Binomial Distributions BPS 7e Chapter 14 © 2015 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? 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? 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 Which of the following settings must hold for a binominal distribution for a count X variable? There are n observations. The observations are independent of each other. Each observation results in a success or a failure. Each observation has the same probability p of a success. All of the above

Binomial Setting (answer) Which of the following settings must hold for a binominal distribution for a count X variable? There are n observations. The observations are independent of each other. Each observation results in a success or a failure. Each observation has the same probability p of a success. All of the above

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? 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? 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? 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? 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 purely by guessing. Does X have a binomial distribution? 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 purely by guessing. Does X have a binomial distribution? 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? 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? 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 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. What are the possible values of X? 0 to 1 1 to 20 0 to 20 0 to 21

Binomial Distribution (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. What are the possible values of X? 0 to 1 1 to 20 0 to 20 0 to 21

Binomial Distribution True or False: When an opinion poll calls residential telephone numbers at random, only 25% of the calls reach a live person. Imagine that you watch the random dialing machine making the phone calls. If X is the number of calls until the first live person answers, then X has a binomial distribution. True False

Binomial Distribution (answer) True or False: When an opinion poll calls residential telephone numbers at random, only 25% of the calls reach a live person. Imagine that you watch the random dialing machine making the phone calls. If X is the number of calls until the first live person answers, then X has a binomial distribution. True False

Binomial Distribution 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, p of all college students are over 21 years old. If X is to have the binomial distribution, p must be in the range: 0 to 1. –1 to 1. –1 to 0. 0 to 20.

Binomial Distribution (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, p of all college students are over 21 years old. If X is to have the binomial distribution, p must be in the range: 0 to 1. –1 to 1. –1 to 0. 0 to 20.

Binomial Probabilities Suppose that, for a randomly selected high school student who has taken a college entrance exam, the probability of scoring above 650 is 0.30. A random sample of n = 9 such students was selected. What is the probability that exactly two of the students scored over 650? (0.30)2 2 (0.30)

Binomial Probabilities (answer) Suppose that, for a randomly selected high school student who has taken a college entrance exam, the probability of scoring above 650 is 0.30. A random sample of n = 9 such students was selected. What is the probability that exactly two of the students scored over 650? (0.30)2 2 (0.30)

Binomial Probabilities Suppose that, for a randomly selected high school student who has taken a college entrance exam, the probability of scoring above 650 is 0.30. A random sample of n = 9 such students was selected. X is the number in the sample who scored over 650. What does the binomial coefficient give? the probability that X is greater than 9 the number of students in the population the number of students in the sample the number of ways of arranging X successes among 9 observations

Binomial Probabilities (answer) Suppose that, for a randomly selected high school student who has taken a college entrance exam, the probability of scoring above 650 is 0.30. A random sample of n = 9 such students was selected. X is the number in the sample who scored over 650. What does the binomial coefficient give? the probability that X is greater than 9 the number of students in the population the number of students in the sample the number of ways of arranging X successes among 9 observations

Binomial Mean and Standard Deviation Suppose that, for a randomly selected high school student who has taken a college entrance exam, the probability of scoring above 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 over 650?  = (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;  =

Binomial Mean and Standard Deviation (answer) Suppose that for a randomly selected high school student who has taken a college entrance exam, the probability of scoring above 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 over 650?  = (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;  =

Normal Approximation True or False: The Normal approximation to the binomial distribution says that, if X is a count having the binomial distribution with parameters n and p, then when n is large, X is approximately . True False

Normal Approximation (answer) True or False: The Normal approximation to the binomial distribution says that, if X is a count having the binomial distribution with parameters n and p, then when n is large, X is approximately . True False

Normal Approximation Suppose that, for a randomly selected high school student who has taken a college entrance exam, the probability of scoring above 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 650, what probability distribution would be easiest to use? Note that np = 27 and binomial (n = 90, p = 0.30) binomial ( = 27,  = 0.435) Normal (n = 90, p = 0.30) Normal ( = 27,  = 0.435)

Normal Approximation (answer) Suppose that, for a randomly selected high school student who has taken a college entrance exam, the probability of scoring above 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 650, what probability distribution would be easiest to use? Note that np = 27 and binomial (n = 90, p = 0.30) binomial ( = 27,  = 0.435) Normal (n = 90, p = 0.30) Normal ( = 27,  = 0.435)

Normal Approximation When does the Normal approximation to the binomial work best? when n is big when n is small when p is big when p is small

Normal Approximation (answer) When does the Normal approximation to the binomial work best? when n is big when n is small when p is big when p is small

Normal Approximation Why would we want to use the Normal approximation to the binomial instead of just using the binomial distribution? The Normal distribution is more accurate. The Normal distribution uses the mean and the standard deviation. The Normal distribution works all the time, so it should be used for everything. The binomial distribution is awkward and computationally intense. However, 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 (answer) Why would we want to use the Normal approximation to the binomial instead of just using the binomial distribution? The Normal distribution is more accurate. The Normal distribution uses the mean and the standard deviation. The Normal distribution works all the time, so it should be used for everything. The binomial distribution is awkward and computationally intense. However, 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.