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MM207 Statistics Welcome to the Unit 7 Seminar Prof. Charles Whiffen

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Statistical Significance A set of measurements or observations in a statistical study is said to be statistically significant if it is unlikely to have occurred by chance. A detective in Detroit finds that 25 of the 62 guns used in crimes during the past week were sold by the same gun shop. Statistically Significant? The team with the worst win-loss record in basketball wins one game against the defending league champions. Statistically Significant?

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Quantifying Statistical Significance In general, we determine statistical significance by using probability to quantify the likelihood that a result may have occurred by chance. We therefore ask a question like this one: Is the probability that the observed difference occurred by chance less than or equal to 0.05 (or 1 in 20)?

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Quantifying Statistical Significance If the answer is yes (the probability is less than or equal to 0.05), then we say that the difference is statistically significant at the 0.05 level. If the answer is no, the observed difference is reasonably likely to have occurred by chance, so we say that it is not statistically significant. The choice of 0.05 is somewhat arbitrary, but it’s a figure that statisticians frequently use. Other probabilities, such as 0.1 or 0.01, are also used,.

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Slide 6.1- 5 Copyright © 2009 Pearson Education, Inc. In the test of the Salk polio vaccine, 33 of the 200,000 children in the treatment group got paralytic polio, while 115 of the 200,000 in the control group got paralytic polio. Calculations show that the probability of this difference between the groups occurring by chance is much less than 0.01. Describe the implications of this result. EXAMPLE 2 Polio Vaccine Significance

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Three Approaches to Finding Probability A theoretical probability is based on assuming that all outcomes are equally likely. It is determined by dividing the number of ways an event can occur by the total number of possible outcomes. A relative frequency probability is based on observations or experiments. It is the relative frequency of the event of interest. A subjective probability is an estimate based on experience or intuition.

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Theoretical Probability Experiment: Rolling a single die Sample Space: All possible outcomes from experiment S = {1, 2, 3, 4, 5, 6} Outcomes are the most basic possible results of observations or experiments Event: a collection of one or more outcomes (denoted by capital letter) Event A = {3} Event B = {even number} Probability = (number of favorable outcomes) / (total number of outcomes) P(A) = 1/6 P(B) = 3/6 = ½

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Counting Possible Outcomes Suppose process A has a possible outcomes and process B has b possible outcomes. Assuming the outcomes of the processes do not affect each other, the number of different outcomes for the two processes combined is: a × b This idea extends to any number of processes. If a third process C has c possible outcomes, the number of possible outcomes for the three processes combined is: a × b × c.

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Applying the Counting Rule How many outcomes are there if you roll a fair die and toss a fair coin? The first process, rolling a fair die, has six outcomes (1, 2, 3, 4, 5, 6). The second process, tossing a fair coin, has two outcomes (H, T). Therefore, there are 6 × 2 = 12 outcomes for the two processes together (1H, 1T, 2H, 2T,..., 6H, 6T).

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Relative Frequency (Empirical) Probability

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Probability of an Event Not Occurring If the probability of an event A is P(A), then the probability that event A does not occur is P(not A). Because the event must either occur or not occur, we can write: P(A) + P(not A) = 1 or P(not A) = 1 – P(A) The event not A is called the complement of the event A; the “not” is often designated by a bar, so Ā means not A.

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Probability Distributions A probability distribution represents the probabilities of all possible events. Do the following to make a display of a probability distribution: 1.List all possible outcomes. Use a table or figure if it is helpful. 2.Identify outcomes that represent the same event. Find the probability of each event. 3.Make a table in which one column lists each event and another column lists each probability. The sum of all the probabilities must be 1.

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Creating a Probability Distribution What is the probability distribution for the number of heads that occurs when three coins are tossed simultaneously? The number of different outcomes when three coins are tossed is 2 × 2 × 2 = 8.

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3 Coin Probability Distribution

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Law of Large Numbers The law of large numbers (or law of averages) applies to a process for which the probability of an event A is P(A) and the results of repeated trials do not depend on results of earlier trials (they are independent). It states: If the process is repeated through many trials, the proportion of the trials in which event A occurs will be close to the probability P(A). The larger the number of trials, the closer the proportion should be to P(A).

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Roulette Example

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Expected Value The expected value of a variable is the weighted average of all its possible events. Because it is an average, we should expect to find the “expected value” only when there are a large number of events, so that the law of large numbers comes into play. Consider two events, each with its own value and probability. The expected value is: expected value = (value of event 1) * (probability of event 1) + (value of event 2) * (probability of event 2) This formula can be extended to any number of events by including more terms in the sum.

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Winning the Lottery A $1 lottery tickets have the following probabilities: 1 in 5 to win a free ticket (worth $1); 1 in 100 to win $5; 1 in 100,000 to win $1,000; and 1 in 10 million to win $1 million. What is the expected value of a lottery ticket? Thus, averaged over many tickets, you should expect to lose 64¢ for each lottery ticket that you buy. If you buy, say, 1,000 tickets, you should expect to lose about 1,000 × $0.64 = $640. EventValueProbabilityValue * ProbabilityResult Purchase ticket-$11-$1 x 1-$1.00 Win free ticket$11/5$1 x 1/5$0.20 Win $5$51/100$5 x $100$0.05 Win $1,000$1,0001/100,000$1,000 x 1/100,000$0.01 Win $1,000,000 $1,000,0001/10,000,000$1,000,000/10,000,000$.10 Expected Value =$-0.64

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Gambler’s Fallacy The gambler’s fallacy is the mistaken belief that a streak of bad luck makes a person “due” for a streak of good luck. Assume you are playing coin flipping game and you win $1 each time the coin lands Heads. You have just lost 7 flips in a row. You might think that, given the run of heads, a tail is “due” on the next toss. But the probability of a head or a tail on the next toss is still 0.50; the coin has no memory of previous tosses.

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Accident Rates Travel risk is often expressed in terms of an accident rate or death rate. For example, suppose an annual accident rate is 750 accidents per 100,000 people. This means that, within a group of 100,000 people, on average 750 will have an accident over the period of a year. The statement is in essence an expected value, which means it also represents a probability: It tells us that the probability of a person being involved in an accident (in one year) is 750 in 100,000, or 0.0075.

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Death Rates

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Vital Statistics

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Life Expectancy Life expectancy is the number of years a person with a given age today can expect to live on average. As we would expect, life expectancy is higher for younger people because, on average, they have longer left to live. At birth, the life expectancy of Americans today is about 78 years (75 years for men and 81 years for women)

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Using StatCrunch

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What About StatCrunch? Many of the calculations of simple probabilities are best done with just a calculator. However, StatCrunch can be a great help with contingency tables used to calculate compound probabilities. More information on contingency tables can be found on pp. 418-419 (Chapter 10) of the text.

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Example – Appendix A Data Set 18, Homes Sold in Dutchess County

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Stat>Tables>Contingency>with data

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Contingency Table with data

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What this tells us (for example): P(3BR)=19/40 P(2Bath)=21/40 P(1or2Bath)=33/40 P(2Bath & 3BR)=8/40 P(2Bath|3BR)=8/19 P(3BR|2Bath)=8/21

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Contingency Table with data Now you try it! P(4BR)= P(3Bath)= P(3or4Bath)= P(3Bath & 2BR)= P(3Bath|2BR)= P(2BR|3Bath)=

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QUESTIONS?

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Review of Unit 7 Work By Tuesday at Midnight you must complete: Initial post to one discussion question Two responses to other student posts to discussion questions Live Binder MSL HW MSL Quiz 32

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From Randomness to Probability

From Randomness to Probability

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