normal curve, with known values of μ and σ and for a given area under the curve the x value is calculated as x = μ + zσ