7.6 The Normal Distribution 1. Normal Curve 2. Normally Distributed Outcomes 3. Properties of Normal Curve 4. Standard Normal Curve 5. The Normal Distribution 6. Percentile 7. Probability for General Normal Distribution 1
7.6 The Normal Distribution
1. Normal Curve2. Normally Distributed Outcomes3. Properties of Normal Curve4. Standard Normal Curve5. The Normal Distribution 6. Percentile7. Probability for General Normal
Distribution
1
Normal Curve The bell-shaped curve, as shown below, is
call a normal curve.
2
Normally Distributed Outcomes
Examples of experiments that have normally distributed outcomes:
1. Choose an individual at random and observe his/her IQ.
2. Choose a 1-day-old infant and observe his/her weight.
3. Choose a leaf at random from a particular tree and observe its length.
3
Properties of Normal Curve
4
Example Properties of Normal Curve
A certain experiment has normally distributed outcomes with mean equal to 1. Shade the region corresponding to the probability that the outcome
(a) lies between 1 and 3; (b) lies between 0 and 2; (c) is less than .5; (d) is greater than 2.
5
Example Properties of Normal Curve (2)
6
Standard Normal Curve
The equation of the normal curve is
21
21
2where 3.1416 and 2.7183.
x
y e
e
7
The standard normal curve has 0 and 1.
The Normal Distribution
8
A(z) is the area under the standard normal curve to the left of a normally distributed random variable z.
Example The Normal Distribution
Use the normal distribution table to determine the area corresponding to
(a) z < -.5; (b) 1< z < 2; (c) z > 1.5.
9
Example The Normal Distribution (2)
(a) A(-.5) = .3085
(b) A(2) - A(1) = .9772 - 8413
= .1359
(c) 1 - A(1.5) = 1 - .9332 = .0668
10
Percentile
If a score S is the pth percentile of a normal distribution, then p% of all scores fall below S, and (100 - p)% of all scores fall above S. The pth percentile is written as zp.
11
Example Percentile
What is the 95th percentile of the standard normal distribution?
In the normal distribution, find the value of z such that A(z) = .95.
A(1.65) = .9506 Therefore, z95 = 1.65.
12
Probability for General Normal Distribution
If X is a random variable having a normal distribution with mean and standard deviation then
where Z has the standard normal distribution and A(z) is the area under that distribution to the left of z.
Pr( ) Pr
and Pr( ) Pr
a b b aa X b Z A A
x xX x Z A
13
,
Example Probability Normal Distribution
Find the 95th percentile of infant birth weights if infant birth weights are normally distributed with = 7.75 and = 1.25 pounds.
The value for the standard normal random variable is z95 = 1.65.
Then x95 = 7.75 + (1.65)(1.25) = 9.81 pounds.
14
Summary Section 7.6 - Part 1
A normal curve is identified by its mean ( ) and its standard deviation ( ). The standard normal curve has = 0 and = 1. Areas of the region under the standard normal curve can be obtained with the aid of a table or graphing calculator.
15
Summary Section 7.6 - Part 2
A random variable is said to be normally distributed if the probability that an outcome lies between a and b is the area of the region under a normal curve from x = a to x = b. After the numbers a and b are converted to standard deviations from the mean, the sought-after probability can be obtained as an area under the standard normal curve.
16