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5.1 Sampling Distributions for Counts and Proportions

5.1 Sampling Distributions for Counts and Proportions. Binomial Experiments. Consider the familiar discrete probability distribution of a fair 6-sided die. Now suppose we are interested in obtaining a 3 or a 4 over 100 rolls. Here we say that n=100 and p=1/3.

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5.1 Sampling Distributions for Counts and Proportions

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  1. 5.1 Sampling Distributions for Counts and Proportions

  2. Binomial Experiments Consider the familiar discrete probability distribution of a fair 6-sided die. Now suppose we are interested in obtaining a 3 or a 4 over 100 rolls. Here we say that n=100 and p=1/3. It seems like over 100 rolls, roughly 100(1/3)≈33 would turn up a 3 or 4. But what if we want the probability of exactly seven 3s? Or more than ten 4s? This is an example of a binomial experiment, which we denote B(100,1/3).

  3. What is it? • We define a binomial experiment, B(n,p), to satisfy the following properties: • The experiment consists of a fixed number n of trials. (100 rolls of a die) • Each trial has exactly two outcomes, success (S) or failure (F). We write P(S)=p and P(F)=q. Then p+q=1 so q=1-p and p=1-q. Each trial is identical and hence p and q do not change from one trial to another. (S= obtaining a 3 or 4. F= obtaining 1,2,5,6) • The trials are independent of one another; that is, the outcome of one trial has no bearing on the outcome of any other trial. This is referred to as a Bernoulli trial. Then the random variable x, called the binomial random variable, is the number of successes in the n trials. The probability distribution of x is a binomial probability distribution.

  4. Suppose we roll a fair die 20 times. What is the probability of obtaining exactly four 3s? • Not quite obvious… • Table C (pp. T-6 to T-10 in the book) gives a binomial probability distribution for various values of n, k, and various probabilities. • Now we may answer this question.

  5. 40% of voters are opposed to a proposed development. If 15 voters are randomly selected, find the probability that • 7 are opposed to the development. • Fewer than 4 are opposed to the development. • More than 2 are opposed to the development.

  6. Practice Problem • A person has a 5% chance of winning a free ticket in a state lottery. If she plays the game 12times, what is the probability she will win more than 1 free ticket?

  7. Expected value • Recall that for a discrete probability distribution, the expected value was the mean. One can show that for a binomial distribution, • It can also be shown that

  8. Thus, we can now compute the mean, variance, and standard deviation for both the voting example and the lottery example. Let’s do that.

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