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# STAT2332 EXAM 2 Study Guide STAT 2332

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This 18 page Study Guide was uploaded by veronica on Tuesday October 11, 2016. The Study Guide belongs to STAT 2332 at University of Texas at Dallas taught by Dr. Chen in Fall 2016. Since its upload, it has received 119 views. For similar materials see Introductory Statistics for Life Sciences in Statistics at University of Texas at Dallas.

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Date Created: 10/11/16

EXAM #2 STUDY GUIDE Topics covered in Exam 2: CH 10-12: The SD and Regression Line - SD line - Regression, Residuals, RMSE CH 13 & 14: Intro to Probability - Conditional probability - Independence - Mutually exclusive events - Multiplication Rule - The Addition Rule CH 15: Finding probability in specific circumstances - Permutations - Combinations - Binomial Distributions - Geometric Distributions *(Poisson and Exponential Distributions will be on Exam 3 not Exam 2) SD line: - Both the SD line and the regression line go through the point of averages (AVGx, AVGy) - The sign of the slope of the SD line is the same as the sign of r ???????????? You’ll be finding the - The size of the slope of SD line = slope of the ???????????? regression line more often than this one Regression line: - Describes how a response variable y changes as an explanatory variable x changes. - Used to predict values of y - Always less steep/smaller slope than the SD line because −1 ≤ r ≤ 1 Correlation Coefficient ( r ): measures the strength and direction of a linear relationship. r = 0.95 r = - 0.98 If r is positive positive correlation + Correlation If r is negative negative correlation – ≠ The closer r is to -1 or +1, the stronger Causation the relationship/association r = 0 r = 0.90 (nonlinear) You can also rewrite this as Equation for regression line: ŷ = m x − AVGx + AVGy ŷ = ???????? + ???? or ???????????? ŷ = ???? ???????????? ???? + ???????????????? − ???????????????????? Where ŷ = predicted value ???????????? m = slope = ???? ???????????? ???????????? b = y-intercept = ???????????????? − ???????????????????? or ???????????????? − ???? ???????????? ???????????????? RMSE: Root-Mean-Square-Error 2 ???????????????? = √ 1 − ???? ∙ ???????????? Residual: Difference between observed value and predicted value ???????????????????????????????? = ???? − ŷ Extrapolation: Predicting beyond the range of predictor variables; predicting outside our data; risky & not good probability: describes chance of occurrence or outcome. P(event) = favorable outcome total outcomes sample space (S): collection of all possible outcomes. Represented w/ “S” Ex: roll a die S= {1,2,3,4,5} sample space Event: any collection of outcomes from the sample space. Represented w/ a letter (A, B, C, etc.) Ex: roll a die, Event A is rolling a prime # A= {2,3,5} Complement: the complement of an event is the event NOT occurring. Represented by A’ or Ā Ex: A= roll an even # A= {2,4,6} Ā= do not roll an even # Ā= {1,3,5} Complement Rule: P(A) + P(Ā) = 1 Conditional Probability: probability that takes into account a given condition; gives the probability of one event “given that” another event has happened. Notation is P(B|A) probability of B “given that” A has occurred; “Event B given Event A” st nd *A is the 1 event and B is the 2 event. * ???? ???? ???????????? ???? ) ???? ????|???? = ???? ????) Or if you have INDEPENDENT events ???? ????|???? = ????(????) Independent: event has no effect on the probability of another event occurring. Ex: drawing with replacement Ex:In 3 tosses of a coin, what is the probability that you will get 3 tails in a row (T)? P(T and T and T) = 0.5 × 0.5 × 0.5 = 1/8 or 0.125 General Multiplication Rule: ???? ???? ???????????? ???? = ????(????) × ????(????|????) Or if you have INDEPENDENT events ???? ???? ???????????? ???? = ????(????) × ????(????) The event A and B happens= consists of all outcomes that are in both events. Ex: A= drawing a red card A and B is the same as A∩B (called intersection) B= drawing a 3 A or B is the same as A and B = {2 hearts, 2 diamonds} A∪B (called union) P(A and B) = 2/52 cards = 1/26 Addition Rule: ???? ???? ???????? ???? = ???? ???? + ???? ???? − ????(???? ???????????? ????) Or if you have MUTUALLY EXCLUSIVE events ???? ???? ???????? ???? = ???? ???? + ????(????) The event A or B happens= consists of all outcomes that are in at least one of the 2 events. Shortcut for finding probability of at least 1 event: ???? ???????? ???????????????????? ???????????? = 1 − ????(???????????????? ????????????ℎ???? ???????????????????????? ????????????????????) mutually exclusive/disjoint: occurrence of one event prevents the occurrence of the other. Events cannot happen together at once. Ex: A= vegetable is a carrot B= vegetable is a lettuce P(A and B) = 0 or Ø (called an empty set, Ø means no outcomes) Ex: Toss coin 1 time. A= Head B=Tail P(A and B) = Ø Mutually exclusive events a.k.a. disjoint events will always have P(A and B) equal zero. Independent events are NOT the same thing as mutually exclusive events; independent events can happen at once. Permutation: ordered arrangement of “n” distinct items; how many ways we can rearrange a set in order. Ex: {a,b,c} can be rearranged as{b,a,c}{b,c,a}{a,c,b}{c,a,b}{c,b,a} - ORDER matters Permutations are for lists (order matters) - Notation: and combinations are for groups (order doesn’t matter). P(n,r) orn r Combination Ex: Choose 3 desserts from a menu of 10 C(10,3) = 120 Combination: an arrangement of things selected in which order of selection doesn’t matter. - Order doesn’t matter - Notation: C(n,r) orn r **Note: We will be using k instead of r in class; replace r with k for combinations. Venn Diagram comparing Binomial vs. Geometric Distributions: Binomial Geometric Distribution Distribution Each observation either success or failure Fixed n (total # Observations are No limit to of trials) independent # of trials k = # of Probability of success p m = # of successes stays constant for each trials (minimum # of trial successful (minimum Discrete (k = 1,2,3etc.) # of trial is event k = 0) m = 1) For Binomial Distribution: The probability of k successes out of n trials is ???? ???? = ????! ???? ???? − ???? )????−???? ????! ???? − ???? ! Ex: Ask 10 people if they eat fried chicken. The chance that each person says yes is 60%. What is the probability that 7 eat fried chicken? ???? ???? = ???? =) ????????! (????.????????) ????.???????? )???? = ????.???????????? ????! ???????? − ???? !) (a) Probability that more than 3 eat fried chicken = ???? ???? < ???? = ???? ???? = ???? + ???? ???? = ???? + ????(???? = ????) ???????? ???? ???? ≤ ???? = ???? ???? = ???? + ???? ???? = ???? + ????(???? = ????) (b)Probability that at least 1 eats fried chicken = ???? ???? ≥ ???? = ???? ???? = ???? + ???? ???? = ???? + ⋯+ ????(???? = ????????) or use the “at least one” rule 1-P(none) ???? − ????(???? = ????) (c) Probability that more than 1 eats fried chicken = ???? ???? > ???? = ???? ???? = ???? + ???? ???? = ???? + ⋯+ ????(???? = ????????) For more examples on using the binomial and geometric probability formula, please see uploaded CH 15 week 6 notes (week 7 for geometric) For Geometric Distribution: The probability that the first success occurs atthtrial is ????−???? ???? ???? = ???? = ???? − ????( ) ???? Or P(m) but P(X=m) is more common The probability that at least m trials are needed to get the first success is ???? ???? ≥ ???? = ???? − ???? ( )????−???? Lack of Memory Property: - Information from the past/what happened before doesn’t affect the probability; process resets itself even after consecutive failures Geometric Prob. Ex: A cereal manufacturer puts a special prize in 1/20 of the boxes. 1. What is the probability that you have to purchase 3 boxes to get a prize? ???? ???? = ???? = ???? − ????.???????? )????−???? ????.???????? 2. What is the probability that you have to purchase at least 3 boxes to get a prize? ???? ???? ≥ ???? = ???? − ????.???????? )????−???? 3. What is the probability of getting a prize before purchasing 4 boxes? ???? ???? < ???? = ???? ???? = ???? + ???? ???? = ???? + ????(???? = ????) or ???? ???? ≤ ???? = ???? ???? = ???? + ???? ???? = ???? + ????(???? = ????) 4. Suppose you have already purchased 5 boxes and didn’t get a prize. What is the probability that you have to purchase at least 3 more boxes before getting a prize? This is a lack of memory problem (previous purchases don’t matter) ???? ???? ≥ ???? = ???? − ????.???????? )????−???? -end- EXAM #2 STUDY GUIDE Topics covered in Exam 2: CH 10-12: The SD and Regression Line - SD line - Regression, Residuals, RMSE CH 13 & 14: Intro to Probability - Conditional probability - Independence - Mutually exclusive events - Multiplication Rule - The Addition Rule CH 15: Finding probability in specific circumstances - Permutations - Combinations - Binomial Distributions - Geometric Distributions *(Poisson and Exponential Distributions will be on Exam 3 not Exam 2) SD line: - Both the SD line and the regression line go through the point of averages (AVGx, AVGy) - The sign of the slope of the SD line is the same as the sign of r ???????????? You’ll be finding the - The size of the slope of SD line = slope of the ???????????? regression line more often than this one Regression line: - Describes how a response variable y changes as an explanatory variable x changes. - Used to predict values of y - Always less steep/smaller slope than the SD line because −1 ≤ r ≤ 1 Correlation Coefficient ( r ): measures the strength and direction of a linear relationship. r = 0.95 r = - 0.98 If r is positive positive correlation + Correlation If r is negative negative correlation – ≠ The closer r is to -1 or +1, the stronger Causation the relationship/association r = 0 r = 0.90 (nonlinear) You can also rewrite this as Equation for regression line: ŷ = m x − AVGx + AVGy ŷ = ???????? + ???? or ???????????? ŷ = ???? ???????????? ???? + ???????????????? − ???????????????????? Where ŷ = predicted value ???????????? m = slope = ???? ???????????? ???????????? b = y-intercept = ???????????????? − ???????????????????? or ???????????????? − ???? ???????????? ???????????????? RMSE: Root-Mean-Square-Error 2 ???????????????? = √ 1 − ???? ∙ ???????????? Residual: Difference between observed value and predicted value ???????????????????????????????? = ???? − ŷ Extrapolation: Predicting beyond the range of predictor variables; predicting outside our data; risky & not good probability: describes chance of occurrence or outcome. P(event) = favorable outcome total outcomes sample space (S): collection of all possible outcomes. Represented w/ “S” Ex: roll a die S= {1,2,3,4,5} sample space Event: any collection of outcomes from the sample space. Represented w/ a letter (A, B, C, etc.) Ex: roll a die, Event A is rolling a prime # A= {2,3,5} Complement: the complement of an event is the event NOT occurring. Represented by A’ or Ā Ex: A= roll an even # A= {2,4,6} Ā= do not roll an even # Ā= {1,3,5} Complement Rule: P(A) + P(Ā) = 1 Conditional Probability: probability that takes into account a given condition; gives the probability of one event “given that” another event has happened. Notation is P(B|A) probability of B “given that” A has occurred; “Event B given Event A” st nd *A is the 1 event and B is the 2 event. * ???? ???? ???????????? ???? ) ???? ????|???? = ???? ????) Or if you have INDEPENDENT events ???? ????|???? = ????(????) Independent: event has no effect on the probability of another event occurring. Ex: drawing with replacement Ex:In 3 tosses of a coin, what is the probability that you will get 3 tails in a row (T)? P(T and T and T) = 0.5 × 0.5 × 0.5 = 1/8 or 0.125 General Multiplication Rule: ???? ???? ???????????? ???? = ????(????) × ????(????|????) Or if you have INDEPENDENT events ???? ???? ???????????? ???? = ????(????) × ????(????) The event A and B happens= consists of all outcomes that are in both events. Ex: A= drawing a red card A and B is the same as A∩B (called intersection) B= drawing a 3 A or B is the same as A and B = {2 hearts, 2 diamonds} A∪B (called union) P(A and B) = 2/52 cards = 1/26 Addition Rule: ???? ???? ???????? ???? = ???? ???? + ???? ???? − ????(???? ???????????? ????) Or if you have MUTUALLY EXCLUSIVE events ???? ???? ???????? ???? = ???? ???? + ????(????) The event A or B happens= consists of all outcomes that are in at least one of the 2 events. Shortcut for finding probability of at least 1 event: ???? ???????? ???????????????????? ???????????? = 1 − ????(???????????????? ????????????ℎ???? ???????????????????????? ????????????????????) mutually exclusive/disjoint: occurrence of one event prevents the occurrence of the other. Events cannot happen together at once. Ex: A= vegetable is a carrot B= vegetable is a lettuce P(A and B) = 0 or Ø (called an empty set, Ø means no outcomes) Ex: Toss coin 1 time. A= Head B=Tail P(A and B) = Ø Mutually exclusive events a.k.a. disjoint events will always have P(A and B) equal zero. Independent events are NOT the same thing as mutually exclusive events; independent events can happen at once. Permutation: ordered arrangement of “n” distinct items; how many ways we can rearrange a set in order. Ex: {a,b,c} can be rearranged as{b,a,c}{b,c,a}{a,c,b}{c,a,b}{c,b,a} - ORDER matters Permutations are for lists (order matters) - Notation: and combinations are for groups (order doesn’t matter). P(n,r) orn r Combination Ex: Choose 3 desserts from a menu of 10 C(10,3) = 120 Combination: an arrangement of things selected in which order of selection doesn’t matter. - Order doesn’t matter - Notation: C(n,r) orn r **Note: We will be using k instead of r in class; replace r with k for combinations. Venn Diagram comparing Binomial vs. Geometric Distributions: Binomial Geometric Distribution Distribution Each observation either success or failure Fixed n (total # Observations are No limit to of trials) independent # of trials k = # of Probability of success p m = # of successes stays constant for each trials (minimum # of trial successful (minimum Discrete (k = 1,2,3etc.) # of trial is event k = 0) m = 1) For Binomial Distribution: The probability of k successes out of n trials is ???? ???? = ????! ???? ???? − ???? )????−???? ????! ???? − ???? ! Ex: Ask 10 people if they eat fried chicken. The chance that each person says yes is 60%. What is the probability that 7 eat fried chicken? ???? ???? = ???? =) ????????! (????.????????) ????.???????? )???? = ????.???????????? ????! ???????? − ???? !) (a) Probability that more than 3 eat fried chicken = ???? ???? < ???? = ???? ???? = ???? + ???? ???? = ???? + ????(???? = ????) ???????? ???? ???? ≤ ???? = ???? ???? = ???? + ???? ???? = ???? + ????(???? = ????) (b)Probability that at least 1 eats fried chicken = ???? ???? ≥ ???? = ???? ???? = ???? + ???? ???? = ???? + ⋯+ ????(???? = ????????) or use the “at least one” rule 1-P(none) ???? − ????(???? = ????) (c) Probability that more than 1 eats fried chicken = ???? ???? > ???? = ???? ???? = ???? + ???? ???? = ???? + ⋯+ ????(???? = ????????) For more examples on using the binomial and geometric probability formula, please see uploaded CH 15 week 6 notes (week 7 for geometric) For Geometric Distribution: The probability that the first success occurs atthtrial is ????−???? ???? ???? = ???? = ???? − ????( ) ???? Or P(m) but P(X=m) is more common The probability that at least m trials are needed to get the first success is ???? ???? ≥ ???? = ???? − ???? ( )????−???? Lack of Memory Property: - Information from the past/what happened before doesn’t affect the probability; process resets itself even after consecutive failures Geometric Prob. Ex: A cereal manufacturer puts a special prize in 1/20 of the boxes. 1. What is the probability that you have to purchase 3 boxes to get a prize? ???? ???? = ???? = ???? − ????.???????? )????−???? ????.???????? 2. What is the probability that you have to purchase at least 3 boxes to get a prize? ???? ???? ≥ ???? = ???? − ????.???????? )????−???? 3. What is the probability of getting a prize before purchasing 4 boxes? ???? ???? < ???? = ???? ???? = ???? + ???? ???? = ???? + ????(???? = ????) or ???? ???? ≤ ???? = ???? ???? = ???? + ???? ???? = ???? + ????(???? = ????) 4. Suppose you have already purchased 5 boxes and didn’t get a prize. What is the probability that you have to purchase at least 3 more boxes before getting a prize? This is a lack of memory problem (previous purchases don’t matter) ???? ???? ≥ ???? = ???? − ????.???????? )????−???? -end-

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