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
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
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)
