Normal Curve Area Problems involving only z (not x yet) To accompany Hawkes lesson 6.2 A few slides are from Hawkes Plus much original content by D.R.S.

The probability of a random variable having a value in a given range is equal to the area under the curve in that region. HAWKES LEARNING SYSTEMS math courseware specialists Probability of a Normal Curve: Continuous Random Variables 6.2 Reading a Normal Curve Table The Key Idea behind all of this is that Probability IS Area !!! Shaded area Is (out of total area ) Probability is , too!

The probability of a random variable having a value in a given range is equal to the area under the curve in that region. HAWKES LEARNING SYSTEMS math courseware specialists Probability of a Normal Curve: Continuous Random Variables 6.2 Reading a Normal Curve Table This picture shows a normal distribution with mean μ=75 and std deviation σ=5. The probability that X > 80 is the same as the area under the curve to the right of x = 80.

This first simple basic batch of problems about area and probability These are all z problems, not involving x They are all worked with the Standard Normal Distribution curve – Where mean μ is the standard normals mean = 0 – And std. dev. σ is the standard normals st.dev = 1 Two ways to find the areas 1.With a printed table of values 2.With the TI-84 normalcdf( ) function

There are three basic problem types 1.What is the area to the LEFT of z = _____ ? – This is the same as probability P(z < ___ ) 2.What is the area to the RIGHT of z = ____ ? – This is the same as probability P(z > ___ ) 3.What is the area BETWEEN z = __ and __? – This is the same as probability P( ___ < z < ___ )

HAWKES LEARNING SYSTEMS math courseware specialists Standard Normal Distribution Table: Standard Normal Distribution Table from – to positive z z Continuous Random Variables 6.2 Reading a Normal Curve Table

HAWKES LEARNING SYSTEMS math courseware specialists Standard Normal Distribution Table (continued): Continuous Random Variables 6.2 Reading a Normal Curve Table 1.The standard normal tables reflect a z-value that is rounded to two decimal places. 2.The first decimal place of the z-value is listed down the left-hand column. 3.The second decimal place is listed across the top row. 4.Where the appropriate row and column intersect, we find the amount of area under the standard normal curve to the left of that particular z-value. When calculating the area under the curve, round your answers to four decimal places.

HAWKES LEARNING SYSTEMS math courseware specialists Area to the Left of z: Continuous Random Variables 6.2 Reading a Normal Curve Table

Area to left: P(z < 1.69); P(z < -2.03) With the printed table For P(z < 1.69) You should know to expect something > Look down to row 1.6 Look across to column 0.09 For P(z < -2.03) You should expect <.5000 Look down to row -2.0 Look across to column 0.03 With the TI-84

Area to the left of z=0: P(z < 0) You should know instantly that its.5000 because of – Total area = – Symmetry But just confirm it with table and TI-84 for now Note insignificant rounding error in TI-84

Area to the left of z = 4.2, z = -4.2 Very very little area way out in the extremities of the tails Almost 100% to the left of z = 4.2 Almost 0% to the left of z = -4.2

HAWKES LEARNING SYSTEMS math courseware specialists Area to the Right of z: Continuous Random Variables 6.2 Reading a Normal Curve Table

Finding area to the right of some z Probability P(z > ___ ) With the printed table 1.Find the area to the LEFT of that z value 2.Subtract total area minus area to the left equals area to the right With the TI-84 Its just normalcdf again Your z value is the low z Except this time its positive infinity for the high z

Find area to right of z = 3.02; z=-1.70 With the printed table Lookup area to the left of z = 3.02 is _____ So area to the right of z = 3.02 is _____ = _____ Lookup area to the left of z = is _____ So area to the right of z = is _____ = _____ With the TI-84

Find area to the right of z = 0, z = 5.1 P(z > 0) should be instantly known as P(z > 5.1) should be instantly known as 0 How about area to right of z = -5.1 ?

HAWKES LEARNING SYSTEMS math courseware specialists Area Between z 1 and z 2 : Continuous Random Variables 6.2 Reading a Normal Curve Table

Area between z = 1.16 and z = 2.31 With the printed table Area to the left of the higher z, ______ Minus area to the left of the lower z, ______ Equals the area between the two z values, ______ With the TI-84

Area between z = and z = 0.31 With the printed table Area to the left of the higher z, ______ Minus area to the left of the lower z, ______ Equals the area between the two z values, ______ With the TI-84

Area between z = and z = With the printed table Area to the left of the higher z, ______ Minus area to the left of the lower z, ______ Equals the area between the two z values, ______ With the TI-84

Area in two tails, outside of z=1.25 and z = 2.31 With the printed tables minus area between the two z values Or another way, area to left of lower z + area to right of higher z With the TI-84

Special: z = -1 and z = +1 Agrees with The Empirical Rule value of ____% So the area in the two tails is ____ % And the area in each is tail is ____%

Special: z = -2 and z = +2 Agrees with The Empirical Rule value of ____% And the area in the two tails is _____ % Therefore ____ % in each tail.

Special: z = -3 and z = +3 Agrees with The Empirical Rule value of ____% And the area in the two tails is _____ % Therefore ____ % in each tail.