I Topic 7 Binomial Distribution A Important Probability Distribution 1 Binomial distribution 2 Normal distribution B Binomial Probability Distribution 1 Trivial examples a 10 coins flipped heads or tails each flip b 20 rolls of die even or odd each roll c Heaving three children boy or girl each time 2 Important Example a Random sample of n 20 items each item either defective or ok b Survey of 1200 registered voters each respondent either prefers A or B C Conditions of Binomial Distribution 1 Lets label an events occurrence a Success event occurred b Failure event doesnt occur 2 Trials a Repetitions of random experiment under identical conditions 3 The random variable of interest is X number of successes in n trials 4 The probability of success is the same for every trial a Denoted by pie b Probability of failure is denoted by 1 pie D Summary of Binomial Probability Distribution 1 The two possible outcomes or each trial are success and failure 2 The experiment has n identical trials 3 The n trials are independent of each other The outcome of one trial does not afect the outcome of the next trial 4 The probability of success pie and the probability of failure 1 pie remain the same from trial to trial 5 The random variable x represents the number of successes 6 The probability of success pie which represents the proportion of items in the population or process that we would like to know is often unknown 7 We sample in order to estimate pie E When sampling the best estimate of pie is the sample proportion p where p equals the number of successes X found in the sample divided by the toral number of trials n 1 Ex if success is defined as a defective part and one defective part is found 2 Binomial distribution creates a discrete probability number of successes out of 400 p 1 400 or 0025 must be a whole number 3 We let X denote the number of successes observed within n number of 4 trials In summary we need three pieces of information for a binomial distribution x pie and n F Binomial example problems 1 What is the probability of exactly 2 heads within 3 coin flips a x 2 pie 5 n 3 b Using what we already know there are three combinations out of eight 1 THH HTH HHT 2 ORDER MATTERS c Probability is and the distribution for the three coin flips is X P X 0 1 2 3 1 8 3 8 3 8 1 8 d Did problem on paper 1 G Binomial Distribution Probability Function 1 Once we know the parameters n and pie we can apply the binomial probability function to find the probability that X equals a chosen value 2 Just as we obtained expected values variances and standard deviations from quantitative data we can also obtain these measures from our qualitative in the binomial distributions 3 For example if we threw a fair die 1000 times how many even numbers would you expect to get 2 on paper a By multiplying the probability of getting an even number pie 5 by the number of trials 1000 our expected value is 500 H Example 1 Recently 1 out of 100 parts made by a manufacturer are defective On a typical day the company manufactures 3000 parts Answers on paper a Justification 1 3000 identical trials 2 Success or failure of each trial 3 X represents number of success 4 ASSUME outcomes of trials are independent 5 Some probability of success on each trial II Topic 8 Normal Distriubtion A Important probability distributions 1 Binomial distribution 2 Normal distribution a Paramter Statistic mean u o Standard deviation 3 Normal Curves a On page 84 4 Charcateristics of the normal distribution x s a Bell shaped unimodel symmetric b Mean median mode all at center c Parameters are mean u and standard deviation o d Two tails extend indefinitely and never touch horizontal axis 1 Axis is continuous and ranges from negative to positive B The standard normal distribution infinity 1 The mean is 0 2 Standard deviation is 1 3 Is used to find probabilities for any normal distribution without resorting to calc Is sometimes referred to as the z distribution since the probabilities are tabulated from standardized z values 4 5 To find probabilities proportions or percentages for a normal curve a Convert x to z b Find z in the table c Look in the body of the table for the appropriate probability 1 Difference between normal distribution and standard normal distribution page 86 6 Example if x is a normal random variable with a mean of 50 and a standard deviation of 8 how many standard deviation away from the mean is x 66 a Answer on paper 5 C Application of the normal probability distribution 1 Example problem a On page 88 b Answer is on paper 6 D To find probabilities proportions or percentages for a normal curve we 1 Convert x to z 2 Look at z table 3 Look in the body of the table for the appropriate probability E to find the percentiles of a normal curve we use a reverse process 1 Loook up the given probability in the body of the z table 2 Read off the appropriate z value 3 Using that z value solve for x using the formula G Important Note PERCENTILES ARE VALUES NOT PERCENTAGES OR XXXXPROBABILTIES H Sleeping times normal curve problem cintunued I using Quantile Z values to determine x value 1 Pages 94 95 1 Pages 95 96 III Topic 9 Sampling Distriubtion for the Sample Mean A Parameters and Estimators a Parameter i ii Value that summarizes a characteristic of a population or process descriptive measure of a population 1 Parameter of the binomial distribution n and pie 2 Parameter of the normal distribution u and o Usually we dont know the parameter values We use sample stats called estimators to estimate them B Concepts of Estimators a Estimators vs estimate i An estimator is a sample stat used to estimate a population parameter 1 Ex x estimates u and s estimates o ii An estimate is a specific observed value of a stat 1 Ex x 50 and s 10 C Unbiased estimator equal to that parameter a An unbiased estimator of a pop parameter is an estimator whose expected value is i For example the sample mean is an unbiased estimator of the population mean because the expected value of the sample mean is equal to the population mean b Unbiaseness required that the average value of the estimator equals the parameter being estimated i We want estimators to be unbiased D Consitant Estimators and closer to the parameter values i x and s are consistent estimators of u and o E Point and Interval Estimates a As you increase the size of a random sample the values of the estimator get closer a A point estimate is a single number that is used to estimate an unknown pop b