7.2The StandardNormal Distribution

Standard NormalThe 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 NormalThere 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 calculationsFind the area to the left ofFind the area to the right ofFind 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 1.6 + .08 Find the row 1.6 Find the column .08

(Table is IV on back cover)

● The probability is 0.9535

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 is1 – 0.9535 = 0.0465

“Between”

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

BetweenBetween 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 -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

A different “Between”Between Z = – 0.51 and Z = 1.87

We want

We delete theextra on the left

We delete theextra on the right

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

Z-Score●We did the problem:

Z-Score Area●Now we will do the reverse of

thatArea 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 0.3192 … a Z-Score of -.47

Z-Score"To the right of" – using a tableFind the Z-score for which the

area to the right of it is 0.4332Right of it is .4332 … left of it

would be .5668A value of .17

Middle RangeWe 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

Middle90% in the middle is 10% outside

the middle, i.e. 5% off each endThese problems can be solved in

either of two equivalent waysWe could find

◦The number for which 5% is to the left, or

◦The number for which 5% is to the right

MiddleThe 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

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●P(Z < 1.68) Normalcdf(small number, z,0,1)

Menu, 5:Probability, 2:Normal Cdf Lower Bound: Upper Bound: µ : :Normalcdf(

Calculator Method● "To the right of“ 1.68 – using a

calculator P(Z > 1.68)

Normalcdf(

BetweenBetween Z = – 0.51 and Z = 1.87P(-0.51 < Z < 1.87)

NormalCdf(

OrFind the area to the left of -1.56

or to the right of .79P(Z < -1.56) or P(Z > .79)

Find a Z-Score if given a probability● “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"To the right of" – using a

calculatorFind the Z-score for which the

area to the right of it is 0.4332Important: Calculator can only do

“left of” for inverse normal functions

Therefore, we need to convert this to a “left of”

Fun StuffSpend Time on this stuff…there is

a lot to remember and keep organized!

Practice makes perfect!