Chapter 7 Section 2

1. Chapter 7Section 2 The Standard Normal Distribution

2. 1 2 3 Chapter 7 – Section 2 • Learning objectives • Find the area under the standard normal curve • Find Z-scores for a given area • Interpret the area under the standard normal curve as a probability

3. 1 2 3 Chapter 7 – Section 2 • Learning objectives • Find the area under the standard normal curve • Find Z-scores for a given area • Interpret the area under the standard normal curve as a probability

4. Chapter 7 – Section 2 • The standard normal curve is the one with mean μ = 0 and standard deviation σ = 1 • We have related the general normal random variable to the standard normal random variable through the Z-score • In this section, we discuss how to compute with the standard normal random variable

5. Chapter 7 – Section 2 • There are several ways to calculate the area under the standard normal curve • What does not work – some kind of a simple formula • We can use a table (such as Table IV on the inside back cover) • We can use technology (a calculator or software) • Using technology is preferred

6. Chapter 7 – Section 2 • Three different area calculations • Find the area to the left of • Find the area to the right of • Find the area between • Three different area calculations • Find the area to the left of • Find the area to the right of • Find the area between • Three different methods shown here • From a table • Using Excel • Using Statistical software

7. "To the left of" – using a table • Calculate the area to the left of Z = 1.68 • Break up 1.68 as 1.6 + .08 • Find the row 1.6 • Find the column .08 • Read answer at intersection of the two. • The probability is 0.9535 Enter Enter Read

8. Here is what is looks like on the normal distribution curve. • The blue area is 0.9535

9. To find area greater than 1.68 is 1 – .9535 = .0465 The red area is 0.0465 • The blue area is 0.9535

10. Chapter 7 – Section 2 • "To the right of" – using a table • The area to the left of Z = 1.68 is 0.9535 • The area to the left of Z = 1.68 is 0.9535 • The right of … that’s the remaining amount • The two add up to 1, so the right of is 1 – 0.9535 = 0.0465 Enter Enter Read

11. Chapter 7 – Section 2 • “Between” • Between Z = – 0.51 and Z = 1.87 • This is not a one step calculation

12. Included too much Included too much Chapter 7 – Section 2 • The left hand picture … to the left of 1.87 … includes too much • It is too much by the right hand picture … to the left of -0.51

13. We want We start out with, but it’s too much We correct by Chapter 7 – Section 2 • Between Z = – 0.51 and Z = 1.87

14. This area for 1.87 is 0.9693 This area for -.51 Is 0.3050 Chapter 7 – Section 2 • Between Z = – 0.51 and Z = 1.87 .9693- .3050= .6643

15. Area left of 1.87 is 0.9693 so area to the right is 1- 0.9693 This area for -.51 Is 0.3050 • We can use any of the three methods to compute the normal probabilities to get: • The area to the left is read directly from the chart • The area to the right of 1.87 is 1 minus area to the left.

16. The area between -0.51 and 1.87 • The area to the left of 1.87, or 0.9693 … minus • The area to the left of -0.51, or 0.3050 … which equals • The difference of 0.6643 • Thus the area under the standard normal curve between -0.51 and 1.87 is 0.6643 .9693- .3050= .6643

17. This area for -.51 Is 0.3050 Area left of 1.87 is 0.9693 so area to the right is 1- 0.9693 • We can use any of the three methods to compute the normal probabilities to get: • The area to the left is read directly from the chart • The area to the right of 1.87 is 1 minus area to the left. • The area between -0.51 and 1.87 The area to the left of 1.87= 0.9693 Minus area to the left of -0.51= 0.3050 Which equals the difference of 0.6643 .9693- .3050= .6643

18. 1 2 3 Chapter 7 – Section 2 • Learning objectives • Find the area under the standard normal curve • Find Z-scores for a given area • Interpret the area under the standard normal curve as a probability

19. Chapter 7 – Section 2 • We did the problem: Z-Score  Area • Now we will do the reverse of that Area  Z-Score • We did the problem: Z-Score  Area • Now we will do the reverse of that Area  Z-Score • This is finding the Z-score (value) that corresponds to a specified area (percentile) • And … no surprise … we can do this with a table, with Excel, with StatCrunch, with …

20. Find the Z-score for which the area to the left of it is 0.32 • Look in the middle of the table … find 0.32 • Find the Z-score for which the area to the left of it is 0.32 • Look in the middle of the table … find 0.32 • The nearest to 0.32 is 0.3192 … a Z-Score of -.47 Read Find Read Chapter 7 – Section 2 • “To the left of” – using a table • Find the Z-score for which the area to the left of it is 0.32

21. Read Read Enter Chapter 7 – Section 2 • "To the right of" – using a table • Find the Z-score for which the area to the right of it is 0.4332 • Right of it is .4332 … left of it would be .5668 • A value of .17

22. We will often want to find a middle range, to find the middle 90% or the middle 95% or the middle 99%, of the standard normal • The middle 90% would be

23. Chapter 7 – Section 2 • 90% in the middle is 10% outside the middle, i.e. 5% off each end • These problems can be solved in either of two equivalent ways • We could find • The number for which 5% is to the left, or • The number for which 5% is to the right

24. The two possible ways • The number for which 5% is to the left, or • The number for which 5% is to the right 5% is to the left 5% is to the right

25. Chapter 7 – Section 2 • The number zα is the Z-score such that the area to the right of zα is α • The number zα is the Z-score such that the area to the right of zα is α • Some useful values are • z.10 = 1.28, the area between -1.28 and 1.28 is 0.80 • z.05 = 1.64, the area between -1.64 and 1.64 is 0.90 • z.025 = 1.96, the area between -1.96 and 1.96 is 0.95 • z.01 = 2.33, the area between -2.33 and 2.33 is 0.98 • z.005 = 2.58, the area between -2.58 and 2.58 is 0.99

26. 1 2 3 Chapter 7 – Section 2 • Learning objectives • Find the area under the standard normal curve • Find Z-scores for a given area • Interpret the area under the standard normal curve as a probability

27. The area under a normal curve can be interpreted as a probability • The standard normal curve can be interpreted as a probability density function • We will use Z to represent a standard normal random variable, so it has probabilities such as • P(a < Z < b) The probability between two numbers • P(Z < a) The probability less than a number • P(Z > a) The probability greater than a number • The area under a normal curve can be interpreted as a probability • The standard normal curve can be interpreted as a probability density function

28. Summary: Chapter 7 – Section 2 • Calculations for the standard normal curve can be done using tables or using technology • One can calculate the area under the standard normal curve, to the left of or to the right of each Z-score • One can calculate the Z-score so that the area to the left of it or to the right of it is a certain value • Areas and probabilities are two different representations of the same concept

29. Were are done.