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Lecture 11



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  • 1. Introduction to Statistics STA250 Lecture 11 - April 21st, 2010 1
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  • 4. Probability ✤ How we express likelihood mathematically ✤ For an event “A”, the probability of A occurring is denoted “P(A)” ✤ Always number between 0 and 1 ✤ P(A) = 0 means that A never happens ✤ P(A) = 1 means that A always happens 4
  • 5. Independence & Exclusivity ✤ independence - A and B are independent if the occurrence of one does not affect the probability of the other: ✤ P(A|B) = P(A) = P(A|not B) ✤ P(B|A) = P(B) = P(B|not A) ✤ mutually exclusive - A and B are mutually exclusive if it is impossible for both of them to occur: ✤ P(A and B) = 0 5
  • 6. Probability Rules ✤ Probability of not happening is 1 minus probability of occurring ✤ P(not A) = 1 - P(A) ✤ When A and B are independent: ✤ P(A and B) = P(A) × P(B) ✤ P(A or B) = P(A) + P(B) - P(A and B) 6
  • 7. Probability Fundamentals ✤ Sum of probabilities of all possible outcomes is 1 ✤ Flip a coin and you get either heads or tails: ✤ P(heads) + P(tails) = 1 = P(heads or tails) ✤ With mutually exclusive outcomes A, B, C, and D ✤ P(A) + P(B) + P(C) + P(D) = 1 = P(A or B or C or D) 7
  • 8. Conditional Probability ✤ With non-independent events, knowing one has happened may change the likelihood of the other occurring ✤ Conditional probability - what is the probability of A given that B has already happened? ✤ P(A|B) ✤ Bayes Rule for conditional probability: P (A and B) P (B|A) × P (B) P (A|B) = = P (B) P (A) 8
  • 9. Conditional Probability Hoedown ✤ At John Jay, 62.5% of all students hate statistics while 25% of all students hate statistics and passed the class. What is the probability that a student passes stats given that the student hates statistics? ✤ Two fair dice are rolled, what is the (conditional) probability that exactly one die’s value is a 1 or 2 given that they show different numbers? 9
  • 10. Something Really Important ✤ Classic stats problem emerged from the game show Let’s Make a Deal, often called the Monty Hall Problem after the show’s host ✤ Has ended many friendships and caused bitter internet arguments 10
  • 11. The Game ✤ There are 3 doors labeled “1”, “2”, and “3”, behind one of these doors is a fabulous prize that Monty has hidden ✤ You get to choose a door, which may or may not have the prize ✤ Monty opens another door without revealing the prize ✤ You now have the option to stay with your door or switch to another, should you stick with your original choice or switch? 11
  • 12. Choosing The First Door ✤ Three doors and one prize so you’ll pick the right door one out of three times, i.e. P(right first choice) = 1/3 ✤ Likewise, you’ll pick the wrong door with P(wrong first choice) = 2/3 12
  • 13. The Reveal ✤ No matter how you choose, there are two other doors one prize. This means there is at least one of the two unchosen doors with nothing behind it. ✤ Monty knows where the prize is and opens the door that DOESN’T have the prize behind it. ✤ This leaves your door and one other. One of them has the prize and the other doesn’t, should you switch? 13
  • 14. To Switch, or Not To Switch ✤ You don’t know if you have the right door! ✤ What’s the probability that your door has the prize? ✤ What’s the probability that the other door has the prize? ✤ What’s the probability that your door doesn’t have the prize? 14
  • 15. Example of the Game ✤ As an example, the prize is hidden behind door “3”. ✤ If you choose door “3” initially, switching can only lose you the prize ✤ If you choose door “2” initially, Monty must open door “1” and switching will get you the prize ✤ If you choose door “1” initially, Monty must open door “2” and switching will get you the prize 15
  • 16. Switch Already! ✤ Switching is a way of saying “I don’t think the prize is behind this door” ✤ Since the probability is 1/3 that the prize is behind any one door, the probability is 2/3 that the prize is not behind that door ✤ Always switch and you’ll win 2/3’s of the time! 16
  • 17. Expected Values ✤ Probability can be used to estimate rewards in a game of chance ✤ Expected Value = P(A)×Reward(A) + P(B)×Reward(B) + ... ✤ Silly coin-flipping game: If you can flip a coin three times and have exactly one Heads, you get a dollar. If not, you give me a dollar. ✤ Should you take the bet? 17
  • 18. Normal Distribution ✤ The distribution is ✤ unimodal ✤ symmetric ✤ “light tailed” ✤ Notation: X ~ N(μ, σ) means “the random variable X has a normal distribution with mean μ and standard deviation σ“ 18
  • 19. Area Under the Curve Equals 1 0.8 N(!3,0.5) N(2,1) N(!1,3) 0.6 f(x) 0.4 0.2 0.0 !4 !2 0 2 4 19
  • 20. Rules of Thumb ✤ P(within one standard deviation) = 0.68 ✤ P(within 1.68 standard deviations) = 0.95 ✤ P(within three standard deviations) = 0.997 ✤ With “real” normal distributions, you just don’t get outliers! 20
  • 21. Standard Normal Distribution ✤ standard normal distribution is the normal distribution with mean μ = 0 and standard deviation σ = 1: Z ~ N(0, 1) ✤ Any normal distribution can be transformed into a standard normal distribution. If X ~ N(μ, σ), then: X −µ =Z σ ✤ 21
  • 22. Z - Scores & the Standard Normal ✤ Each observation has an associated z-score, which is the number of standard deviations that observation is away from the mean ✤ Converting a sample from a normal distribution to z-scores transforms it to a standard normal distribution ✤ z-score = (observation - mean) ÷ standard deviation ✤ If the observation is above the mean then the Z-score is positive, if below then the Z-score is negative 22
  • 23. Interval Estimation ✤ We might estimate the mean for an entire population using the mean for a small sample, this is called a point estimate. ✤ A confidence interval gives a range of “plausible” values for the population mean ✤ Usually reported as "mean ± wiggle room" ✤ Each interval has an associated level of confidence, usually written as a percent (95% being the most common) ✤ "I am 95% confident that the population mean is in this range, with the sample mean being the most likely guess" 23
  • 24. Two-Sided: 1.96 Std. Dev.’s 24
  • 25. Normal Critical Deviates the point for which the area und ht is γ. how many you wanted to find the middle X% of to travel Critical normal deviate: If ✤ distribution, standard deviations would you have the in each direction. ✤ Define zγ to be the point for which the area under the normal curve to matical notation, zγ is the point f the right is γ. ✤ In more mathematical notation, zγ is the point for which: P (Z > zγ ) = γ, 25
  • 26. Interpreting Confidence Intervals ✤ The width of a confidence interval indicates precision ✤ An observation's z-score can test if an observation is similar to others, bigger than ±1.96 means 95% likely to be different ✤ 95% confidence intervals are by far the most common, but any level of confidence interval can be computed: ✤ 90%: mean ± (1.645 × standard deviation) ✤ 95%: mean ± (1.96 × standard deviation) ✤ 99%: mean ± (2.58 × standard deviation) 26
  • 27. Components of Confidence ✤ How might a confidence interval change as: ✤ Ȳ increases ✤ σ increases ✤ n increases ✤ the confidence level increases (e.g., from 95% to 99%) 27
  • 28. Conflicting Hypotheses ✤ In statistical inference, there are always two conflicting hypotheses: ✤ null hypothesis “H0” - often states “no effect” or “no difference”. This is the hypothesis that we will assume to be true unless we have convincing evidence to the contrary. ✤ alternative hypothesis “H1” or “Ha” - The hypothesis that we will believe only if the evidence strongly supports it. ✤ The null hypothesis typically has “=” in it 28
  • 29. Hypothesis as Metaphor ✤ Hypothesis tests are like U.S. criminal trials ✤ The judicial system is structured such that the accused person is presumed innocent until proven guilty. In such a system the absence of convincing evidence (“beyond a reasonable doubt”) results in the person being set free. ✤ H0: innocent ✤ Ha: guilty 29
  • 30. P-values ✤ In each hypothesis testing situation we will compute a p-value. This is the probability that the null hypothesis is correct given the data. ✤ Accept H0 if the p-value is large ✤ Reject H0 if the p-value is small, go with Ha ✤ How small is small enough? It depends... (usually p < 0.05) 30
  • 31. Notes on Hypothesis Testing ✤ “Statistical significance” is not the same as “clinical significance”. A tiny effect may be “statistically significant” if the sample size is huge. ✤ The p-value does not describe the magnitude of the effect! ✤ When reporting analysis results, a confidence interval should always be provided along with the results of a hypothesis test. ✤ The choice of 0.05 is arbitrary. (p = 0.051 and p = 0.049 should lead to similar conclusions, in practice they often do not) ✤ Never report results as “p < 0.05”, report the p-value and let the reader decide if they agree with your interpretation. 31
  • 32. • Type I Error: Reject H0 when H0 is actually true. – For example, to conclude there is an effect (or a difference) when there really isn’t one. – Also called “false positive”. • Type II Error: Accept H0 when H0 is actually false. – For example, to fail to find an effect (or a difference) when there really is one. – Also called “false negative”. State of nature Decision H0 is true Ha is true Accept H0 qh q Type II qh q Reject H0 Type I 32
  • 33. Probabilities of Errors of Type I and Ty Probabilities ✤ Each of the errors has an associated probability: associated Each of the errors has an probabilit • α = P (Type I Error) • β = P (Type II Error) ✤ Hypothesis testing is set up to control Type I error rate (α) Hypothesis testing is set up to control Type I The experimenter chooses α - everything else follows from this! The experimenter chooses α — everything else ✤ Most common (by far) choice for α is 0.05. ✤ (Also, 0.01 and 0.10most common The on occasion) (by far) choice for α is 0.05 33
  • 34. Comparing Means ✤ Tests: ✤ Single group versus a fixed mean ✤ Two groups with the same variable ✤ Two groups with pairwise observation ✤ Hypotheses: ✤ H0 : the two groups have equal means ( mean A = mean B ) ✤ Ha : the means of the groups are different 34
  • 35. Assumptions for t-Tests ✤ The group (sample) is the Independent Variable (dichotomous) ✤ The outcome of interest is the Dependent Variable ✤ t-Tests are only valid if these assumptions are not violated: ✤ The research question DOES involve the comparison of 2 means ✤ The Dependent Variable is a quantitative scale ✤ The distribution of the Dependent Variable is normal ✤ Independent Variable assigned randomly (independently) 35
  • 36. Met Assumptions, but Which Test? ✤ Only one group with data: One-Sample t-Test ✤ Two groups: ✤ Not related to each other: Independent-Samples t-Test ✤ Related samples (e.g. before & after): Paired-Samples t-Test 36
  • 37. One-Sample t-Test ✤ Compares a sample mean to a known population mean. ✤ Need to know the population mean! ✤ Example: Is there a difference between the population mean IQ (100) and the mean IQ for a sample of 50 John Jay students (125)? 37
  • 38. Paired-Samples t-Test ✤ Sometimes we have two sets of measurements that are related: ✤ Each subject is measured before and after treatment ✤ With pairs of identical twins ✤ Subject has different treatment on left & right arms ✤ For each observation in one group there is exactly one closely related observation in the other groups (can make pairs, one of each group) 38
  • 39. Independent-Samples t-Test ✤ Compares the means of two groups or samples. ✤ One of the most common situations in statistical inference is that of comparing two means from independent samples ✤ Clinical trials - treatment group vs. placebo group ✤ Exposed vs. unexposed ✤ Males vs. females ✤ General population vs. specific subpopulation 39
  • 40. Review: Hypotheses ✤ Null Hypothesis: there is no relationship between the independent and dependent variables ✤ p-value: the probability of the null hypothesis (H0) being true ✤ Reject H0 if p is too small (usually p < 0.05) ✤ If we reject H0, we must instead choose the alternative (Ha) 40
  • 41. Review: t-Tests ✤ Compare the means of exactly two groups ✤ Only one group (with data) compared to a fixed number: ✤ One-Sample t-Test ✤ Two groups (with data): ✤ Not related to each other: Independent-Samples t-Test ✤ Related samples (e.g. before & after): Paired-Samples t-Test 41