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# INTR STATISTCL REASONING STAT 110

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This 25 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 110 at University of South Carolina - Columbia taught by L. Hendrix in Fall. Since its upload, it has received 34 views. For similar materials see /class/229650/stat-110-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia.

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Date Created: 10/26/15

Chapter 17 Thinking About Chance Exploring Data on a Quantitative Variable What s the chance of getting killed by lightning How could we come up with a number US population about 300 million An average of 80 people per year are killed by lightning Chance of being killed by lightning 80 300000000 0000027 STAT 110 7 Introduction to Descriptive Statistics Exploring Data on a Quantitative Variable What does a baseball player s batting average really mean Let s say your favorite player is batting 300 If a player gets 3 hits out of 10 atbats he has a batting average of 3 10 30 For baseball stats we multiply by 10 So we have player with a batting average of 300 So when our player steps up to the plate he has only a 30 chance of getting a hit STAT 110 7 Introduction to Descriptive Statistics Idea of Probability Why is random good Why do we use random samples and randomized experiments Why is tossing a coin before a football game reasonable Is it random Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run STAT 110 7 Introduction to Descriptive Statistics Randomness and Probability random when individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repe ons probability a number between 0 and 1 that describes the proportion of times an outcome would occur in a very long series of repetitions STAT 110 7 Introduction to Descriptive Statistics Randomness and Probability Random in statistics does not mean haphazard In statistics random describes a kind of order that emerges only in the long run Probability describes the longterm regularity of events Probability describes what happens in very many trials STAT 110 7 Introduction to Descriptive Statistics Probability probability O 9 the outcome never occurs probability 1 9 the outcome happens on every repetition probability 12 9 the outcome happens half the time in a very long series of trials Probability gives us a language to describe the longterm regularity of random behavior STAT 110 7 Introduction to Descriptive Statistics Probability Relative frequency repeat an experiment many times and calculate proportion of time each outcome occurs This is also called experimental or empirical probability Theoretical if we make some assumptions we can calculate probability based on theory STAT 110 7 Introduction to Descriptive Statistics Probability Can begin to think of probability in the following way of ways the outcome occurs Probability of an outcome total of poss1ble outcomes Prob roll 2 on a die 2 P2 1667 STAT 110 7 Introduction to Descriptive Statistics Probability How many possible outcomes when rolling a fair die twice DE BE US EU E U E D E U E U Em IZJ E i m m E D D E B E E E E 6 ways for first die 6 ways for second die 36 STAT 110 7 Introduction to Descriptive Statistics Myth Short Run Regularity The idea of probability is that randomness is regular in the long run Our intuition tells us that randomness should also be regular in the short run When regularity in the short run is absent we look for some explanation other than chance vana on Toss a fair coin six times Which of these outcomes is more probable HTHTTH TTTHHH STAT 110 7 Introduction to Descriptive Statistics Myth Short Run Regularity In basketball what s a hot hand If a player has a hot hand is heshe more likely to make the next shot Players perform consistently not in streaks If a player makes half hisher shots in the long run hits and misses behave like a tossed coin Runs of hits and misses are more common than out intuition expects S 1A1 1 10 7 introduction to Descriptive Statistics Myth Surprising Coincidence When something unusual happens we look back and say Wow what were the chances Say you re spending the summer in London You re at the Tower of London and you run into an acquaintance from college What are the chances STAT 110 7 Introduction to Descriptive Statistics Example Cancer accounts for more than 23 of all deaths in the US Of 300 million people about 500000 will die of cancer per year 18 out of 10000 people How many of those people live in the sample neighborhood There are bound to be clusters of cancer cases simply by chance But when a cluster happens in our neighborhood we tend to be upset and not think statistically STAT 110 7 Introduction to Descriptive Statistics 14 Myth Interpretation of Law of Averages aka The Law of Large Numbers In a large number of independent repetitions of a random phenomenon such as coin tossing averages or proportions are likely to become more stable as the number of trials increases This does not happen by compensation for a run of bad luck because by independent we mean that knowing the outcome of one trial does not change the probabilities for the outcomes of any other trial It is not uncommon to see the Law of Averages misstated in terms of the sumscounts rather than meansproportions l STAT 110 7 Introduction to Descriptive Statistics 15 Myth Interpretation of Law of Averages 10 53 00 l probability 05 Proportion of heads c a 1 S 4 I 0392 I l I I 1 5 10 50 100 Number of tosses l 1 500 1000 STAT 110 7 Introduction to Descriptive Statistics Myth Interpretation of Law of Averages The myth comes into play when believers in the law of averages think that future outcomes must make up for an imbalance For example if you toss a coin six times and get TTTTTT the law of averages believers will tell you the next toss will be a H just so it evens out Coins and dice have no memories Getting a result of 6 tails in a row is not compensated for by the next tosses it is simply overwhelmed by the results of the many tosses in the long run I STAT 110 7 Introduction to Descriptive Statistics 17 Example A couple gets married and decides to start a family They decide to have four children All four are girls Wanting a boy they try again What happens For this couple having children is like tossing a coin Eight girls in a row is highly unlikely but once seven girls have been born it s not at all unlikely that the next child will be a girl I STAT 110 7 Introduction to Descriptive Statistics 18 Personal Probability personal probability an outcome is a number between 0 and 1 that expresses an individual s judgment of how likely the outcome is Personal probabilities are not limited to repeatable settings They re useful because we base decisions on them STAT 110 7 Introduction to Descriptive Statistics Personal Probability Personal probability expresses individual opinion It can t be said to be right or wrong It is NOT based on many repetitions There is no reason why a person s degree of confidence in the outcome of one try must agree with the results of many tries What s the probability that USC will win the football game this weekend STAT 110 7 Introduction to Descriptive Statistics I 20 Risk and Probability We have two types of probability One looks at personal judgment of how likely The other looks at what happens in many repetitions Experts tend to look at what happens in many repetitions and the public looks to personal judgment So what s considered risky STAT 110 7 Introduction to Descriptive Statistics I 21 Risk and Probability Approximately 115 people die in an automobile accident every day in the US that s one person every 13 minutes Yet we all get into cars on a regular basis Approximately 1400 college students will die this year due to alcohol related accidents Yet there are plenty of college students attending parties with alcohol on any given night I E TAT 110 7 Introduction to Descriptive Statistics 22 Risk and Probability Approximately 1400 college students will die this year due to alcohol related accidents this is approximately 4 deaths per day What is the probability of a college student dying in an alcohol related accident this year 1440 deaths 15 9 million college students 91 deaths per million college students Why are there still plenty of parties on any given night where college students are drinking I STAT 110 7 Introduction to Descriptive Statistics 23 Risk and Probability Why is there such a difference between what we consider risky and what the experts consider risky We feel safer when a risk seems under our control than when we can t control it It s hard to comprehend small probabilities The probabilities for some risks are estimated by experts from complicated statistical studies I STAT 110 7 Introduction to Descriptive Statistics 24 Odds What are the odds Odds ofAto B against an outcome means that the probability of that outcome is B A B So odds of 5 to 1 is another way of saying probability of 16 Odds range from O to infinity STAT 110 7 Introduction to Descriptive Statistics I 25

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