Transcript Clicker_chapter13.ppt

Binomial Distributions
BPS chapter 13
© 2006 W.H. Freeman and Company
Binomial setting
A manufacturing company takes a sample of n = 100 bolts from their
production line. X is the number of bolts that are found defective in
the sample. It is known that the probability of a bolt being defective
is 0.003.
Does X have a binomial distribution?
a)
b)
c)
d)
e)
Yes.
No, because there is not a fixed number of observations.
No, because the observations are not all independent.
No, because there are more than two possible outcomes for each
observation.
No, because the probability of success for each observation is not
the same.
Binomial setting (answer)
A manufacturing company takes a sample of n = 100 bolts from their
production line. X is the number of bolts that are found defective in
the sample. It is known that the probability of a bolt being defective
is 0.003.
Does X have a binomial distribution?
a)
b)
c)
d)
e)
Yes.
No, because there is not a fixed number of observations.
No, because the observations are not all independent.
No, because there are more than two possible outcomes for each
observation.
No, because the probability of success for each observation is not
the same.
Binomial setting
A survey-taker asks the age of each person in a random sample of 20
people. X is the age for the individuals.
Does X have a binomial distribution?
a)
b)
c)
d)
Yes.
No, because there is not a fixed number of observations.
No, because the observations are not all independent.
No, because there are more than two possible outcomes for each
observation.
Binomial setting (answer)
A survey-taker asks the age of each person in a random sample of 20
people. X is the age for the individuals.
Does X have a binomial distribution?
a)
b)
c)
d)
Yes.
No, because there is not a fixed number of observations.
No, because the observations are not all independent.
No, because there are more than two possible outcomes for
each observation.
Binomial setting
A survey-taker asks whether each person in a random sample of 20
college students is over the age of 21. X is the number of people
who are over 21. According to university records, 35% of all college
students are over 21 years old.
Does X have a binomial distribution?
a)
b)
c)
d)
e)
Yes.
No, because there is not a fixed number of observations.
No, because the observations are not all independent.
No, because there are more than two possible outcomes for each
observation.
No, because the probability of success for each observation is not
the same.
Binomial setting (answer)
A survey-taker asks whether each person in a random sample of 20
college students is over the age of 21. X is the number of people
who are over 21. According to university records, 35% of all college
students are over 21 years old.
Does X have a binomial distribution?
a)
b)
c)
d)
e)
Yes.
No, because there is not a fixed number of observations.
No, because the observations are not all independent.
No, because there are more than two possible outcomes for each
observation.
No, because the probability of success for each observation is not
the same.
Binomial setting
A certain test contains 10 multiple-choice problems. For five of the
problems, there are four possible answers, and for the other five
there are only three possible answers. X is the number of correct
answers a student gets by simply guessing.
Does X have a binomial distribution?
a)
b)
c)
d)
e)
Yes.
No, because there is not a fixed number of observations.
No, because the observations are not all independent.
No, because there are more than two possible outcomes for each
observation.
No, because the probability of success for each observation is not
the same.
Binomial setting (answer)
A certain test contains 10 multiple-choice problems. For five of the
problems, there are four possible answers, and for the other five
there are only three possible answers. X is the number of correct
answers a student gets by simply guessing.
Does X have a binomial distribution?
a)
b)
c)
d)
e)
Yes.
No, because there is not a fixed number of observations.
No, because the observations are not all independent.
No, because there are more than two possible outcomes for each
observation.
No, because the probability of success for each observation is
not the same.
Binomial setting
A fair die is rolled and the number of dots on the top face is noted. X is
the number of times we have to roll in order to have the face of the
die show a 2.
Does X have a binomial distribution?
a)
b)
c)
d)
e)
Yes.
No, because there is not a fixed number of observations.
No, because the observations are not all independent.
No, because there are more than two possible outcomes for each
observation.
No, because the probability of success for each observation is not
the same.
Binomial setting (answer)
A fair die is rolled and the number of dots on the top face is noted. X is
the number of times we have to roll in order to have the face of the
die show a 2.
Does X have a binomial distribution?
a)
b)
c)
d)
e)
Yes.
No, because there is not a fixed number of observations.
No, because the observations are not all independent.
No, because there are more than two possible outcomes for each
observation.
No, because the probability of success for each observation is not
the same.
Binomial distribution
Suppose that for a randomly selected high school student who has
taken a college entrance exam, the probability of scoring above a
650 is 0.30. A random sample of n = 9 students was selected.
What is the probability that exactly two of the students scored over
650 points?
a)
b)
c)
d)
(0.30)2
2 (0.30)
9
2
(0.3)
 
 2
9
2
7
(0.3)
(0.7)
 
 2
Binomial distribution (answer)
Suppose that for a randomly selected high school student who has
taken a college entrance exam, the probability of scoring above a
650 is 0.30. A random sample of n = 9 students was selected.
What is the probability that exactly two of the students scored over
650 points?
a)
b)
c)
d)
(0.30)2
2 (0.30)
9
2
(0.3)
 
 2
9
2
7
(0.3)
(0.7)
 
 2
Binomial distribution
Suppose that for a randomly selected high school student who has
taken a college entrance exam, the probability of scoring above a
650 is 0.30. A random sample of n = 9 students was selected.
What are the mean  and standard deviation  of the number of
students in the sample who have scores above 650?
a)
b)
c)
d)
 = (9)(0.3) = 2.7,  = 0.30
 = 3,  = (9)(0.3)
 = (9)(0.3) = 2.7,  = (9)(0.7)(0.3)
 = (9)(0.3) = 2.7,  = (9)(0.3)(0.7)
Binomial distribution (answer)
Suppose that for a randomly selected high school student who has
taken a college entrance exam, the probability of scoring above a
650 is 0.30. A random sample of n = 9 students was selected.
What are the mean  and standard deviation  of the number of
students in the sample who have scores above 650?
a)
b)
c)
d)
 = (9)(0.3) = 2.7,  = 0.30
 = 3,  = (9)(0.3)
 = (9)(0.3) = 2.7,  = (9)(0.7)(0.3)
 = (9)(0.3) = 2.7,  = (9)(0.3)(0.7)
Normal approximation to the binomial
Suppose that for a randomly selected high school student who has
taken a college entrance exam, the probability of scoring above a
650 is 0.30. A random sample of n = 90 high school students was
selected. If we want to calculate the probability that at least 20 of
these students scored above a 650, what distribution would be the
best for us to use?
Note that np  27 and
np(1  p)  0.435
a)
b)
c)
d)
Binomial (n = 90, p = 0.30)
Binomial ( = 27,  = 0.435)
Normal (n = 90, p = 0.30)
Normal ( = 27,  = 0.435)
Normal approximation to the binomial (answer)
Suppose that for a randomly selected high school student who has
taken a college entrance exam, the probability of scoring above a
650 is 0.30. A random sample of n = 90 high school students was
selected. If we want to calculate the probability that at least 20 of
these students scored above a 650, what distribution would be the
best for us to use?
Note that np  27 and
np(1  p)  0.435
a)
b)
c)
d)
Binomial (n = 90, p = 0.30)
Binomial ( = 27,  = 0.435)
Normal (n = 90, p = 0.30)
Normal ( = 27,  = 0.435)
Normal approximation to the binomial
When does the normal approximation to the binomial give the most
accurate answer?
a)
b)
c)
d)
When n is big.
When n is small.
When p is big.
When p is small.
Normal approximation to the binomial (answer)
When does the normal approximation to the binomial give the most
accurate answer?
a)
b)
c)
d)
When n is big.
When n is small.
When p is big.
When p is small.
Normal approximation to the binomial
Why would we want to use the normal approximation to the binomial
instead of just using the binomial distribution?
a)
b)
c)
d)
e)
The normal distribution is more accurate.
The normal distribution uses the mean and the standard deviation.
The normal distribution works all the time, so use it for everything.
The binomial distribution is awkward and takes too long if you have
to sum up many probabilities. The binomial distribution looks like
the normal distribution if n is large.
The binomial distribution is awkward and takes too long if you have
to multiply many probabilities. The binomial distribution looks like
the normal distribution if n is large.
Normal approximation to the binomial (answer)
Why would we want to use the normal approximation to the binomial
instead of just using the binomial distribution?
a)
b)
c)
d)
e)
The normal distribution is more accurate.
The normal distribution uses the mean and the standard deviation.
The normal distribution works all the time, so use it for everything.
The binomial distribution is awkward and takes too long if you
have to sum up many probabilities. The binomial distribution
looks like the normal distribution if n is large.
The binomial distribution is awkward and takes too long if you have
to multiply many probabilities. The binomial distribution looks like
the normal distribution if n is large.