Lesson 8 - 1 Discrete Distribution Binomial Taken from http://www.pendragoncove.info/statistics/statistics.htm And modified slightly
Knowledge Objectives • Describe the conditions that need to be present to have a binomial setting. • Define a binomial distribution. • Explain when it might be all right to assume a binomial setting even though the independence condition is not satisfied. • Explain what is meant by the sampling distribution of a count. • State the mathematical expression that gives the value of a binomial coefficient. Explain how to find the value of that expression. • State the mathematical expression used to calculate the value of binomial probability.
Construction Objectives • Evaluate a binomial probability by using the mathematical formula for P(X = k). • Explain the difference between binompdf(n, p, X) and binomcdf(n, p, X). • Use your calculator to help evaluate a binomial probability. • If X is B(n, p), find µx and x (that is, calculate the mean and variance of a binomial distribution). • Use a Normal approximation for a binomial distribution to solve questions involving binomial probability
Vocabulary • Binomial Setting – random variable meets binomial conditions • Trial – each repetition of an experiment • Success – one assigned result of a binomial experiment • Failure – the other result of a binomial experiment • PDF – probability distribution function; assigns a probability to each value of X • CDF – cumulative (probability) distribution function; assigns the sum of probabilities less than or equal to X • Binomial Coefficient – combination of k success in n trials • Factorial – n! is n (n-1) (n-2) … 2 1
Binomial Mean and Std Dev A binomial experiment with n independent trials and probability of success p has Mean μx = np Standard Deviation σx = √np(1-p)
Example 1 Find the mean and standard deviation of a binomial distribution with n = 10 and p = 0.1 Mean: μx = np = 10(0.1) = 1 Standard Deviation: σx = √np(1-p) = 10(0.1)(0.9) = 0.9 = 0.9487
Using Normal Apx to Binomials As binominal’s number of trials increases the formula for a binomial becomes unworkable (a situation alleviated with statistical software). So statisticians developed a procedure to use a continuous distribution, the normal, to estimate a discrete distribution. This procedure is used later with proportions.
Example 2 Sample surveys show that fewer people enjoy shopping than in the past. A survey asked a nationwide random sample of 2500 adults if shopping was often frustrating and time-consuming. Assume that 60% of all US adults would agree if asked the same question, what is the probability that 1520 or more of the sample would agree? P(X ≥ 1520) = 1 – P(X ≤ 1519) = 1 – binomcdf(2500, 0.6, 1519) = 0.2131 𝜇x = np = 2500 (0.6) = 1500 σx= √np(1-p) = √ 2500(0.6)(0.4) = 600 = 24.49 P(X ≥ 1520) = using normCDF(1520, E99, 1500, 24.49) = 0.2071 a difference of 0.0061 or less than 0.6%
Example 2 cont Histogram showing normal apx to binomial
Simulating Binomial Events • To simulate a binomial event, we must know: • how random variable X and “success” is defined • probability of success • number of trials • Calculator has a randbin(1,p,n) function that will generate results with “1”s as successes • We can store the results in lists and do statistics on the lists as usual
Example 3 Each entry in a table of random digits like Table B in our book has a probability of 0.1 of being a zero. • Find probability of finding exactly 4 zeros in a line 40 digits long. • What is the probability that a group of five digits from the table will contain at least 1 zero? P(X = 4) = binompdf(40, 0.1, 4) = 0.206 P(X ≥ 1) = 1 – P(X=0) = 1 ‒ binompdf (5, 0.1, 0) = .4095 = 1 – P(no zeros)5 = 0.4095 (before binomials) = 1 – (0.9)5 = 0.4095
Example 4 A university claims that 80% of its basketball players get their degree. An investigation examines the fates of a random sample of 24 players who entered the program over a period of several years. Of these players, 12 graduated and 12 are no longer in school. If the university's claim is true, • What is the probability that at most 12 out of 24 players graduate? b) What is the probability that all 24 graduate? P (x < 12) = binomcdf (24, .8, 12) = .000978 binompdf (24, .8, 24) = .0047
Summary and Homework • Summary • Binomial experiments have 4 specific criteria that must be met • Means and Variance for a Binomial • E(X) = np and V(X) = np(1-p) • Normal distribution (continuous) can approximate a Binomial (discrete) • Calculator has a random binomial generator