Transcript 8-1 Day 1

AP STATISTICS
LESSON 8 - 1
THE BINOMIAL DISTRIBUTION
ESSENTIAL QUESTION:
What is a binomial setting and
how can binomial
distributions be solved?
Objectives:
• To identify binomial settings.
• To become familiar with the binomial
formula.
• To solve problems using the binomial
formula.
Introduction
In practice, we frequently encounter
experimental situations where there
are two outcomes of interest.
Some examples are:
• We use a coin toss to answer a
question.
• A basketball player shoots a free
throw.
• A young couple prepares for their first
child.
The Binomial Setting
1.
Each observation falls into one of just two
categories, which for convenience we call
“success” or “failure.”
2.
There is a fixed number n of observations.
3.
The n observations are all independent.
(That is, knowing the results of one
observation tells you nothing about the
other observations).
4.
The probability of success, call it p, is the
same for each observation.
Binomial Distribution
The distribution of the count X of
successes in the binomial settings is the
binomial distribution with parameters n
and p.
The parameter n is the number of
observations, and p is the probability of a
success on any one observation.
The possible values of X are the whole
numbers from 0 to n. As an abbreviation,
we say that X is B(n,p).
Example 8.1
Blood Types
page 440
Blood type is inherited. If both parents
carry genes for the O and A blood
types, each child has probability of 0.25
of getting two O genes and so of having
blood type O. Different children inherit
independently of each other.
Find n, p and X.
Example 8.2
page 440
Deal 10 cards from a shuffled deck
and count the number X of red
cards. There are 10 observations
and each are either a red or a black
card.
Is this a binomial distribution?
If so what are the variables n,p and
X?
Example 8.3
page 440
An engineer chooses an SRS of 10
switches from a shipment of 10,000
switches Suppose that (unknown to
the engineer) 10% of the switches in
the shipment are bad.
The engineer counts the number X of
bad switches in the sample.
Is this a binomial situation?
Why?
pdf
Given a discrete random
variable X, the probability
distribution function assigns a
probability to each value of X.
The probability must satisfy the
rules for probabilities given in
Chapter 6.
Example 8.6
page 443
Corinne is a basketball player who
makes 75% of her free throws over the
course of a season.
In a key game, Corinne shoots 12 free
throws and makes only 7 of them.
The fans think that she failed because
she is nervous.
Is it unusual for Corinne to perform
this poorly?
Example 8.7
page 443
We want to determine the
probability that all 3 children in
a family are girls.
Find n, p, and X.
cdf
The cumulative binomial probability is
useful in a situation of a range of
probabilities.
Given a random variable X, the cumulative
distribution function (cdf) of X calculates
the sum of the probabilities for 0,1,2,…, up
to the value X.
That is , it calculates the probability of
obtaining a most X successes in n trials.
Example 8.8
page 444
If Corrine shoots n = 12 free
throws and makes no more than
7 of them?