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Welcome to MM207 Unit 4 Seminar Binomial and the Discrete Probability Function (w/ Excel)

Welcome to MM207 Unit 4 Seminar Binomial and the Discrete Probability Function (w/ Excel). Definitions. Statistical Experiment : Any process by which we obtain measurements or data. In Unit 3 seminar we discussed dice. Rolling the dice is a statistical experiment.

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Welcome to MM207 Unit 4 Seminar Binomial and the Discrete Probability Function (w/ Excel)

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  1. Welcometo MM207Unit 4 SeminarBinomial and the Discrete Probability Function (w/ Excel)

  2. Definitions • Statistical Experiment: Any process by which we obtain measurements or data. • In Unit 3 seminar we discussed dice. Rolling the dice is a statistical experiment. • Random Variable: A random variable is the outcome of a statistical experiment. We don’t know that this outcome will be before conducting the experiment • Discrete random variable: The possible values of the experiment take on a countable number of results. • For the roll of a die there were 6 possible results. • Discrete variables have to be counted (like eggs) • Continuous random variable: The possible values of the experiment are infinite. • For example, measure the weight of 1 year old cows is continuous. The number of possibilities are uncountable. (500.12 pounds, 534.1534 pounds, etc… • Continuous variables have to be measured (like weight or milk or gas)

  3. Probability Distribution • Probability distribution is the assignment of probabilities to specific values for a random variable or to a range of values for the random variable • Plain English: Each random variable (outcome) from a random experiment has a particular probability of occurring. • See page 196 Example 2

  4. Probability distribution properties • Mean: This is the expected value of a probability distribution. This is the outcome about which the distribution is centered.μ = ∑ x P(x) • Standard deviation: This is the spread of the data around the expected value (mean)σ = √ ∑ (x – μ)2 P(x) • How do you use the equations? Let’s use EXCEL!!!!.

  5. Excel Procedure for Mean & Standard Deviation (Try It Yourself 5 & 6; pp. 198-199)

  6. Features of the Binomial Experiment • Fixed number of trials denoted by n • n trials are independent and performed under identical conditions • Each trial has only two outcomes: success denoted by S and failure denoted by F • For each trial the probability of success is the same and denoted by p. The probability of failure is denote by q and q = 1 - p) • The central problem is to determine the probability of x successes out of n trials. P(x) = ?

  7. Example n = 10 p = 0.4 x = 6 Find P(x = 6) Using the binomial table (Table 2 A8-A10) Using the binomial formula Using Excel

  8. Using the Binomial Formula P(x) = nCx px qn-x x = number of successes n = number of trials p = probability of one success q = probability of one failure (1 – p) nCx is the binomial coefficient give by nCx = n! / [x! (n-x)!] Remember 4! = 4*3*2*1 = 24 and is called factorial notation.

  9. Using the Binomial Formula n = 10 p = 0.4 x = 6 Find P(x = 6) P(x) = nCx px qn-x nCx= 10C6 = 210 px = 0.46 = 0.004096 qn-x = 0.610-6 = 0.64 =0.1296 210* 0.004096 * 0.1296 = 0.111476736 ≈ 0.111

  10. Using Excel to Calculate Binomial Probabilities

  11. Using Excel n = 10 p = 0.4 x = 6 Find P(x = 6) Click on the cell where you want the answer. Under fx, find BINOMDIST Number_s: Enter 6 Trials: Enter 10 Probability: Enter 0.4 Cumulative: False P(x = 6) = 0.111476736 ≈ 0.111

  12. Finding Cumulative Probabilities n = 10 p = 0.4 x ≤ 6 Find P(x ≤ 6) Find each probability using the binomial table or the formula. P(x ≤ 6) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6) = 0.006 + 0.040 + 0.121 + 0.215 + 0.251 + 0.201 + 0.111 = 0.945 Use the complement P(x ≤ 6) = 1 – P(x > 6) = 1 – [P(x = 7) + P(x = 8) + P(x = 9) + P(x = 10)] = 1 – [0.042 + 0.011 + 0.002 + 0.000] = 1 – 0.055 = 0.945

  13. Finding Cumulative Probabilities con’t n = 10 p = 0.4 x ≤ 6 Find P(x ≤ 6) Use Excel • Number_s: Enter 6 • Trials: Enter 10 • Probability: Enter .4 • Cumulative: True P(x ≤ 6) = 0.945238118 ≈ 0.945

  14. Mean and Standard Deviation of the binomial probability distribution • Mean or expected number of successμ = np • Standard deviationσ = √ npq • Where:n = number of trialsp = probability of successq = probability of failure (q = 1 – p)

  15. Computing the Mean, Standard Deviation, and Variance for a Binomial Distribution n = 10 p = 0.4 Mean μ = np μ = 10 * 0.4 μ = 4 Standard deviation σ = √ npq σ = √ 10 * 0.4 * 0.6 σ = √ 2.4 ≈ 1.549 Variance σ2 = npq = 2.4

  16. Have a great week!

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