7.2 The Standard Normal Distribution

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

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

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

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

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 1 – =

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

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

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

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

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 1 – =

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)

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

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

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

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

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

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

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)

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

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

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

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

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

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