Presentation on theme: "AP Statistics 51 Days until the AP Exam"— Presentation transcript:
1AP Statistics 51 Days until the AP Exam The Binomial DistributionThe Binomial CoefficientMean and Standard Deviation
2The Binomial Setting Each observation falls into one of two categories The probability of success is the same for each observationThe observations are ALL independentThere is a fixed number n of all observations.
3The Binomial Distribution If data is produced in the binomial setting then the random variable X is called the binomial random variable.The probability distribution of X is called the binomial distribution.The parameters of X are n and p.n is the number of observationsp is the probability of the observation occurring.
4The Binomial Coefficient This is read “n choose k”The “!” means factorialn is the total number of observationsk is the number of observations you are interested in
7Developing the Binomial Formula Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 3 children, what is the probability distribution?XP(X)OutcomesProbabilityRewritten
8And now to introduce: The Binomial Probability Formula: If X has the binomial distribution with n observations and probability p of success on each observation, the possible values of X and 0, 1, 2, ... , n. If k is any one of these values,
9Example 1:Twenty-five percent of the customers entering a grocery store between 5 p.m. and 7 p.m. use an express checkout. Consider five randomly selected customers, and let X denote the number among the five who use the express checkout.a. What is the probability that two used express checkout?b. What is the probability that at least four used express checkout?
10Example 2“Do you believe your children will have a higher standard of living than you have?” This question was asked to a national sample of American adults with children in a Time/CNN poll (1/29,96). Assume that the true percentage of all American adults who believe their children will have a higher standard of living is Let X represent the number who believe their children will have a higher standard of living from a random sample of 8 American adults.a. Interpret and find the numerical answer.b. Find the probability that none of the parents believe their children will have a higher standard.
11The Mean and Standard Deviation of a Binomial Random Variable If a count X has the binomial distribution with number of observations n and probability of success p, the mean and standard deviation of X areNow, as the number of trials, n, gets large, the binomial distribution becomes more normal. We can then use the normal distribution to approximate the binomial. The notation B(n,p) becomes N( )