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Announcements
Finite Probability
Wednesday, November 16th
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MyMathLab 9 is Monday Nov 28
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Problem Set 9 is due Monday Nov 28
Today: Sec. 7.6: The Normal Distribution
Find area below a standard normal curve using a table
Transform a normally distributed random variable to the
standard normal distribution.
Next Class: Sec. 7.6: The Normal Distribution II
Cherveny
Nov 16
Math 1004: Probability
Normal Curve
A normal curve (a.k.a. “bell curve”) approximates many situations:
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The heights of adult men in a population.
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The number of heads obtained in 1000 flips of a coin.
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The weight of cereal put in a box by a factory machine.
A normal curve is symmetric about x = µ and σ controls its width.
Cherveny
Nov 16
Math 1004: Probability
Looking Ahead: Normal Approximations
Goal: Sec. 7.7 will use normal curves to approximate probabilities.
It turns out the total area under any normal curve is 1. If a
random variable is “normally distributed”, the probability of an
outcome within a range is the area under the curve on that range.
Cherveny
Nov 16
Math 1004: Probability
Standard Normal Distribution
Definition
The standard normal curve is normal with µ = 0 and σ = 1.
We want to:
I Find areas of regions under the standard normal curve.
I Find areas for other normal curves by translating the problem
to the standard normal curve.
Cherveny
Nov 16
Math 1004: Probability
Finding Areas on the Standard Normal Curve
The area under the standard normal curve to the left of a given
z-value is given in Appendix A.
Example
(a) What is the area to the left of z = .7?
(b) What is the area to the right of z = −1.5?
(c) What is the area between z = .5 and z = 2.2?
Answers:
(a) .7580
(b) 1 - .0668 = .9332
(c) .9861 - .6915 = 0.2946
Cherveny
Nov 16
Math 1004: Probability
“68-95-99.7 Rule”
The “68-95-99.7” rule says that on any normal curve roughly..
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68% of the area is within 1 standard deviation of the mean.
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95% of the area is within 2 standard deviations of the mean.
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99.7% of the area is within 3 standard deviations of the mean.
Cherveny
Nov 16
Math 1004: Probability
Actual GRE Question
Example
The standard deviation on a test was 12 points, and the mean was
70. If the scores fell along a normal distribution and student X
scored 95 points, then student X scored higher than approximately
what percent of students?
(a) 2%
(b) 13%
(c) 48%
(d) 96%
(e) 98%
Answer: E
Cherveny
Nov 16
Math 1004: Probability
Finding Areas on Normal Curves
If X follows a normal curve with mean µ and standard deviation σ,
then
Z=
X −µ
σ
follows a standard normal curve.
Example
(a) What is the area to the left of X = 4 on a normal curve with
mean 5 and standard deviation 2?
(b) The amount of contaminant in a water sample is normally
distributed with mean .5 grams and standard deviation .15
grams. The sample is a health violation if more than .7 grams
are measured. What is the probability a sample is in violation
of health code?
Answer: (a) .3085 (b) ≈ .0885
Cherveny
Nov 16
Math 1004: Probability
Practice
1. What is the area of each shaded region on the standard
normal curve?
2. What is the area of each shaded region on these normal
curves?
3. If test scores are normal with mean 34 and standard deviation
4, what percent of the class made above a 30?
Cherveny
Nov 16
Math 1004: Probability
Practice Answers
1. (a) .8944
(b) .4013
(c) .2417
2. (a) .9772
(b) .6247
3. 84% (Can you do this with only the 68-95-99.7 Rule?)
Cherveny
Nov 16
Math 1004: Probability