Transcript Probability
Probability & Standard Error of the Mean Definition Review Population: all possible cases Parameters describe the population Sample: subset of cases drawn from the population Statistics describe the sample Statistics = Parameters Why Sample???? Can afford it Why Sample???? Can afford it Can do it in reasonable time Why Sample???? Can afford it Can do it in reasonable time Can estimate the amount of error (uncertainty) in statistics, allowing us to generalize (within limits) to our population Even with True Random Selection Some error (inaccuracy) associated with the statistics (will not precisely match the parameters) sampling error: everybody is different The whole measured only if ALL the parts are measured. With unbiased sampling Know that the amount of error is reduced as the n is increased statistics more closely approximate the parameters Amount of error associated with statistics can be evaluated estimate by how much our statistics may differ from the parameters Sample size Rules of thumb Larger n the better law of diminishing returns ie 100 to 200 vs 1500 to 1600 $$$ and time constraints Less variability in population => better estimate in statistics reduce factors affecting variability control and standardization Human beings are terrible randomizers True Random sampling: rare What population is the investigator interested in??? Getting a true random sample of any population is difficult if not impossible subject refusal to participate Catch 22 NEVER know our true population parameters, so we are ALWAYS at risk of making an error in generalization Probability Backbone of inferential stats Probability: the number of times some event is likely to occur out of the total possible events # particular event p= # of possible events Backbone of inferential stats The classic: flip a coin heads vs tails: each at 1/2 (50%) flip 8x: what possible events (outcomes)?? flip it 8 million times: what probable distribution of heads/tails? Wayne Gretzky Wayne Gretzky & probability What is the probability that a geeky looking kid from Brantford, Ontario, Canada would meet, much less marry, a movie star? Wayne’s famous quote: Wayne Gretzky redux. life insurance rates car insurance rates obesity smoking age previous accidents driving demerits flood insurance All life depends on probabilities Voltaire (1756) Life with Probability The Ever-Changing Nature of %s Never go for a 50-50 ball unless you're 80-20 sure of winning it. Ian Darke The 50/50/90 Rule: whenever you have a 50/50 chance of guessing at something, there’s a 90% chance you will guess wrong. Menard’s Philosophy How to Count Cards We are going to show you how to count cards. Card counting is not illegal. If caught counting cards you will not be arrested. You will not be taken into the back room and beaten unconscious, then dragged to the desert and buried with the rest of the casino cheaters. You will not get your fingers cut off with a butcher knife by Michael Corleone. However, if caught counting cards you may be banned from playing at that casino. You have to be smart about counting cards and don't be too obvious. You do not want to be banned from the casino that you are sleeping at. If you are going to try your luck at counting cards we suggest you go down the street to a different casino in case you get caught. Use this [email protected] information at yourFrom own risk. One of the most popular card counting systems currently in use is the point count system, also known as Hi-Low. This system is based on assigning a point value of +1, 0, or -1 to every card dealt to all players on the table, including the dealer. Each card is assigned its own specific point value. Aces and 10-point cards are assigned a value of -1. Cards 7, 8, 9 each count as 0. Cards 2, 3, 4, 5, and 6 each count as +1. As the cards are dealt, the player mentally keeps a running count of the cards exposed, and makes wagering decisions based on the current count total. •The higher the plus count, i.e. the higher percentage of ten-point cards and aces remaining to be dealt, means that the advantage is to player and he/she should increase their wager. •If the running count is around zero, the deck or shoe is neutral and neither the player nor the dealer has an advantage. • The higher the minus count, the greater disadvantage it is to the player, as a higher than normal number of 'stiff' cards remains to be dealt. In this case a player should be making their minimum wager or leave the As the dealing of the cards progresses, the credibility of the count becomes more accurate, and the size of the player's wager can be increased or decreased with a better probability of winning when the deck or shoe is rich in face cards and aces, and betting and losing less when the deck is rich in 'stiff' cards. It is important to note that a player's decision process, when to hit, stand, double down, etc. is still based on basic strategy. Remember, you MUST learn basic strategy. However, alterations in basic strategy play is sometimes recommended based on the current card count. For example, if the running count is +2 or greater and you have a hard 16 against a dealer's up card of ten, you should stand, which is a direct violation of basic strategy. But considering that the deck or shoe is rich in face cards you are more likely to bust in this situation, thus you ignore basic strategy and stand. Another example is to always take insurance when the count is +3 or greater. For the most part however, you should stick with basic strategy and use the card count as an indication of when to increase or decrease the amount of your bet, as that is the whole strategy behind card counting. Probability & the Normal Curve Normal Curve mathematical abstraction unimodal symmetrical (Mean = Mode = Md) Asymptotic (any score possible) a family of curves Means the same, SDs are different Means are different, SDs the same both Means & SDs are different Dice Roll Outcomes Each dice has six equal possible outcomes when thrown numbers one through six. The two dice thrown together have a total of 36 possible outcomes, the six combinations of one dice by the six combination of the other. Dice Roll Outcomes Numbers 2 3 4 5 6 7 8 9 10 11 12 Combinations one two three four five six five four three two one Dice 1 1 1 2, 1 3, 1 4, 1 5, 1 6, 2 6, 3 6, 4 6, 5 6, 6 6 Combinations 2 3 4 5 6 6 6 6 6 1 1, 1, 1, 1, 2, 3, 4, 5 Notice how certain totals have more possibilities of being thrown, or are more probable of occurring by random throw of the two dice. 2 2 2 2 3 4 5 2 3, 4, 5, 5, 5, 5 3 4 5 5 5 2 2, 3 3 2, 3 4, 4 3 3, 4 4 4 Probability & the Normal Curve 99.7% of ALL cases within plus or minus 3 Standard Deviations Any score is possible but some more likely than others (which one?) Using the NC table Mean = 50 SD = 7 What is probability of getting a score > 64? one-tailed probability Probability & the Normal Curve Using the NC table What is probability of getting a score that is more than one SD above OR more than one SD below the mean? two-tailed probability Defining probable or likely What risk are YOU willing to take? Fly to Europe for $1,000,000 BUT… 50% chance plane will crash 25% chance 1%chance .001% chance .000000001% chance Defining probable or likely In science, we accept as unlikely to have occurred at random (by chance) 5% (0.05) 1% (0.01) 10% (0.10) May be one-tailed or two-tailed Serious people take seriously probabilities, not mere possibilities. George Will, 11/2/2000 Six monkeys fail to write Shakespeare Pantagraph, May 2003 Probability & the Normal Curve Any score is possible, but some more likely than others Key to any problem in statistical inference is to discover what sample values will occur in repeated sampling and with what probability. With what probability will a score arise by chance that is as extreme as a certain value???? Statistics Humour A man who travels a lot was concerned about the possibility of a bomb on board his plane. He determined the probability of this, found it to be low but not low enough for him. So now he always travels with a bomb in his suitcase. He reasons that the probability of two bombs being on board would be infinitesimal. Sampling Distributions: Standard error of the mean Recall With sampling, we EXPECT error in our statistics statistics not equal to parameters cause: random (chance) errors Recall With sampling, we EXPECT error in our statistics statistics not equal to parameters cause: random (chance) errors Unbiased sampling: no factor(s) systematically pushing estimate in a particular direction Recall With sampling, we EXPECT error in our statistics statistics not equal to parameters cause: random (chance) errors Unbiased sampling: no factors systematically pushing estimate in a particular direction Larger sample = less error Central Limit Theorem Consider (conceptualize) a distribution of sample means drawn from a distribution repeated sampling (calculating mean) from the same population produces a distribution of sample means Central Limit Theorem A distribution of sample means drawn from a distribution (the sampling distribution of means) will be a normal distribution class: from list of 51 state taxes, each student create 5 random samples of n = 6. Look at distribution in SPSS Mp = 32.7 cents, SD = 18.1 cents Central Limit Theorem Mean of distribution of sampling means equals population mean if the n of means is large Central Limit Theorem Mean of distribution of sampling means equals population mean if the n of means is large true even when population is skewed if sample is large (n > 60) Central Limit Theorem Mean of distribution of sampling means equals population mean if the n of means is large true if population when skewed if sample is large (n > 60) SD of the distribution of sampling means is the Standard Error of the Mean Take home lesson We have quantified the expected error (estimate of uncertainty) associated with our sample mean Standard Error of the Mean SD of the distribution of sampling means Typical procedure Sample calculate mean & SD Typical procedure Sample calculate mean & SD KNOW & RECOGNIZE that Typical procedure Sample calculate mean & SD KNOW & RECOGNIZE that statistics are not exact estimates of parameters Typical procedure Sample calculate mean & SD KNOW & RECOGNIZE that statistics are not exact estimates of parameters a larger n provides a less variable measure of the mean Central Limit Theorem Typical procedure Sample, calculate mean & SD KNOW & RECOGNIZE that statistics are not exact estimates of the parameters a larger n provides a less variable measure of the mean sampling from a population with low variability gives a more precise estimate of the mean Estimating Sample SEm Example Calculation •Mean = 75 •SDp = 16 •n = 64 •SEm = ??? Confidence Interval for the Mean •Mean = 75 •SDp = 16 •n = 64 •SEm = 2 Distribution of sampling means 68% Confidence Interval for the Mean •Mean = 75 •SDp = 16 •n = 64 •SEm = 2 We are about 68% sure that population mean lies between 73 and 77 Sample mean 73 75 68% 77 Confidence Interval for the Mean •Mean = 75 •SDp = 16 •n = 64 •SEm = 2 73 and 77 are the upper and lower limits of the 68% confidence interval for the population mean Sample mean 73 75 68% 77 Example Calculation •Mean = 75 •SDp = 16 •n = 16 •SEm = ??? Example Calculation •Mean = 75 •SDp = 16 •n = 640 •SEm = ??? Example Calculation •Mean = 75 •SDp = 160 •n = 16 •SEm = ??? Example Calculation •Mean = 75 •SDp = 160 •n = 640 •SEm = ??? Explain how SD and n affect the error inherent in estimating the population mean 95 % Confidence Interval for the Mean •Mean = 80 •SDp = 20 •n = 36 •SEm = ?? Distribution of sampling means ?? ?? 80 ?? ?? 95 % Confidence Interval for the Mean •Mean = 80 •SDp = 20 •n = 36 •SEm = 3.33 Limits X 1.96 SE M 1.96 * 3.33 = 6.53 Up = 80 + 6.53 Lo = 80 - 6.53 73.34 76.67 80 95% 83.33 86.66 95 % Confidence Interval for the Mean •Mean = 80 •SDp = 20 •n = 36 •SEm = 3.33 Sample mean 86.53 73.47 73.47 and 86.53 are the upper and lower limits of the 95% confidence interval 73.34 for the population mean 76.67 80 95% 83.33 86.66 Key to any problem in statistical inference is to discover what sample values will occur in repeated sampling and with what probability.