1 22S 105 Statistical Methods and Computing 2 Why do we want to study probability So far we have studied descriptive statistics methods of describing or summarizing a sample Introduction to Probability We want to move ahead to inferential statistics methods for using the data in a sample to draw conclusions about the population from which the sample is drawn Lecture 9 February 21 2011 Kate Cowles 374 SH 335 0727 kcowles stat uiowa edu Methods of inferential statistics are based on the question How often would this method give a correct answer if I used it very very many times The laws of probability relate to this question 3 Parameters and statistics A parameter is a numeric quantity that describes a characteristic of a population We almost never can know the exact value of a parameter because we would have to measure every member of the population Example We would like to know the average percent body fat of all Chinese males aged 21 65 years We generally use Greek letters to refer to population parameters is the standard symbol for a population mean 4 A statistic is a numeric value that can be computed directly from sample data Example we draw a sample of 10 Chinese males aged 21 65 years and measure the percent body fat of each one The sample mean x of the 10 data values is a statistic We do not need to use unknown parameters to compute a statistic We often use a statistic to estimate and unknown parameter But the exact value of a particular statistic will be different in different samples drawn from the same population sampling variability 5 6 Randomness French naturalist Count Buffon 1707 1788 tossed a coin 4040 times and got 2048 heads Chance behavior is unpredictable in the short run but has a predictable pattern in the long run proportion heads 2048 4040 0 5069 While imprisoned by the Germans during World War II South African mathematician John Kerrich tossed a coin 10 000 times and got 5067 heads Example tossing a coin proportion heads The proportion of heads in a small number of coin tosses is very variable As more and more tosses are done the proportion settles down It gets close to 0 5 and stays there 5067 10 000 0 5067 In 1900 English statistician Karl Pearson tossed a coin 24 000 times and got 12 012 heads proportion heads 12012 24000 0 5005 American statistician Kate Cowles 19 20 tossed a coin 5 times and got 4 heads proportion heads 4 5 0 8 She repeated the experiment and got 2 heads proportion heads 7 Randomness An experiment or observation is called random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of independent repetitions Examples 2 5 0 4 8 The sample space S is the set of all possible outcomes of a random experiment Examples We flip a coin and record the outcome as a head or tail We flip a coin and record the outcome as a head or tail We draw an 18 year old American male at random and follow up to find out whether he lives to be 65 We draw an 18 year old American male at random and follow up to find out whether he lives to be 65 We draw an American child at random and record birth order We draw an American child at random and record his her position in birth order of children in the family A researcher feeds a baby rat a particular diet and records the rat s weight gain from birth to age 30 days A researcher feeds a baby rat a particular diet and records the rat s weight gain from birth to age 30 days 9 10 An event is any outcome or set of outcomes of a random experiment Capital letters near the beginning of the alphabet often are used to denote events Example At random we draw a child born in the US and record his her live birth order We would observe one of the following events Example A might represent the event that the child is a 1st child B might represent the event that the child is not a first child She is 1st child 2nd child 3rd child 4th child 5th child 6th or later Or we might lump certain outcomes together into a single event of interest Child is 1st child or not 1st child 11 The probability of an event is the proportion of times the outcome would occur in a very long series of repetitions under the same conditions This is the long run frequency definition of probability coin tosses the probability of getting a head is 0 5 birth order of randomly drawn American child Birth 1st 2nd 3rd 4th 5th 6 order Probability 0 416 0 330 0 158 0 058 0 021 0 017 The probability that an event occurs is often denoted with the letter P P A is the probability of event A 12 More probability terminology The event A does not occur is called the complement of A and represented by Ac If A is the event that the randomly drawn child is a first born child then what is Ac Two events A and B that cannot occur simultaneously are disjoint or mutually exclusive The union of two events is the event that one or the other or both occur The union of events A and B is the event A or B or both The intersection of two events is the event that both occur 13 14 The intersection of events A and B is the event A and B Example 3 I have a deck of cards I draw a card at random Without putting it back I draw a second card at random The event A is that the first card is a heart The event B is that the second card is a heart Are events A and B independent Two events A and B are independent if the probability that one occurs does not change the probability that the other one occurs Example Suppose one person tosses a penny and another person tosses a dime The outcomes of the two tosses are independent Each has a probability of 12 of being a head The outcome for one of the coins has no effect on the probabilities of the two possible outcomes for the other coin Example 2 What if the same person tossed the same coin twice 15 Probability models mathematical models for randomness consist of two parts a sample space S a way of assigning probabilities to events 16 Probability rules 1 Any probability is a number between 0 and 1 If P A is the probability of any event A then 0 P A 1 2 All possible outcomes taken together must have probability 1 P S 1 One of the possible outcomes has to happen 17 3 The probability that an event does not occur is 1 minus the probability that the event does occur P Ac 1 P A 18 4 Addition rule …