7.2. The Standard Normal Distribution

Transcription

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 Find the row 1.6 Find the column.08 (Table is IV on back cover) The probability is

6 Table Method "To the right of" using a table The area to the left of Z = 1.68 is The right of that s the remaining amount The two add up to 1, so the right of is =

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 and 1.87 The area to the left of 1.87, or minus The area to the left of -0.51, or which equals The difference of Thus the area under the standard normal curve between and 1.87 is

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 and 1.87 The area to the left of -0.51, or plus The area to the right of 1.87, or.0307 which equals The total area to get rid of which equals Thus the area under the standard normal curve between and 1.87 is =

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 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 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 and 1.28 is 0.80 z.05 = 1.64, the area between and 1.64 is 0.90 z.025 = 1.96, the area between and 1.96 is 0.95 z.01 = 2.33, the area between and 2.33 is 0.98 z.005 = 2.58, the area between 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

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

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

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25 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

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

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