Transcript original powerpoint presentation

Normal distribution
and intro to continuous probability
density functions...
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), 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 .
• Spread is determined by .
• 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
Probability 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
0.00
55
60
65
70
Grades
75
80
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- )/
• Z is called the standard normal. Its mean
 is 0 and standard deviation  is 1.
• Then, use probability table for Z.
Using Z Table
Standard Normal Curve
0.4
Density
0.3
0.2
Tail probability
P(Z > z)
0.1
0.0
-4
-3
-2
-1
0
Z
1
2
3
4
Reading Z Table
p. 484, Appendix A
• Carry out Z calculations to two decimal
places, that is X.XX
• 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.
Probability 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
0.00
55
60
65
70
Grades
75
80
85
Probability below 65?
0.08
0.07
Density
0.06
0.05
0.04
0.03
0.02
P(X < 65)
0.01
0.00
55
65
75
Grades
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.”
• Always check this assumption! (We’ll learn
how to next class.)