lecture 10.4 b bt

Today’s Agenda

Attendance / Announcements

Sections 10.4b

Have a good Holiday!

Exam Schedule

Exam 5 (Ch 10)

Thur 12/5

Final Exam (All)

Thur 12/12 1 hour, 50 mins

The Normal Curve

The Normal Curve

The Normal Curve

The Normal Curve

The Normal Curve

Z-scores:

The Normal Curve

The following are synonymous

when it comes to the normal curve:

• Find the area under the curve …

• Find the percentage of the population

…

• Find the probability that …

The Normal Curve

Using a Z-Table to find probabilitiesNote: Our Z-table only gives area to the left

(or probabilities less than z)

The Normal CurveFind Probability that

z < 0.97

Z-scores: -2 -1 0 1 2 3-3

0.97

Find area under the

curve to the left of

z = 0.97)97.0(zP

Using a Z-Table to find probabilitiesFind Probability that

z < 0.97

Since z > 0, use

positive side

The Normal CurveFind Probability that

z < -2.91

Z-scores: -2 -1 0 1 2 3-3

-2.91

Find area under the

curve to the left of

z = -2.91

Using a Z-Table to find probabilitiesFind Probability that

z < -2.91

Since z < 0, use

negative side

Using a Z-Table to find probabilities

• Not all Z-Tables are alike!

Using a Z-Table to find probabilities

• But we can still use our z-table to

find areas to the right (probability

greater than), as well as areas

between two values (probability

between two values).

The Normal CurveFind Probability that

z > 0.75

Z-scores: -2 -1 0 1 2 3-3

0.75

Find area under the

curve to the right of

z = 0.75

Finding Area to the Right

• Two Methods

–Using the Complement of 1

–Using Symmetry

Complement MethodFind Probability that

z > 0.75

Z-scores: -2 -1 0 1 2 3-3

0.75

Find area under the

curve to the right of

z = 0.75

Complement MethodUse fact that area

under entire curve is 1.

And that we can find

area to the left

Z-scores: -2 -1 0 1 2 3-3

0.75

Find area under the

curve to the right of

z = 0.75

Complement MethodUse fact that area

under entire curve is 1.

And that we can find

area to the left

Z-scores: -2 -1 0 1 2 3-3

0.75

Find area under the

curve to the right of

z = 0.75

The Normal CurveFind Probability that

z > 0.75

Z-scores: -2 -1 0 1 2 3-3

0.75

Find area under the

curve to the right of

z = 0.75

Symmetry Method

Symmetry MethodUse symmetry of the

normal curve to find

area

Z-scores: -2 -1 0 1 2 3-3

0.75

Find area under the

curve to the right of

z = 0.75

- 0.75

Finding Area between two values

• Just use difference of the two

areas

Difference of AreaFind Probability that

-1.25 < z < 0.75

Z-scores: -2 -1 0 1 2 3-3

0.75

Find area under the

curve between -1.25

and 0.75

-1.25

Finding Probabilities of Normal Distributions

1. For data that is normally distributed, find the percentage of data items that are:

a) below z = 0.6b) above z = –1.8c) between z = –2 and –0.5

Finding Probabilities of Normal Distributions

2. Given a data set that is normally distributed, find the following probabilities:

a) P(0.32 ≤ z ≤ 3.18)b) P(z ≥ 0.98)

Solving Applications of Normal Distributions

Before solving real world applications of data that is normally distributed, we need to first calculate any appropriate z-scores based on the data. This is called normalizing the data.

Solving Applications of Normal Distributions

Systolic blood pressure readings are normally distributed with a mean of 121 and a standard deviation of 15. After converting each reading to its z-score, find the percentage of people with the following blood pressure readings:

a) below 142b) above 130c)between 142 and 154

Solving Applications of Normal Distributions

A machine produces bolts with an average diameter of 7 mm and a standard deviation of 0.25 mm. What is the probability that a bolt will have a diameter greater than 7.1 mm? Assume the distribution is normal.

Solving Applications of Normal Distributions

The placement test for a college has scores that are normally distributed with = 500 and = 100. If the college accepts only the top 10% of examinees, what is the cutoff score on the test for admission? (hint: you’ll need to use the table first, and work backwards)

Classwork / Homework

• 10.4 Worksheet

• Page 638

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