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BU255: Statistics Exam-AID

BU255: Statistics Exam-AID. By: Ryan Pink Some images used from course slides. Agenda. Chapter 2-8.. Go through them all.. Show you the formulas.. Use examples for each.. Answer any questions you have.. Leave you with a sick package.. Then, tell your friends to come support! .

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BU255: Statistics Exam-AID

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  1. BU255: Statistics Exam-AID By: Ryan Pink Some images used from course slides

  2. Agenda • Chapter 2-8.. • Go through them all.. • Show you the formulas.. • Use examples for each.. • Answer any questions you have.. • Leave you with a sick package.. • Then, tell your friends to come support! 

  3. Chapter 2 • What is statistics?: • A way of getting information from data • Is the science of estimating info about a POP based on analysis from a SAMPLE. • Population vs Sample • POP: complete set • SAM: subset of the POP • We make estimates or inferences about the POP from the sample data.

  4. Chapter 2 • Parameter and Statistics • PARA: Describes the population ie pop. mean (μ) or pop. variance (σ2) • STAT: describes a sample, an estimate of the population parameter. ie sample mean or sample variance (s2)

  5. Chapter 2 • Descriptive Statistics: Uses data collected on a group to describe or reach conclusions on that same group. • Inferential Statistics: Uses data collected on a sample to describe or reach conclusions on the population that the sample represents. • Types of Data: • Nominal • Ordinal • Interval

  6. Chapter 2 • NOMINAL: can only be used to classify or categorize • Frequency: how many times did it occur? • Relative Frequency: what percentage of the time did it occur? • Only Pie Graphs and Bar graphs

  7. Chapter 2 • ORDINAL:can be used to rank or order objects • Nominal and Ordinal level data are referred to as nonmetric or qualitative data

  8. Chapter 2 • INTERVAL: distances b/w numbers have meaning • Can draw Histograms, to get probability, proportions. • Ie. average daily temperature or change in stock price • Skewness: a distribution lacks symmetry BIMODAL Negatively Skewed Positively Skewed

  9. Chapter 2 • INTERVAL: • Relationships between two interval variables: • SCATTER DIAGRAM • We are interested in 1) Linearity and 2) Direction

  10. Chapter 4 • Measure of Central Location • Mean, Median, Mode • Measure of Variability • Range, Standard Deviation, Variance, Coefficient of Variation. • Measure of Linear Relationship • Covariance, Correlation, Coefficient of determination

  11. Chapter 4 • Measure of Central Location • Arithmetic Mean • Only for Interval data • Simple Average • 1, 1, 1, 4, 4, 7, 7, 10, 30 • Sum = 65, n = 9. Mean = 65/9 = 7.22 • Median • Value that falls in the middle of the set • 1, 1, 1, 4, 4, 7, 7, 10, 30. • Median = 4 • Mode • Most frequent number • 1 was present three times. NOTATION: N = number in POP n = number in SAM u = mean of POP x = mean of SAM

  12. Chapter 4 • Geometric Mean (diff from Arithmetic) • If you invested in 2006 in RIM ($70), you doubled in 2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately, you lost 64% in 2008 from your 2007 level). • Arithmetic mean = [1.00 + (-.64) ] / 2 = 18% • BUT WRONG! (cause you went from $70 down to $50..) • Geometric mean: • R1 = 100% (OR 1) • R2 = -64% (OR -.64) • Rg = -% • (your annual return is a loss of 15% - DON’T MESS THIS UP!)

  13. Chapter 4 VARIENCE FOR SAMPLE • Measures of Variability • Measures spread • Range: difference between largest and smallest, but doesn’t tell you anything about the points in between. • Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0, 10, and 20 (10-0, 10-10 and 20-10 = 0, but mean of 10 with points 9,10,11 is MUCH tighter, but still sum to 0) VARIENCE FOR POP

  14. Chapter 4 • Standard Deviation • Square root of the variation • Used to compare variability in several pop’s and to make statements about the general shape of a dist. • EMPIRICAL RULE: 1 stdev encompasses 68% of points • 2 stdev’s  95% and 3  99.7% • CHEBYSHEFF’s THEOREM: k stdev encompasses of points (so for 2  1-(1/2)^2 = .75 or 75%) • DIFFERENCE: Empirical Rule is about NORMAL distributions, if NOT NORMAL (or if you don’t know), use Chebysheff to be safe!

  15. Chapter 4 • Example: • If the midterm average of those who attended an SOS session is 80 with a standard deviation of 5 marks, if dist is normal, what range would include 95% of all marks? • Empirical, 2 stdev’s, so 70 - 90 • What range would include 88.9% of marks if the dist was not normal? • ChebySheff, 2 stdev’s is 75%, 3 is 88.9% (try it!) • SO a range of 65 – 95 would include 88.9% of marks.

  16. Chapter 4 • Measure of Linear Relationship: • Three ways to infer strength and direction • Covariance • Coefficient of Correlation • Coefficient of Determination

  17. Chapter 4 • Covariance • If sets are positively correlated, then positive. • If sets are negatively related, then negative. • If no real relationship, then around 0 = Sxy = σxy

  18. Chapter 4 • Coefficient of Correlation • Covariance says ‘what is the relationship? + or – • Coeff. Of Corr says ‘how strong is that relationship? Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)

  19. Chapter 4

  20. Chapter 4 • Coefficient of Correlation: • If -1, 0, or 1 you can definitely indicate the relationship between the two (perfectly +ve etc) • But for all the others between, you don’t know the exact amount that they are affected by each other • Coefficient of Determination! • measures the amount of variation in the dependent variable that is explained by the variation in the independent variable. • Denoted by R2 so just square the coefficient of correlation.

  21. EXAMPLE • How much of Obama’s change in popularity is directly attributed to the length of SNL skits of Palin? (assume normal) END GOAL: NEED Coefficient of Determination!! To get that: need Coefficient of Correlation To get that: need Covariance and both standard deviation’s To get that: need variance

  22. Chapter 4 VARIENCE FOR SAMPLE = Sxy Length Mean = 35 / 5 = 7 mins Obama Mean = 220 / 5 = 44 % of votes (-4)(-9) +(-1)(-2)+(-3)(-6) +(1)(6) +(7)(11) = 139/4 = 34.75 LEN Variance: sx2 (4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76/n-1 = 76/4 = 19 Obama Var: Obama Variance: : sy2 (9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278/n-1 = 278/4 = 69.5

  23. Chapter 4 Sxy = 34.75 Sx = √S2 = √19 = (len stdev) = 4.35 Sy = √69.5 = (obmam stdev) = 8.33 r = 34.75 / (4.35) * (8.33) = 0.959 (between -1 and 1) We know that there is a strong relationship, but since it is not 1 exactly, how much of the variance is due to the length of palin’s skits? – Coefficient of Determination R2 = 0.9592 = .92 92% of the variation in Obama’s % is due to the direct length of Palin’s skits. BOTTOM LINE: See if you can get Palin’s skits extended by any means necessary!!

  24. CHAPTER 5 • 1. Data Collection • 1a. Published data • 1b. Observational and Experimental data • 1c. Surveys • 1d. Sampling • 2. Sampling Methods • 2a. Non-probability sampling • 2b. Probability Sampling • 3. Errors • 3a. Sampling Errors • 3b. Non-sampling Errors

  25. Chapter 5 • Reliability and accuracy depend on the method of collection, and affect the validity of the results. • Three most popular sources: • Published data (revenue can) • PRIMARY = done yourself! • SECONDARY = taking from another source • Observational studies • Uncontrolled recorded of results • Experimental studies • Recording of results while controlling factors

  26. Chapter 5 • Survey • Solicit info from people • Personal / Phone / self-administered • Sampling • Why Sampling: • Lower Cost • Impossible population size • Possible destructive nature of the sampling process • Probability Sampling: 100% random selection • Non-probability sampling: selecting on researcher's judgment that they are representative.

  27. Chapter 5 • Sampling: • Three Different Types: • Simple Random Sampling: Assign numbers, generate random numbers and sample! • Stratified Random Sampling: classify pop into strat’s and then selected randomly within each (age, education, race, province..) • Can get info about whole pop, about relationship between strata’s and among each strata! • Cluster Sampling: if you can’t get a full pop list, or they are hard to question, then take a cluster (GTA, or people on facebook) and sample them • Issue: may increase sampling error due to similarities in cluster!

  28. Chapter 5 • Sampling Errors: • When the distribution of the sample is not the same as the population (means or stdev are different) • INCREASE SAMPLE SIZE to minimize this error!! • Non-sampling Error: • Mistakes made in data acquisition. • Inc sample size does NOT fix this. • 3 types: • Error in Data Acquisition • Non-Response Errors • Selection Bias

  29. Chapter 6 • Introduction to Probability • Assigning Probabilities • Basic Relationships of Events • Joint, Marginal, Conditional Probability • Rules

  30. Probability • Assigning Probabilities • Classical: assume equally likely and independent. • Rolling dice (1/6 chance) • Relative Frequency: assigning probabilities on experimental or historic data. • Forecasting based on previous demand. If you sold 1 computer 20% of all working days, use that going forward. • Subjective: assign on assignor’s judgment • When historic measure aren’t good enough, often used in conjunction with benchmarks. (WEATHER FORECASTING!) • Theoretical: use known probabilities. • Based on a calculated probability (like arrivals at Tim Horton’s in queue theory)

  31. Events • 4 different type of events: • Complement of an Event • Union of Two Events • Intersection of Two Events • Mutually Exclusive Events

  32. Chapter 6 • Joint Probability • Intersection of two events. • P(A and B) • Question: Odds you passed and you came to an SOS session? P(Pass and SOS)

  33. Chapter 6 • Marginal Probability • The summation of a particular event • Add up each row and column (make new r/c) • Question: Probability that you will pass the exam?

  34. Marginal Probability • The summation of a particular event • Add up each row and column (make new r/c) • P(A1) = P(A1 + B1) + P(A1 + B2) • Question: Probability that you will pass the exam?

  35. Conditional Probability • The probability of an event GIVEN another event • P(A | B) = P(A B) / P(B) • Question: Probability that you passed given you came to an SOS session? • P(passed | attended SOS) = P(passed and came) / P(attended) • .40/ .45 = 88.8% U

  36. RULES • No empty set • The probability of A is 1 minus its complement • Union is all of A + all of B, subtract what they have in common (don’t double count!) • If A and B are mutually exclusive (no touching of circles) then it is just P(A) + P(B) • Set of A is smaller or equal to set of B if A is a subset of B.

  37. RULES • Independent Events • Events A and B are independent if P(A|B) = P(A) • If there is a 30% chance that it is going to rain on your exam day. • Question: Probability that you passed given that it rained? • P(passed| rained) = They are independent, no correlation, so • = P(passed) = 85%

  38. Bayes Theorem • Start with your initial or prior probabilities. • You get new info. • So now with new info, you calculate revised or posterior probabilities • This process is Bayes Theorem

  39. Bayes Theorem • Bayes’ theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space KEY DIFFERENCE: You are just now, adding up all the partitions that contain B on the bottom, since you have them all split up. Conditional Probability: P(Ai|B) = P(Ai)*P(B|Ai) P(B)

  40. Bayes Theorem • Example: • Two printer cartridge companies, Alamo and Jersey. • Alamo makes 65% of the cartridges • Jersey makes 35%. • Alamo has a defective rate of 8% • Jersey has a defective rate of 12% • Customer purchases a cartridge, prob that Alamo made it? - Cartridge is tested, and it is defective. (new info) b) What is the probability that Alamo made the cartridge? c) What is the probability that Jersey made the cartridge?

  41. ANSWER • The knowledge of the producer breakdown is the prior probability: • Alamo = 65% P(E1) • Jersey = 35% P(E2) • We know the conditional probabilities of the defective rates: • Alamo = 8% P(D|E1) • Jersey = 12% P(D|E2)

  42. ANSWER 1: TABLE Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random Given that you got a defective cartridge, since there is a 9.4% chance of getting a defective one, and 5.2% of that 9.4% is Alamo’s, then you have a 55.3% of it being Alamo’s!

  43. ANSWER 2: TREE Defective .08 .052 Alamo .65 .094 Acceptable .92 .598 Defective .12 .042 Jersey .35 Acceptable .88 .308 Revised Probabilty: Alamo = .052 / .094 = .553 Revised Probabilty: Jersey = .042 / .094 = .447

  44. ANSWER 3: FORMULA • Chance defective will be an Alamo: Probably of defective given an Alamo (.08) Probably of an Alamo (.65) P(Alamo | D) = The summation of all the cartridge types * their defective probability (find out in total how many defective ones are there?) = (.094) P(Alamo) * P(D|Alamo) P(Alamo)*(P(D|Alamo) + P(Jersey)* P(D|Jersey) P(Alamo | D) = .65 * .08 / (.65*.08) + (.35*.12) P(Alamo | D) = .052 / .094 = 55.3%

  45. CHAPTER 7 1. Random Variables and Probability • Distributions: Introduction 2. Discrete Probability Distributions • A. Introduction • B. Mean and Variance • C. Laws of Mean and Variance 3. Bivariate Distributions • A. Introduction, Marginal probability distribution • B. Mean, Variance, covariance, coefficient of correlation • C. Conditional probability, independence • D. Laws of summation

  46. Random Variable • Random variable definition: a variable that contains the outcomes of a chance experiment. • Two types: • Discrete Random Variable • Countable number of values (students in a class) • Continuous Random Variable • Takes on an uncountable number of possible outcomes • Time in 100m sprint (could be 9.5s, or 9.51s, or 9.519s…)

  47. Discrete Prob. Distributions • Table / Graph that lists all the outcomes and their probabilities = Discrete Prob. Dist. • You can calculate the prob of a certain outcome • P(x) • RULES: • P(x) MUST be between 0 and 1 • Sum of all P(xi) = 1

  48. Continuous Prob. Distribution • This represents a population (since infinite amount of outcomes), and need to calculate parameters to depict distribution: • Need Pop mean and Pop variance. • Population Mean • (using discrete variables to determine parameter about pop): • Population Variance • (using discrete variables to determine parameter about pop): OR

  49. Example • With dice: • What is the probability distribution? • What is the mean?What is the variance? • What is the stdev? Mean = 1 (1/6) + 2 (1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5 Variance = [12(1/6)+ 22 (1/6) + 32 (1/6) + 42 (1/6) + 52 (1/6) + 62 (1/6) ] – 3.52 Variance = 15.166 – 12.25 = 2.91 Standard Deviation = sqrt (2.91) = 1.70

  50. Laws Of Expected Value / Var Example: Long Distance phone company bills customers 50 cents per call plus 2 cents a minute. A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 . Determine the mean and variance for each call: E(cX + c) = E(cX) + c E(cX) + c = cE(X) + c = 2*E(X) + 50 = 2*10 + 50 E(X) = 70 cents V(cX + c) = V(cX) V(cX) = c2 V(X) = 22 (9) V(X) = 36 Stdev(X) = 6 cents

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