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Chapter 13

Chapter 13.odt

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Simon Fraser University
STAT 101
Qian( Michelle) Zhou

Chapter 13 – Binomial Distributions The Binomial Setting and Binomial Distribution • Distribution of a count depends on how data is produced • The Binomial Setting 1. There are fixed number of “n” observations ◦ “count observations until the first success” is NOT FIXED, so it won't be binomial distribution 2. The “n”observations are all independent; result of one observation does not change the probabilities we assign to other observations 3. Each observations falls into one of just two categories, “success” and “failure” 4. Probability of success, let's call it p, is the same for each observation • Ex) coin toss – knowing outcome of one toss doesn't change probability of a head on any other toss – it's independent ◦ number of heads we count is a discrete random variable X ◦ distribution of X is called binomial distribution ◦ ex) P(H) = 0.5 ▪ P(HH) = 0.5 x 0.5 = 0.25 ▪ P(T) P(H) will happen = P(T) x P(H) = 0.5 x 0.5 = 0.25 • The count X of successes in a binomial setting has the binomial distribution w/ parameters n and p ◦ n = number of trials ◦ p = probability of success on any trial • Warning: not all counts have binomial distributions Ex 13.1) Blood Types Each child has probability of 0.25 of having type blood O. If a pair of parents have 5 children, the number who have blood type O is the count X of successes in 5 independent observations w/ 0.25 probability of success. So X has the binomial distribution, n = 5 and p = 0.25 Binomial Distributions in Statistical Sampling Example 13.3) Choosing an SRS of Cds Music inspector inspects SRS of 10 Cds from shipment of 10,000 Cds. Suppose 10% of the Cds in shipment are defective. Count the number of X of bad CDS in sample. This is not quite a binomial setting. Removing 1 CD changes the proportion of bad CDS remaining in shipment. • When population is much larger than the sample, a count of successes in SRS of size n has approximately the binomial distribution with n equal to the sample size and p equal to the proportion of successes in the population • Sampling Distribution of a Count ◦ Choose an SRS of size n from a population w/ proportion p of successes. When population much larger than sample, the count X of successes in sample has approx the binomial distribution w/ parameters n and p Ex 13.1) Opinion poll calls residential telephon numbers @ random, only 20% of calls reach a live person. You watch random sampling machine make 15 calls. X is number that reach a live person. This is binomial distribution. P = 0.25; n is fixed, so it's n = 15. X (success) = success of reaching a live person. And each call is independent of another. Binomial Probabilities • formula for probability that a binomial random variable takes any value by adding probabilities for the different ways of getting exactly that many successes in n observations Example 13.4) Inheriting blood type Blood types of successive children born to the same parents are independent and have fix probabilities that depend on genetic makeup of parents. Each child born to certain set of parents have 0.25 probability of having “O”. If parents have 5 children
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