Exam 1 Review (Notes 1-8)

Transcription

1 1 / 17 Exam 1 Review (Notes 1-8) Shiwen Shen Department of Statistics University of South Carolina Elementary Statistics for the Biological and Life Sciences (STAT 205)

2 Basic Concepts 2 / 17 Type of studies: Case report, prospective cohort, retrospective case-control, experiment Treatment (exposure) variable, outcome variable (endpoint) (Random) Sample v.s. Population Observational unit/experimental unit Variables: categorical (nominal, ordinal), numeric (discrete, continuous)

3 Graphic Representation 3 / 17 Dotplot Barchart Histogram Density (mode, symmetry, skew to the left, skew to the right, unimodal, bimodal, multimodal, heavy tail, light tail, etc) Boxplot

4 Descriptive Statistics 4 / 17 mean, median (difference?) first and third quartiles quantile min, max range, interquartile range (IQR), variance, standard deviation five number summary outlier

5 Describing relation between variables 5 / 17 categorical-categorical: contingency table categorical-numeric: side-by-side dotplot, side-by-side boxplot, etc. numeric-numeric: scatter plot

6 Concept of Inferential Statistics 6 / 17 Goal of statistical inference Population proportion vs. sample proportion Population mean vs. sample mean Parameter vs. statistic

7 Probability and Probability Rules 7 / 17 Event, probability Eight basic probability rules: 1 0 Pr(E) 1 2 k i=1 Pr(E i) = 1 if E i s are all possible experimental outcomes 3 pr(e c ) = 1 pr(e). 4 If E 1 and E 2 are disjoint, Pr(E 1 ore 2 ) = Pr(E 1 ) + Pr(E 2 ) 5 Pr(E 1 ore 2 ) = pr(e 1 ) + pr(e 2 ) pr(e 1 ande 2 ) 6 Multiplicative rule: If E 1 and E 2 are independent then Pr(E 1 ande 2 ) = Pr(E 1 ) Pr(E 2 ) 7 Pr(E 1 ande 2 ) = Pr(E 1 ) Pr(E 2 E 1 ) 8 Rule of total probability: Pr(E 1 ) = Pr(E 2 ) Pr(E 1 E 2 ) + Pr(E2 C) Pr(E 1 E2 C)

8 Rule of total probability 8 / 17 Example: If it rains, the probability of me eating beef noodle soup for dinner is 90%. If it does not rain, the probability of me eating beef noodle soup is 20%. The probability of raining today is 10%. What is the probability of me eating beef noodle today? Pr(E 1 ) = Pr(E 2 ) Pr(E 1 E 2 ) + Pr(E C 2 ) Pr(E 1 E C 2 ) Pr{beef noodle} = Pr{beef noodle and rain } + Pr{ beef noodle and not rain} = Pr{rain} Pr{beef noodle rain} + Pr{not rain} Pr{beef noodle not rain} = ( ) + ( ) = = 0.27 = 27%

9 Probability and Probability Rules 9 / 17 Union and intersection, and the probability of them. Pr(E 1 ore 2 ) = Pr(E 1 ) + Pr(E 2 ) Pr(E 1 and E 2 ) Disjoint Conditional probability: Pr(A B) = Pr(A and B)/Pr(B) Bayes theorem: Pr(A B) = Pr(A)Pr(B A) Pr(A) = Pr(A)Pr(B A) Pr(A)Pr(B A) + Pr(A c )Pr(B A c ) Independence: Pr(A B) = Pr(A) Probability tree Contingency table

10 Continuous and Discrete Random Variable 10 / 17 Continuous random variable Probability density function, density curve Total area under the density curve is 1, or 100% Area under the density curve between two values Pr(Y = a) = 0 for any value a. Discrete random variable Probability mass function, a list of probability of possible events. Mean calculation: µ = y i pr(y = y i ) Variance calculation: σ 2 = (y i µ) 2 pr(y = y i ) Standard deviation calculation: σ = σ 2.

11 Binomial Random Variable 11 / 17 When does a binomial random variable arise? Independent-trials model. Binomial distribution: Y Binomial(n, p). Pr(Y = j) = n C j p j (1 p) n j. mean µ = np variance σ 2 = np(1 p) standard deviation σ = np(1 p) Pr(Y = j) = dbinom(j, n, p) in R Pr(Y j) = pbinom(j, n p) in R

12 Normal Random Variable 12 / 17 Characteristic of the normal curve Normal distribution: Y N(µ, σ). { } 1 (x µ) 2 f (x) = exp 2πσ 2 2σ 2 Area under the curve Empirical rule

13 13 / 17 Conditional Probability What is the probability of rolling a 6-sided dice and it s value is less than 4, knowing that the value is an odd number? E 2 : rolling a dice and it s value is less than 4 E 1 : knowing that the value is an odd number Pr{E 2 E 1 } = Pr{E 2 and E 1 } Pr{E 1 } = Pr(less than 4 and an odd number) Pr(an odd number) = Pr{ } 2 Pr{ } = 6 =

14 Conditional Probability 14 / 17 Sometimes, we are told what is the conditional probablity and ask to find Pr{E 2 and E 1 }. Example: 80% patiens took medicine. Among the patients that take the medicine, 90% felt drowsy. What is the probabilty of taking the medicine and feeling drowsy? 0.9 = Pr{feel drowsy take medicne} Pr{feel drwosy and take medicine} 0.9 = Pr{take medicine} Pr{feel drwosy and take medicine} 0.9 = 0.8 Pr{feel drwosy and take medicine} = = 0.72 = 72% 0.72 = 72% (they are the same).

15 Rule of Total probability 15 / 17 Pr{Drowsy} = Pr{Drowsy and Med} + Pr{Drowsy and No Med} = = 0.8

16 16 / 17 If you meet a drowsy flu patient, what is the probability that he does not take the medicine? Knowing that the the patient is drowsy, what is the probability that he/she did not take the medicine? Pr{No Med Drowsy} = = Pr{No med and Drowsy} Pr{Drowsy} 0.08 = 0.1 = 10% ( )

17 When to add and when to multiply 17 / 17 In general, when calculate Pr(E 1 and E 2 ), multiply. When E 1 and E 2 are independent: What is the probability of 3 yellow and 1 green? Pr(Y and Y and Y and G) Pr(E 1 and E 2 ) = Pr(E 1 ) Pr(E 2 E 1 ): Probability tree. 80% take medicine and 90% of them feel drowsy. Pr(med and drowsy) = Pr(med) Pr(drowsy med) In general, when calculate Pr(E 1 or E 2 ), add. But pay attention whether E 1 and E 2 are disjoint. Disjoint: what is the probability that the 4 peas are the same color. Pr(all four are Y or all four are G) = Pr(all four are Y) + Pr(all four are green) Pr(blue eye or black hair) = Pr (blue eye) + Pr(black hair) Pr(blue eye and black hair)