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Chapter 7
 Density
Curve
- A curve that describes the overall pattern
of a distribution.
- All Density Curves have and area of 1 or
100%
-Entire graph mush be above the x-axis
-Normal Distribution – Bell Shape N (  , )
-The area under the curve is the
proportion of observations that fall into that
interval
 Probability
 The
likelihood or chance that an event will
occur ranges from 0 to 1
-
The sum of all Outcomes in an experiment
is equal to 100%
A
quantity whose value changes.
 Today
you will learn about a few different
types of variables.
a
variable whose value is obtained by
counting
 Examples:
number of students present
 number of red marbles in a jar
 number of heads when flipping coins

a
variable whose value is obtained by
measuring
 Examples:
height of students in class
 weight of text books
 distance traveled between classes

a
variable whose value is a numerical
outcome of a random phenomenon
denoted with a capital letter, X
 can be discrete or continuous
 The probability distribution of a random
variable X tells what the possible values of X
are and how probabilities are assigned to
those values

A
coin is flipped 3 times and the sequence of
heads and tails are recorded. The sample
space for this experiment is:
 Let
the random variable X be the number of
heads in three coin tosses. Thus, X asigns
each outcome a number from the set
(0, 1, 2, 3).
Outcome
X
HHH
HHT
HTH
THH
HTT
THT
TTH
TTT
X
has a countable number of values.
 Probability
distribution of a discrete random
variable X lists the values and their
probabilities:
Value of
X
P(X)

0  P 1
 The
sum of the probabilities is 1
 What
is the probability distribution of the
discrete random variable X that counts the
number of heads in four tosses of a coin?
 The
number of heads, X, has possible values
0, 1, 2, 3, 4.
 These values are not equally likely!
P
(X = 0) =
 P(X
 P(
= 2)=
X > 1)=
P(X = 1) =
P(X = 3) =
P(X = 4) =
P( X  2) 
1. P(X < 4) =
2. P( x  2)
 NC
State posts the grade distributions for its
courses online. Students in Statistics 101 in
fall 2003 semester received 21% A’s, 43% B’s,
30% C’s 5% F’s, and 1% F’s. Choose a
Statistics 101 student at random. What is
the probability that the student got a B or
better?
 Less
than a C?
X
takes all values in a given interval of
numbers



The probability distribution of a continuous
random variable is shown by a density curve.
The probability that X is between an interval
of numbers is the area under the density
curve between the interval endpoints
The probability that a continuous random
variable, X is exactly equal to a number is
zero
 REMEMBER:
N(μ, σ) is our shorthand for a
normal distribution with mean μ and
standard deviation σ. So, if X has the N(μ, σ)
distribution then the we can standardize
using
Z
X

 An
opinion poll asks an SRS of 1500 American
adults what they consider to be the most
serious problem facing our schools. Suppose
that if we could ask all adults this question
30% say “drugs”. (We will learn about p̂ in
the next chapters, so for now go with it…) p̂  0.3
and N(0.3, 0.0118). What is the probability
that the poll results differs from the truth
about the population by more than two
percentage points?
 Let
“X” represent the sum of two dice
 A)
Define the random variables in this
situation.
 Is it a continuous or discrete random variable?
 Draw the density curve (ie: probability
distribution).
X
P(X)
2
3
4
5
6
7
8
9
10
11
12
1. P(X > 4) =
2. P( x  2)