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 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:
...
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...
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 t...
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...
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

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

Z-scores:

-3

-2

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

Z-scores:

-3

-2

Fi...
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...
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 o...
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...
Finding Probabilities of Normal Distributions
1. For data that is normally
distributed, find the percentage
of data items ...
Finding Probabilities of Normal Distributions
2. Given a data set that is
normally distributed, find the
following probabi...
Solving Applications of Normal Distributions
Before solving real world applications of data that
is normally distributed, ...
Solving Applications of Normal Distributions
Systolic blood pressure readings are normally distributed
with a mean of 121 ...
Solving Applications of Normal Distributions
A machine produces bolts with an average diameter of 7
mm and a standard devi...
Solving Applications of Normal Distributions
The placement test for a
college has scores that are
normally distributed
wit...
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. 1. Today’s Agenda  Attendance / Announcements  Sections 10.4b  Have a good Holiday!
  2. 2. Exam Schedule  Exam 5 (Ch 10) Thur 12/5  Final Exam (All) Thur 12/12 1 hour, 50 mins
  3. 3. The Normal Curve
  4. 4. The Normal Curve
  5. 5. The Normal Curve
  6. 6. The Normal Curve
  7. 7. The Normal Curve Z-scores:
  8. 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 …
  9. 9. The Normal Curve
  10. 10. Using a Z-Table to find probabilities Note: Our Z-table only gives area to the left (or probabilities less than z)
  11. 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. 12. Using a Z-Table to find probabilities Find Probability that z < 0.97 Since z > 0, use positive side
  13. 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. 14. Using a Z-Table to find probabilities Find Probability that z < -2.91 Since z < 0, use negative side
  15. 15. Using a Z-Table to find probabilities • Not all Z-Tables are alike!
  16. 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. 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. 18. Finding Area to the Right • Two Methods –Using the Complement of 1 –Using Symmetry
  19. 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. 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. 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. 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
  23. 23. Symmetry Method
  24. 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. 25. Finding Area between two values • Just use difference of the two areas
  26. 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. 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. 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. 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. 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. 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. 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. 33. Classwork / Homework • 10.4 Worksheet • Page 638 • 1 – 4, 9 – 19 odd, 25 – 35 odd
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