Binomial Distributions

BPS chapter 13

© 2006 W.H. Freeman and Company

Binomial settingA 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) Yes.

b) No, because there is not a fixed number of observations.

c) No, because the observations are not all independent.

d) No, because there are more than two possible outcomes for each observation.

e) 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) Yes.

b) No, because there is not a fixed number of observations.

c) No, because the observations are not all independent.

d) No, because there are more than two possible outcomes for each observation.

e) No, because the probability of success for each observation is not the same.

Binomial settingA 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) Yes.

b) No, because there is not a fixed number of observations.

c) No, because the observations are not all independent.

d) 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) Yes.

b) No, because there is not a fixed number of observations.

c) No, because the observations are not all independent.

d) No, because there are more than two possible outcomes for each observation.

Binomial settingA 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) Yes.

b) No, because there is not a fixed number of observations.

c) No, because the observations are not all independent.

d) No, because there are more than two possible outcomes for each observation.

e) 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) Yes.

b) No, because there is not a fixed number of observations.

c) No, because the observations are not all independent.

d) No, because there are more than two possible outcomes for each observation.

e) No, because the probability of success for each observation is not the same.

Binomial settingA 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) Yes.

b) No, because there is not a fixed number of observations.

c) No, because the observations are not all independent.

d) No, because there are more than two possible outcomes for each observation.

e) 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) Yes.

b) No, because there is not a fixed number of observations.

c) No, because the observations are not all independent.

d) No, because there are more than two possible outcomes for each observation.

e) No, because the probability of success for each observation is not the same.

Binomial settingA 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) Yes.

b) No, because there is not a fixed number of observations.

c) No, because the observations are not all independent.

d) No, because there are more than two possible outcomes for each observation.

e) 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) Yes.

b) No, because there is not a fixed number of observations.

c) No, because the observations are not all independent.

d) No, because there are more than two possible outcomes for each observation.

e) No, because the probability of success for each observation is not the same.

Binomial distributionSuppose 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) (0.30)2

b) 2 (0.30)

c)

d)

29(0.3)

2

2 79(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) (0.30)2

b) 2 (0.30)

c)

d)

29(0.3)

2

2 79(0.3) (0.7)

2

Binomial distributionSuppose 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) = (9)(0.3) = 2.7, = 0.30

b) = 3, = (9)(0.3)

c) = (9)(0.3) = 2.7, = (9)(0.7)(0.3)

d) = (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) = (9)(0.3) = 2.7, = 0.30

b) = 3, = (9)(0.3)

c) = (9)(0.3) = 2.7, = (9)(0.7)(0.3)

d) = (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 and

a) Binomial (n = 90, p = 0.30)

b) Binomial ( = 27, = 0.435)

c) Normal (n = 90, p = 0.30)

d) Normal ( = 27, = 0.435)

(1 ) 0.435np p 27np

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 and

a) Binomial (n = 90, p = 0.30)

b) Binomial ( = 27, = 0.435)

c) Normal (n = 90, p = 0.30)

d) Normal ( = 27, = 0.435)

(1 ) 0.435np p 27np

Normal approximation to the binomial

When does the normal approximation to the binomial give the most accurate answer?

a) When n is big.

b) When n is small.

c) When p is big.

d) When p is small.

Normal approximation to the binomial (answer)When does the normal approximation to the binomial give the most

accurate answer?

a) When n is big.

b) When n is small.

c) When p is big.

d) 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) The normal distribution is more accurate.

b) The normal distribution uses the mean and the standard deviation.

c) The normal distribution works all the time, so use it for everything.

d) 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.

e) 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) The normal distribution is more accurate.

b) The normal distribution uses the mean and the standard deviation.

c) The normal distribution works all the time, so use it for everything.

d) 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.

e) 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.