Normal Curve Area Problems involving only z (not x yet) To accompany Hawkes lesson 6.2 A few slides...

Normal Curve Area Problemsinvolving only z (not x yet)

To accompany Hawkes lesson 6.2A 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 thatProbability IS Area !!!

Shaded areaIs 0.1587(out of totalarea 1.0000)

Probability is 0.1587, 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.



Probability of a Normal Curve:



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 normal’s mean = 0– And std. dev. σ is the standard normal’s st.dev = 1

• Two ways to find the areas1. With a printed table of values2. 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 < ___ )



Standard Normal Distribution Table:

Standard Normal Distribution Table from – to positive z

z 0.00 0.01 0.02 0.03 0.04

0.0 0.5000 0.5040 0.5080 0.5120 0.5160

0.1 0.5398 0.5438 0.5478 0.5517 0.5557

0.2 0.5793 0.5832 0.5871 0.5910 0.5948

0.3 0.6179 0.6217 0.6255 0.6293 0.6331

0.4 0.6554 0.6591 0.6628 0.6664 0.6700

0.5 0.6915 0.6950 0.6985 0.7019 0.7054

0.6 0.7257 0.7291 0.7324 0.7357 0.7389

0.7 0.7580 0.7611 0.7642 0.7673 0.7704

0.8 0.7881 0.7910 0.7939 0.7967 0.7995





Standard Normal Distribution Table (continued):



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.



Area to the Left of z:



“Area to left”: P(z < 1.69); P(z < -2.03)

With the printed tableFor P(z < 1.69)• You should know to expect

something > 0.5000• Look down to row 1.6• Look across to column 0.09For 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 it’s .5000 because of– Total area = 1.00000000– 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



Area to the Right of z:



Finding area to the right of some zProbability P(z > ___ )

With the printed table1. Find the area to the LEFT of

that z value2. Subtract 1.0000 total area

minus area to the leftequals area to the right

With the TI-84• It’s just normalcdf again• Your z value is the low z• Except this time it’s 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 1.0000 - _____ = _____

• Lookup area to the left of z = -1.70 is _____

• So area to the right of z = -1.70 is 1.0000 - _____ = _____

With the TI-84

Find area to the right of z = 0, z = 5.1

• P(z > 0) should be instantly known as 0.5000• P(z > 5.1) should be instantly known as ≈0• How about area to right of z = -5.1 ?



Area Between z1 and z2:



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 = -2.76 and z = 0.31





With the TI-84

Area between z = -3.01 and z = -1.33





With the TI-84

Area in two tails, outside of z=1.25 and z = 2.31

With the printed tables• 1.0000 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 ____%


• Agrees with The Empirical Rule value of ____%• And the area in the two tails is _____ %• Therefore ____ % in each tail.

Normal Curve Area Problems involving only z (not x yet) To accompany Hawkes lesson 6.2 A few slides...

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