1. 7.2 The Standard Normal Distribution

2. Standard Normal 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

3. Standard Normal 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

4. Area Calculations ● Three different area calculations  Find the area to the left of  Find the area to the right of  Find the area between

5. Table Method ● "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  (Table is IV on back cover) ● The probability is 0.9535

6. Table Method ● "To the right of" – using a table ● 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

7. “Between” Between Z = – 0.51 and Z = 1.87 This is not a one step calculation

8. Between Between Z = – 0.51 and Z = 1.87 We want We start out with, but it’s too much We correct by

9. Table ● 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

10. A different “Between” Between Z = – 0.51 and Z = 1.87 We want We delete the extra on the left We delete the extra on the right

11. Different “Between” ● Again, we can use any of the three methods to compute the normal probabilities to get ● The area between -0.51 and 1.87  The area to the left of -0.51, or 0.3050 … plus  The area to the right of 1.87, or.0307 … which equals  The total area to get rid of which equals 0.3357 ● Thus the area under the standard normal curve between -0.51 and 1.87 is 1 – 0.3357 = 0.6643

12. 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)

13. Z-Score ● “To the left of” – using a table ● 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

14. Z-Score "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

15. Middle Range 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

16. Middle 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

17. Middle 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

18. Common Z-Scores ● The number z α is the Z-score such that the area to the right of z α 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

19. Terminology ● 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)  P(Z < a)  P(Z > a)

20. Calculator Method ● "To the left of" – using a calculator ● Calculate the area to the left of Z = 1.68  Normalcdf(small number, z,0,1)  2 nd Vars  Normalcdf( ● The probability is 0.9535

21. Calculator Method ● "To the right of“ 1.68 – using a calculator  Normalcdf(Z, big number,0,1)  2 nd Vars  Normalcdf(  0.0465

22. Between Between Z = – 0.51 and Z = 1.87 Normalcdf(low,high,0,1) Normalcdf(-.51,1.87,0,1).6642

23. Z-Score ● “To the left of” – using a Calculator ● Find the Z-score for which the area to the left of it is 0.32  InvNorm(.32,0,1)  Z-Score of -.47

24. Z-Score "To the right of" – using a calculator Find the Z-score for which the area to the right of it is 0.4332 Find the Complement of.4332 (1-.4332) InvNorm that number InvNorm(.5668,0,1) A value of.1682

25. Fun Stuff Spend Time on this stuff…there is a lot to remember and keep organized! Practice makes perfect!