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Section 8.1 Binomial Distributions

Section 8.1 Binomial Distributions. AP Statistics. The Binomial Setting. Each observation falls into one of just two categories, which for convenience we call “success” or “failure” There are a fixed number n of observations The n observations are all independent.

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Section 8.1 Binomial Distributions

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  1. Section 8.1Binomial Distributions AP Statistics

  2. The Binomial Setting • Each observation falls into one of just two categories, which for convenience we call “success” or “failure” • There are a fixed number n of observations • The n observations are all independent. • The probability of success, call it p, is the same for each observation. AP Statistics, Section 8.1.1

  3. The Binomial Setting: Example • Each observation falls into one of just two categories, which for convenience we call “success” or “failure”: Basketball player at the free throw. • There are a fixed number n of observations: The player is given 5 tries. • The n observations are all independent: When the player makes (or misses) it does not change the probability of making the next shot. • The probability of success, call it p, is the same for each observation: The player has an 85% chance of making the shot; p=.85 AP Statistics, Section 8.1.1

  4. Shorthand • Normal distributions can be described using the N(µ,σ) notation; for example, N(65.5,2.5) is a normal distribution with mean 65.5 and standard deviation 2.5. • Binomial distributions can be described using the B(n,p) notation • For example, B(5, .85) describes a binomial distribution with 5 trials and .85 probability of success for each trial. AP Statistics, Section 8.1.1

  5. Example • Blood type is inherited. If both parents carry genes for the O and A blood types, each child has probability 0.25 of getting two O genes and so of having blood type O. Different children inherit independently of each other. The number of O blood types among 5 children of these parents is the count X off successes in 5 independent observations. • How would you describe this with “B” notation? • X=B( ) AP Statistics, Section 8.1.1

  6. Example • Deal 10 cards from a shuffled deck and count the number “X” of red cards. • A “success” is a red card. • How would you describe this using “B” notation? • This is not a Binomial distribution because once you pull one card out, the probabilities change. AP Statistics, Section 8.1.1

  7. Example • Blood type is inherited. If both parents carry genes for the O and A blood types, each child has probability 0.25 of getting two O genes and so of having blood type O. Different children inherit independently of each other. The number of O blood types among 5 children of these parents is the count X off successes in 5 independent observations. • What is the probability that 3 children are type O? AP Statistics, Section 8.1.1

  8. Binomial Coefficient • Sometimes referred to as “n choose k” • For example: “I have 10 students in a class. I need to choose 2 of them.” • In these examples, order is not important. AP Statistics, Section 8.1.1

  9. Binomial Coefficients on the Calculator AP Statistics, Section 8.1.1

  10. Binomial Probabilities AP Statistics, Section 8.1.1

  11. Binomial Distributions on the calculator • Corinne makes 75% of her free throws. • What is the probability of making exactly 7 of 12 free throws. • Binomial Probabilities • B(n,p) with k successes • binompdf(n,p,k) • binompdf(12,.75,7)=.1032 AP Statistics, Section 8.1.1

  12. Binomial Distributions on the calculator • Corinne makes 75% of her free throws. • What is the probability of making at most 7 of 12 free throws. • B(n,p) with k successes • binomcdf(n,p,k) • binomcdf(12,.75,7)=.1576 AP Statistics, Section 8.1.1

  13. Corinne makes 75% of her free throws. • What is the probability of making at least 7 of 12 free throws. • B(n,p) with k successes • binomcdf(n,p,k) • 1-binomcdf(12,.75,6)= AP Statistics, Section 8.1.1

  14. Mean and Standard Deviation of a Binomial Distribution AP Statistics, Section 8.1.1

  15. AP Statistics, Section 8.1.1

  16. AP Statistics, Section 8.1.1

  17. AP Statistics, Section 8.1.1

  18. Example: • A recent survey asked a nationwide random sample of 2500 adults if they agreed or disagreed that “I like buying new clothes, but shopping is often frustrating and time-consuming.” Suppose that in fact 60% of all adults would “agree”. What is the probability that 1520 or more of the sample “agree”. AP Statistics, Section 8.1.1

  19. Normal Approximation of Binomial Distribution • As the number of trials n gets larger, the binomial distribution gets close to a normal distribution. • Question: What value of n is big enough? The book does not say, so let’s see how the close two calculations are… AP Statistics, Section 8.1.1

  20. Calculator • B(2500,.6) or P(X>1520) • 1-binomcdf(2500,.6,1519) • .2131390887 • nCDF(1520, 1E99, 1500, 24.495) • P(X>1520)=.207 AP Statistics, Section 8.1.1

  21. Normal Approximations for Binomial Distributions As a “rule of thumb,” we may use the Normal Approximation when… np ≥ 10 and n (1 – p)≥ 10 AP Statistics, Section 8.1.1

  22. Here are some useful applications of the binomcdf and binomcdf commands: To find P(x = k), use binompdf(n,p,k) To find P(x ≤ k), use binomcdf(n,p,k) To find P(x < k), use binomcdf(n,p,k-1) To find P(x > k), use 1-binomcdf(n,p,k) To find P(x ≥ k), use 1-binomcdf(n,p,k-1) AP Statistics, Section 8.1.1

  23. Homework • Binomial Worksheet AP Statistics, Section 8.1.1

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