Download Normal distribution

Normal distribution
and introduction to continuous
random variables and continuous
probability density functions...
Continuous random variable
• A continuous random variable is a
random variable that takes on any value in
an interval of numbers.
• Example: Let X = the weight of a randomly
selected Big Mac, 0.20  x  0.30 pound
• Example: Let Y = the amount of chocolate
consumed by a randomly selected student
on Valentine’s Day, x  0 pounds
Graph: Percent Histogram
IQ
(Intervals of size 20)
40
Percent
30
20
10
0
55
75
95
IQ
115
135
Histogram
(Area of rectangle = probability)
IQ
(Intervals of size 20)
Density
0.02
0.01
0.00
55
75
95
IQ
115
135
Decrease interval size...
IQ
(Intervals of size 10)
Density
0.02
0.01
0.00
55
65
75
85
95
IQ
105
115
125
135
Decrease interval size more….
IQ
(Intervals of size 5)
0.03
Density
0.02
0.01
0.00
50
60
70
80
90
100
IQ
110
120
130
140
Continuous probability
density functions
• The curve describes probability of getting
any range of values, say P(X > 120),
P(X<100), or P(110 < X < 120)
• Area under the curve = probability.
• Area under whole curve = 1.
• Probability of getting specific number is 0,
e.g. P(X=120) = 0.
Special kind of continuous p.d.f
Bell-shaped curve
0.08
Mean = 70 SD = 5
0.07
Density
0.06
0.05
0.04
Mean = 70 SD = 10
0.03
0.02
0.01
0.00
40
50
60
70
Grades
80
90
100
Characteristics of
normal distribution
• Symmetric, bell-shaped curve.
• Shape of curve depends on population
mean  and standard deviation .
• Center of distribution is the mean .
• Spread determined by standard deviation .
• Most values fall around the mean, but some
values are smaller and some are larger.
Examples of normal
random variables
• testosterone level of male students
• head circumference of adult females
• length of middle finger of Stat 250 students
What is probability a student gets
a grade below 65?
0.08
0.07
Density
0.06
0.05
0.04
0.03
0.02
P(X < 65)
0.01
 = 70
=5
0.00
55
65
75
Grades
85
Probability = Area under curve
• Calculus?! You’re kidding, right?
• But somebody did all the hard work for us!
• We just need a table of probabilities for
every possible normal distribution.
• But there are an infinite number of normal
distributions (one for each  and )!!
• Solution is to “standardize.”
Standardizing
• Take value X and subtract its mean  from
it, and then divide by its standard deviation
. Call the resulting value Z.
• That is, Z = (X- )/ = number of standard
deviations X is above or below the mean.
• Z is called the standard normal. Its mean
 is 0 and standard deviation  is 1.
• Then, use probability table for Z.
Probability given in
Standard Normal Z Table
0.08
0.07
Density
0.06
0.05
P(Z < z)
0.04
0.03
0.02
0.01
0.00
-3
-2
-1
0
Z
1
2
3
How to Read a Standard Normal
Z Table
• Carry out Z calculations to two decimal
places, that is X.XX. Round, if necessary.
• Find the first two digits (X.XX) of Z in
column headed by z.
• Find the third digit of Z (X.XX) in first row.
• P(Z < z) = probability found at the
intersection of the column and row.
What is probability student gets
grade above 75?
Probability student scores higher than 75?
0.08
0.07
Density
0.06
0.05
P(X > 75)
0.04
0.03
0.02
0.01
 = 70
=5
0.00
55
60
65
70
Grades
75
80
85
What is probability student gets
grade between 65 and 70?
0.08
0.07
Density
0.06
0.05
P(65 < X < 70)
0.04
0.03
0.02
0.01
 = 70
=5
0.00
55
60
65
70
Grades
75
80
85
Remember!
• Calculated probabilities are accurate only if
the assumptions made are indeed correct!
• When doing the above calculations, you are
assuming that the data are “normally
distributed.”
• When possible, check this assumption!
(We’ll learn how later.)