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AP Statistics

This chapter notes discuss the binomial setting and geometric setting in statistics, including the calculation of probabilities, mean, and standard deviation. It also covers the use of a normal distribution approximation in binomial distributions.

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AP Statistics

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  1. AP Statistics Chapter 8 Notes

  2. The Binomial Setting • If you roll a die 20 times, how many times will you roll a 4? Will you always roll a 4 that many times? • The previous questions dealt with an example of a random occurrence that takes place in a binomial setting.

  3. Binomial Setting • 1. Each observation falls into one of just two categories (often called “success” and “failure”). • 2. There is a fixed number, n, of observations. • 3. The n observations are all independent. • 4. The probability of “success”, usually called p, is the same for each observation.

  4. Binomial Distribution • The distribution of the count, X, of successes in the binomial setting… • B(n, p) • n # of observations • p probability of success on any one observation.

  5. Example • In 20 rolls of a die, what is the probability of getting exactly 3 fours? • Why is this problem difficult to answer based on what you have already learned? • Is this a binomial setting? • You can’t simply use the multiplication rule, because the fours could be rolled in any 3 of the 20 rolls.

  6. Binomial Coefficient • The number of ways of arranging k successes among n observations can be calculated by… • Read as “n choose k” • In your calculator, n choose k can be found by using the command nCr

  7. Finding Binomial Probabilities • X  binomial distribution • n  # of observations • p  prob of success on each observation

  8. Binomial probabilities on the calculator • P(X = k) = binompdf (n, p, k) • pdf  probability distribution function  • Assigns a probability to each value of a discrete random variable, X. • P(X < k) = binomcdf (n, p, k) • cdf  cumulative distribution function  • for R.V. X, the cdf calculates the sum of the probabilities for 0, 1, 2 … up to k.

  9. Mean and Standard Deviation • For a binomial random variable: • When n is large, a binomial distribution can be approximated by a Normal distribution. • We can use a Normal distribution when. • np > 10 and n(1 – p) > 10 • If these conditions are satisfied, then a binomial distribution can be approximated by…

  10. The Geometric Setting • 1. Each observation falls into one of two categories (“success or “failure”) • 2. The observations are independent. • 3. The probability of success, p, is the same for all observations. • 4. The variable of interest is the number of trials required to obtain the first success.

  11. Calculating Geometric Probabilities • P(X = n) = (1 – p)n – 1p • “Probability that the first success occurs on the nth trial” • P(X < n)  geometcdf (p, n)

  12. Mean and Standard Deviation • If X is a geometric random variable with probability of success p on each trial, then • The probability that it takes more than n trials to the first success is… • P(X > n) = (1 – p)n

  13. Calculator Functions for Ch 8 • Binomial • P(X = k)  binompdf(n, p, k) • P(X < k)  binomcdf(n, p, k) • Simulation  randbin(n, p) • Geometric • P(X < n)  geometcdf(p, n) • Normal • P(min< X< max) = normalcdf(min, max, μ, σ)

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