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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 probabilities
Note: Our Z-table only gives area to the left
(or probabilities less than z)
The Normal Curve
Find Probability that
z < 0.97

P(z

Find area under the
curve to the left of
z = 0.97

0.97)

Z-scores:

-3

-2

-1

0

1
0.97

2

3
Using a Z-Table to find probabilities
Find Probability that
z < 0.97

Since z > 0, use
positive side
The Normal Curve
Find Probability that
z < -2.91

Z-scores:

-3
-2.91

-2

Find area under the
curve to the left of
z = -2.91

-1

0

1

2

3
Using a Z-Table to find probabilities
Find 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 Curve
Find Probability that
z > 0.75

Z-scores:

-3

-2

Find area under the
curve to the right of
z = 0.75

-1

0

1
0.75

2

3
Finding Area to the Right
• Two Methods
–Using the Complement of 1
–Using Symmetry
Complement Method
Find Probability that
z > 0.75

Z-scores:

-3

-2

Find area under the
curve to the right of
z = 0.75

-1

0

1
0.75

2

3
Complement Method
Use fact that area
under entire curve is 1.
And that we can find
area to the left

Z-scores:

-3

-2

Find area under the
curve to the right of
z = 0.75

-1

0

1
0.75

2

3
Complement Method
Use fact that area
under entire curve is 1.
And that we can find
area to the left

Z-scores:

-3

-2

Find area under the
curve to the right of
z = 0.75

-1

0

1
0.75

2

3
The Normal Curve
Find Probability that
z > 0.75

Z-scores:

-3

-2

Find area under the
curve to the right of
z = 0.75

-1

0

1
0.75

2

3
Symmetry Method
Symmetry Method
Use symmetry of the
normal curve to find
area

Z-scores:

-3

-2

Find area under the
curve to the right of
z = 0.75

1
-1
0
- 0.75 0.75

2

3
Finding Area between two values
• Just use difference of the two
areas
Difference of Area
Find Probability that
-1.25 < z < 0.75

Z-scores:

-3

Find area under the
curve between -1.25
and 0.75

-2

-1

-1.25

0

1
0.75

2

3
Finding Probabilities of Normal Distributions
1. For data that is normally
distributed, find the percentage
of data items that are:

a) below z = 0.6
b) above z = –1.8
c) 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 142
b) above 130
c)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
• 1 – 4, 9 – 19 odd, 25 – 35 odd

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Lecture 10.4 b bt

  • 1. Today’s Agenda  Attendance / Announcements  Sections 10.4b  Have a good Holiday!
  • 2. Exam Schedule  Exam 5 (Ch 10) Thur 12/5  Final Exam (All) Thur 12/12 1 hour, 50 mins
  • 8. 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 …
  • 10. Using a Z-Table to find probabilities Note: Our Z-table only gives area to the left (or probabilities less than z)
  • 11. The Normal Curve Find Probability that z < 0.97 P(z Find area under the curve to the left of z = 0.97 0.97) Z-scores: -3 -2 -1 0 1 0.97 2 3
  • 12. Using a Z-Table to find probabilities Find Probability that z < 0.97 Since z > 0, use positive side
  • 13. The Normal Curve Find Probability that z < -2.91 Z-scores: -3 -2.91 -2 Find area under the curve to the left of z = -2.91 -1 0 1 2 3
  • 14. Using a Z-Table to find probabilities Find Probability that z < -2.91 Since z < 0, use negative side
  • 15. Using a Z-Table to find probabilities • Not all Z-Tables are alike!
  • 16. 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).
  • 17. The Normal Curve Find Probability that z > 0.75 Z-scores: -3 -2 Find area under the curve to the right of z = 0.75 -1 0 1 0.75 2 3
  • 18. Finding Area to the Right • Two Methods –Using the Complement of 1 –Using Symmetry
  • 19. Complement Method Find Probability that z > 0.75 Z-scores: -3 -2 Find area under the curve to the right of z = 0.75 -1 0 1 0.75 2 3
  • 20. Complement Method Use fact that area under entire curve is 1. And that we can find area to the left Z-scores: -3 -2 Find area under the curve to the right of z = 0.75 -1 0 1 0.75 2 3
  • 21. Complement Method Use fact that area under entire curve is 1. And that we can find area to the left Z-scores: -3 -2 Find area under the curve to the right of z = 0.75 -1 0 1 0.75 2 3
  • 22. The Normal Curve Find Probability that z > 0.75 Z-scores: -3 -2 Find area under the curve to the right of z = 0.75 -1 0 1 0.75 2 3
  • 24. Symmetry Method Use symmetry of the normal curve to find area Z-scores: -3 -2 Find area under the curve to the right of z = 0.75 1 -1 0 - 0.75 0.75 2 3
  • 25. Finding Area between two values • Just use difference of the two areas
  • 26. Difference of Area Find Probability that -1.25 < z < 0.75 Z-scores: -3 Find area under the curve between -1.25 and 0.75 -2 -1 -1.25 0 1 0.75 2 3
  • 27. Finding Probabilities of Normal Distributions 1. For data that is normally distributed, find the percentage of data items that are: a) below z = 0.6 b) above z = –1.8 c) between z = –2 and –0.5
  • 28. 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)
  • 29. 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.
  • 30. 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 142 b) above 130 c)between 142 and 154
  • 31. 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.
  • 32. 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)
  • 33. Classwork / Homework • 10.4 Worksheet • Page 638 • 1 – 4, 9 – 19 odd, 25 – 35 odd