MATH20 S PRE-CALCULUS UNIT H: PROBABIITY AND STATISTICS CLASS NOTES

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1 1 MATH20 S PRE-CALCULUS UNIT H: PROBABIITY AND STATISTICS CLASS NOTES 1. A considerable portion of this material is a review of Grade 9 and prior grades. Regardless; Probability and Statistics plays such an important role in our lives that it is important to very savvy on it. 2. Definition: Statistics - systematic collection and arrangement of large numbers of observations and quantities of numerical observations, and with ways of drawing useful conclusions from such data. Examples: 3. Definition: Sample: a part of the population that is used to gather data about the whole Examples: Surveys, polls Why do we take samples???:give examples of specific ones.

2 2 4. Definition: Bias - an unwanted influence on a sample that prevents the sample from being truly representative of the entire population from which it is drawn. Bias Examples: Asking a team of basketball players what the best running shoe is. (specialist group) Only phoning people with telephones about an economic issue. ( Dewey Defeats Truman Headline 1948! Only rich people had phones back then!) Only asking people on a city bus if they think public transit is important 5. Definition: Unbiased sample: Sample that is representative of the population from which it is taken

3 3 6. Methods of Sampling Simple random sampling: the sample is chosen randomly from the population Good Examples: Bad Examples:

4 4 7. Stratified random sampling: the population is divided into groups (strata or layers) and the data is collected from the strata by simple random sampling. Eg: by age groups. Need to make sure you take proportional numbers samples from each group depending on the size of each strata. Good Examples: Bad Examples: 8. Systematic sampling: every n th member of the population is sampled. The list being sampled may be ordered (alphabetical, seniority, street number, etc). Good Examples: Bad Examples:

5 5 9. Self-selective sample: A sample of the population provides information by volunteering their opinions. Of course those who do not think an issue important are not going to respond, so these types of samples tend to be biased. Good Examples: Bad Examples 10. Cluster sampling: a particular segment of the population is sampled using existing lists (Constituencies, Wards, Households,...) Good Examples: Bad Examples: See also: Definition: Census: all items or individuals of the population are used to collect data The Canadian Census every 10 years See Stats Canada Website Why use samples instead of full census? Quicker, fewer folks to ask, frequently 5% of population is sufficient provided it is a good sample survey What is a partial census?

6 6 13. Errors and Concerns. Who is doing the Polling? It is pretty easy to make surveys and analyze the data to tell someone who wants the survey done what they want to hear! A poll by the XXXXXX party revealed that Example poll data: in Canada, a 2004 OECD study found there had been no significant expansion of public child care in a decade, outside of Quebec. Despite Liberal promises of action, less than 20% of children aged 0-6 years find a place in a regulated, public child care facility A Poll by the Fraser Institute reveals that. [Fraser Instiute is an Independent Public Forum but evident it has a conservative approach based on the books it advertises for example] Example poll: nearly twice as many Canadians favored parent centered financial assistance for child care over more government assistance to child care programs. CHILD CARE ADVOCACY ASSOCIATION OF CANADA: Example Poll:., polling shows considerable support for increased public investments in child care. 14. Intentional Errors/Double Speak. Sometimes errors are intentional to mudify the situation. Can twist the meaning of words. A logging company might say that that 6% of Canada is covered in forest, an environmental group might say 3%. They might both have different definitions of the word forest though. Example: 15. Source. Not saying who the source is or who and how many were sampled should make you wonder about the validity of a survey. Or who the poll was done for (after all, you get paid better if you tell the people asking you to do a survey what they want to hear!) 16. Neutral Responses. Reporting Neutral responses in a biased manner. Example:

7 7 17. The Questions. Not knowing what the actual questions were should also make you suspicious. Some questions can be very leading. Should we have a curfew for children under 12 because they are destructive and poorly behaved? ; is sort of asking like three different questions and mixing in some bad logic at the same time. Here is an actual survey question: It is estimated that disposable diapers account for 2% of the landfill trash today. In contrast, beverage containers, third-class mail, and yard wastes account for 21%. Given this, do you think it would be fair to ban disposable diapers? 18. The question gives too much information in support of not banning diapers. It doesn t consider how long the types of garbage take to decay, or how many actual tons there are of garbage overall, or that diapers are a biological hazard, etc. 19. Self Selective Bias. Only those concerned with an issue actually complete surveys? Those who favour the status quo don t bother? Or check some blogs! They get people of a certain opinion to swamp polls on CJOB or The Glode and Mail for example! 20. People give questioners politically correct answers? Of course they do. You aren t going to tell a person doing a poll you hate a certain category of thing, or person, even if you do. 21. Ignorant people who don't want to appear like they know nothing about the subject. Example i : In a study educators were asked how they would rank Princeton's undergraduate business program. In every case, it was rated among the top 10 departments in the country, even though Princeton does not even offer an undergraduate business major 22. Timing: in January the National Football League reported a poll that revealed football as the nation's favourite (this is at the time of the Super Bowl in the midst of foootball fever ) 23. Statement of questions: Subtle differences in phrasing make large differences in the results 10% said they would support cutting programs involving "aid to the needy" 39% said they would support cutting programs to "public welfare programs" or try saying no to this question: "Do you favour paying hard-working teachers a little more so that our fine young inspired students can have a decent education and a chance at a fulfilling life?" 24. Interpretation of people's answers: Large survey organizations require that their interviewers follow very explicit procedures, and state questions exactly as written. The

8 8 questions have to be read verbatim to the person being surveyed! Even a small change could change the response. 25. Survey Design. Obviously survey design is very important. See: Presenting Results Results can be presented as tables, charts, pie charts, bar graphs, histograms, line graphs, scatter plots, etc. Hopefully you had studied these in Grade 9 already and are able to produce various forms of charts. 27. Graphs Can be misleading What is the matter with these graphs? 104 IQs of Girls and Boys in a Region Doesn t it look like girls are twice as smart as guys? Girls Boys Scatter Plots of Bivariate Data 28. Definitions: Bivariate: a statistics investigation with two variables Bivariate data: data that involves a relationship between two measures Scatterplot: a graph of a set of points representing the relationship between two sets of numbers or data 29. Table of data to prepare a Scatter Plot: Distance From Basket Jump Shots Made Jump shots made in 25 attempts

9 9 Jump Shots Made in 25 Shots 25 Jump Shots Made Distance From Basket 30. Independent Variable (sometimes called the controlled variable) goes on the x- axis (bottom). Dependent variable (sometimes called the response variable) on the y- axis (vertical). The independent variable is the thing over which you have control in your experiment or the thing you are measuring first. The dependent variable is really a measured function of the independent variable: if I am given x, then y occurs. Elapsed time is always an independent variable because you cannot make time happen. 31. Be careful of charts that try to show some causation, which is likely actually coincidence (or association). Example: a chart that shows teachers pay over the years on the bottom and students marks over the years on the vertical. Just because there is a negative correlation doesn t mean the two are actually related or that one causes the other! Another example: plot the number of seagulls in PEI vs the Average marks of your school over the course of several years. There may be a good and strong correlation, but that doesn t mean that one causes the other. There could be other causes for an apparent correlation, called lurking variables or hidden reasons. Example: There are always two reasons for something: a good one and the real one Author unknown

10 10 Correlation 32. Definitions: Correlation: the relationship, or lack there of, that exists between two variables. Positive correlation: both sets of data increase together. [Data moves up to the right] Negative correlation: one set of data decreases as the other set of data increases Strong correlation: when the data is clustered closely along an obvious line No correlation: the two data sets are not related [the line of best fit is a horizontal or vertical line or very weak] Strong and Positive Weak and Negative No Correlation???? Line of Best Fit 33. Definitions: Line of best fit: a line drawn through the points of a scatter plot that best estimates the relationship between the two variables Outlier: a point in a sample separated from the main body of the sample. It can really throw off a line of best fit if it just an erroneous point. It can have undue influence on a line of best fit. Features of a Line of Best Fit (Manually) o The line shows the pattern and direction of the data o The Line passes through as many points as possible or as near to as many points as possible, with the remaining points being equally above and below the line, and spread out along the line rather than concentrated at one end. o The line allows you to make predictions

11 11 Jump Shots Made in 25 Shots Jump Shots Made The line of best shows a negative correlation. The higher the distance from the basket, the fewer jump shots you make Distance From Basket 33. Demo of Line of Best fit on TI83 Graphing Calculator. Demo only, we don t get into TI83 Calculator for a couple weeks. Also see the several on-line demonstrations on the computer lab website. Examples of Outliers ii : Interpolation and Extrapolation 34. Definitions Interpolate: to read information directly from a graph [ in between data points] Extrapolate: to use information on a graph to estimate values that go beyond the graph. Must be very very careful doing this!!! Extrapolation is seldom a good thing!

12 A scatter plot with a strong positive correlation. But can you actually extrapolate to the left? Does that mean in the year 500AD that the Romans could only jump 2 cms?? PROBABILITY 36. Probability is a very common everyday influence in your life. You use it every day whether you know it or not. Deciding whether to run across the street against the red flashing hand, deciding whether to wear that new suede jacket when there is a possibility of rain, or deciding whether to loan a friend some money. 37. The Mathematical Model of Probability. The likelihood that a certain outcome will happen can be expressed as the following: Pr obability of a favoured outcome = 38. Examples of a simple Probability Calculation. number of favourable outcomes total number outcomes a. probability of flipping a coin and getting the outcome of a head. number of ways to get heads 1 Pr ob ( Heads) = = total possible number of outcomes 2 b. probability with one die of rolling a 6. number of outcomes that are a 6 1 Pr ob (6) = = total possible outcomes 6

13 13 c. You calculate the probability that in a class of 12 students, with nine girls and three guys, that the first one to walk through the door for class is a girl. 39. Fairness and Randomness. The simple calculations we did above assumed that there was fairness and randomness. That is, the coin wasn t a trick coin, the die didn t have two sixes on it, and that the arrival time of students to class doesn t depend on the gendre of the student. (Eg: tends to be moms that have to drop off the kids at daycare first!). No process is exactly fair, none of life s events are completely random! 40. Expected Value. We use probability often to calculate a payoff. Is it worth it to do something. It is often possible, provide you can quantify the outcomes, to calculate whether something is worthwhile or not. 41. Demonstration of Expected Value. There is a 50 / 50 draw. Half the pot goes to the winning ticket, half to the organization running the 50 / 50. One hundred tickets are for sale at 1$ each. There are two things that could happen: you buy a ticket and lose. So you lost $1. Or; you could buy a ticket and win (so you win $50) but you had paid $1, so you really win $49 as a profit.. But also notice, there is a 99/100 chance you will lose and only a 1/100 chance you will win. 42. The expected value formula. The value of your expected return is given by: EV = Pr ob( win) * Pr ize Pr ob( lose) * Cost So in the 50/50 case. EV = 0.01* $ *$1 = $0.50 $0.99 = $0. 49 In other words you can expect to lose 49 cents every ticket you buy! So it is really not a smart investment unless you are feeling charitable. 43. A negative EV means you will lose. Any expected value that is negative guarantees that on average you will lose. What happens if all the tickets are not sold??? Say only 20 are sold.

14 Expected Value Example. You try one now: A hospital is offering tickets for a lottery. The tickets are $100 each and there is one prize. $200, ,000 tickets have been printed. EV = What happens if for some reason only 3,000 tickets are sold? EV = 45. Spirituality and Probability. A famous mathematician once used the Expected Value reasoning to justify faith in there existing a heaven. He weighed the probability of heaven actually existing with the payoff of eternal bliss. It makes for an interesting calculation. But it assumes you can quantify the value of eternal bliss and the related probabilities. 46. Expected Payoff and Morality and the Justice System. Expected value and payoff suggests some moral issues also which we wrestle with every day. What sort of Expected Value and pay-off goes through a criminal s mind? What value is placed on Pride, Virtue, and Respect? Is the penalty (loss) sufficient in any social justice Expected Value? Just curious questions. i ii