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This 7 page Class Notes was uploaded by Alyssa Leathers on Thursday March 12, 2015. The Class Notes belongs to Stats 1350 at Ohio State University taught by Ali Miller in Winter2015. Since its upload, it has received 87 views.
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Date Created: 03/12/15
STATS 1350 31215 134 PM Week Nine Talking About ChanceProbability Models Chance random behavior is unpredictable in the short run but has regular and predictable pattern in the long run Random does NOT mean Haphazard A phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large of repetitions 0 Random is the kind of order that emerges only in the long run 0 Ex 0 Gender of baby 0 Coin toss 0 Die Roll 0 Spinning roulette wheel 0 Random samplerandomized experiment Probability The probability of any outcome of a random phenomenon is a number between 0 and 1 that describes the proportion of times the outcome would occur in a vary long series of repetition Probability 0 it NEVER occurs impossible 0 Probability 1 CERTAIN will always occur Coin Tossing Each time you flip a coin there is a 12 chance of getting heads and 12 chance of getting tails assuming fair coin Flip a coin 30 times 0 Proportion of heads in 30 coin tosses o P heads of headstotal of tosses Example 0 Probability of an outcome of a random phenomenon is the proportion of a very long series of repetition on which the outcome occurs Babies 0 Not everything that is random with 2 options has a 5 probability Long run vs short run 0 Probability tells us that randomness is regular in the long run 0 If n is small crazy things can happen 0 Random phenomenon is not necessarily regular in the short run 0 Consider tossing a coin 10 times possible outcomes 0 All of the outcomes are equally likely memory lessquot Example 0 Winning the lottery o All equally likely Myth of the law of averages Flip a coin 9 times and get the following results 0 TITITITI39 0 What s the probability of the 10th coin toss being heads 0 Still 12 the coin doesn t look at the first 9 trials Law of Averages really tell us Averages and proportions more stable 0 Sums or counts more variable Example 0 60 of the customers at a gas station fill up their tanks 0 More likely 0 Between 58 62 of the next customers fill their tank 1000 customers Because averages or proportions are more stable when there are more trials 0 48 of the babies born at a large hospital are girls 0 More likely o A majority more than 50 of the next babies are girls 20 0 Couples eating at a restaurant on Valentine s Day spend an average of50 o More likely o The average amount spent by the next couples is over 60 10 differs from expectation Successive deals are independent of each other Personal Probability guessing 0 Not based on data 0 Not scientific Probability Model 0 For a random phenomenon describes all the possible outcomes and says how to assign probabilities to any collection of outcomes 0 Event collection of outcomes 0 All outcomes and a probability for each Probability Rules 0 Number between 0 and 1 o All possible outcomes together must have probability 1 sum of all probabilities 1 The probability that an event does not occur is 1 minus the probability that the event does occur complement o Pnot A 1 PA o If two events have no outcomes in common the probability that one or the other occurs is the sum of their individual probabilities o If A B have no outcomes in common 0 OR Plus o PA or B PA PB Grades In a Course 0 Add up probabilities Fair Dice Probabilities 0 Odd 36 0 At least 4 36 0 More than 4 26 Example 0 Probability model 0 List of all possible outcomes 0 Probability of family with 7 or more people 0 All add up to 1 o 1 all other probabilities 02 7 or more people 2 0 family with more than 2 people 0 Add probabilities of 37 families 0 family with 3 or 4 people 0 Add probabilities of 3 and 4 families 0 OR as long as mutually exclusive can t be in both categories Example Reese s 50 candied 20 O 12 B 18 Y 0 Probability model 0 O 2050 0 B 1250 0 Y 1850 Randomly select one single orange 2050 0 Not choose a brown candy if you randomly select 3850 0 Yellow or brown candy 3050 What happens when we focus on samples 0 We consider a sample of many candies The proportion of orange candies in the whole bag is a sample statistic a sample proportion A sampling distribution shows us how sample statistics can vary It is a probability model in that it assigns probabilities to the values the sample statistics can take 0 Statistics mean or proportion Sampling Distributions Imagine choosing a random sample Example Reese s Sample proportion 2050 0 Represent the process many times 1000 samples 0 A regular pattern begins to emerge o In the population approximately 50 of the candies in the bag should be orange Sampling Distribution of a statistic tells us what values the statistic takes in repeated samples as opposed to individuals from the sample population and how often it takes those values Example 0 Mean 20 0 Standard deviation 04 0 Sampling distribution of the proportion is normal 0 Probability greater than 28 Z28204 0 Z table 100 z score 227 0 Percentage of samples less than 16 0 Z 1 0 1587 0 More than 26 in sample size of 1000 o 26204 o 100zscore 668 Characteristics of a Sampling Distribution 0 Can be described by its shape center and spread variability 0 Normal shape normal distribution 0 Sampling distribution is centered where the population is centered Expected sample result is population proportion The mean of the mean is the mean 0 Sampling distribution is less spread out than the population standard deviation for x bar is smaller than standard deviation for X Keeping things simple for now Example 0 Mean 50 0 Standard deviation 07 0 Probability of greater than 60 o Zscore 6507 look at chart 0 14 zscore 9192 0 1009192 808 0 Less than 36 OR greater than 57 0 ADD 0 36507 57507 0 look at zscore chart Examples 0 Between 36 z2and 64 22 sample size 50 o 95 0 Greater than 71 o 100 9985 15 31215 134 PM 31215 134 PM
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