1. 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions 4. The Standard Normal Distribution

2. 2 Lecture 6 1. Random Variables Two kinds of random variables: a. Discrete (DRV)  Outcomes have countable values  Possible values can be listed  E.g., # of people in this room  Possible values can be listed: might be …28 or 29 or 30…

3. 3 Lecture 6 1. Random Variables Two kinds of random variables: b. Continuous (CRV)  Not countable  Consists of points in an interval  E.g., time till coffee break

4. 4 Lecture 6 1. Random Variables  The form of the probability distribution for a CRV is a smooth curve. Such a distribution may also be called a  Frequency Distribution  Probability Density Function

5. 5 Lecture 6 1. Random Variables  In the graph of a CRV, the X axis is whatever you are measuring (e.g., exam scores, depression scores, # of widgets produced per hour).  The Y axis measures the frequency of scores.

6. 6 Lecture 6 X The Y-axis measures frequency. It is usually not shown.

7. 7 Lecture 6 2. Symmetric Distributions In a symmetric CRV, 50% of the area under the curve is in each half of the distribution. P(x ≤  ) = P(x ≥  ) =.5  Note: Because points are infinitely thin, we can only measure the probability of intervals of X values – not of individual X values.

8. 8 Lecture 6 µ 50% of area

9. 9 Lecture 6 3. Normal Distributions  A particularly important set of CRVs have probability distributions of a particular shape: mound-shaped and symmetric. These are “normal distributions”  Many naturally-occurring variables are normally distributed.

10. 10 Lecture 6 Normal Distributions  are perfectly symmetrical around their mean, .  have the standard deviation, , which measures the “spread” of a distribution – an index of variability around the mean.

11. 11 Lecture 6 µ 

12. 12 Lecture 6 Standard Normal Distribution  The area under the curve between  and some value X ≥  has been calculated for the “standard normal distribution” and is given in the Z table (Table IV).  E.g., for Z = 1.62, area =.4474  (Note that for the mean, Z = 0.)

13. 13 Lecture 6  XZ = 1.62 Z = 0 Area gives the probability of finding a score between the mean and X when you make an observation.4474

14. 14 Lecture 6 Using the Standard Normal Distribution  Suppose average height for Canadian women is 160 cm, with  = 15 cm.  What is the probability that the next Canadian woman we meet is more than 175 cm tall?  Note that this is a question about a single case and that it specifies an interval.

15. 15 Lecture 6 Using the Standard Normal Distribution 160175 We need this areaTable gives this area

16. 16 Lecture 6 Remember that area above the mean, , is half (.5) of the distribution. µ

17. 17 Lecture 6 Using the Standard Normal Distribution 160175 Call this shaded area P. We can get P from Table IV

18. 18 Lecture 6 Using the Standard Normal Distribution Z = X -  = 175-160  15 = 1.00 Now, look up Z = 1.00 in the table. Corresponding area (= probability) is P =.3413.

19. 19 Lecture 6 Using the Standard Normal Distribution 160175 This area is.3413 So this area must be.5 –.3413 =.1587

20. 20 Lecture 6 Using the Standard Normal Distribution Z = 0Z = 1.0 This area is.3413 So this area must be.5 –.3413 =.1587

21. 21 Lecture 6 Using the Standard Normal Distribution  What is the probability that the next Canadian woman we meet is more than 175 cm tall?  Answer:.1587

22. 22 Lecture 6 Review  Area under curve gives probability of finding X in a given interval.  Area under the curve for Standard Normal Distribution is given in Table IV.  For area under the curve for other normally- distributed variables first compute: Z = X -   Then look up Z in Table IV.