Random Sampling - what did we learn?

Transcription

1 Homework Assignment Chapter 1, Problems 6, 15 Chapter 2, Problems 6, 8, 9, 12 Chapter 3, Problems 4, 6, 15 Chapter 4, Problem 16 Due a week from Friday: Sept. 22, 12 noon. Your TA will tell you where to hand these in Random Sampling - what did we learn? It s difficult to do properly Why not just point? Computers and random numbers Can you tell if your numbers were random? Sampling distribution of the mean Estimating error of the mean Confidence interval Sampling distribution of the mean How confident can we be about this one estimate of the mean? Hard method: take a few MORE random samples, and get more estimates for the mean Easy method: use the formula: SE Y = s n Confidence interval a range of values surrounding the sample estimate that is likely to contain the population parameter We are 95% confident that the true mean lies in this interval µ = 5.14 Y = 5.26 What if we calculate 95% confidence intervals? Approximately ± 2 S.E. Expect that 95% of the intervals from the class will contain the true population mean, invervals * 5% = 3.5 Expect that 3 or 4 will not contain the mean, and the rest will Mean ± 95% C.I.

2 What if we took larger samples? Say, n=20 instead of n=10? Probability Probability Mutually exclusive The proportion of times the event occurs if we repeat a random trial over and over again under the same conditions Pr[A] Two events are mutually exclusive if they cannot both be true. A Sample space Possible outcome The probability of event A (cannot both occur simultaneously) B Pr[B] proportional to area Venn diagram Mutually exclusive Pr(A and B) = 0 Mutually exclusive Not mutually exclusive Pr(A and B)! 0 Pr(purple AND square)! 0 Visual definition - areas do not overlap in Venn diagram

3 For example Event A: First child is female Probability distribution Probability distribution for the outcome of a roll of a die Event B: Second child is female A probability distribution describes the true relative P(A) = 0.48 frequency of all possible values of a random variable. Frequency P(B) = 0.48 But P(A and B)! 0, so these events are NOT mutually exclusive. Random variable - a measurement that changes from one observation to the next because of chance Number rolled Probability distribution for the sum of a roll of two dice The addition rule Sum of areas Addition Rule The addition principle: If two events A and B are mutually Frequency exclusive, then Pr[A OR B] = Pr[A] + Pr[B] Sum of two dice Pr[1 or 2] = Pr[1]+Pr[2] The probability of a range The probability of a range For families of 8 children, Pr[Number of boys! 6] =? For families of 8 children, Pr[Number of boys! 6] = Pr[6 or 7 or 8] = Pr[6]+Pr[7]+Pr[8] The probabilities of all possibilities add to 1.

4 Addition Rule Probability of Not The addition rule Pr[NOT rolling a 2] = 1 - Pr[2] = 5/6 Pr[not A] = 1-Pr[A] The addition principle: If two events A and B are mutually exclusive, then Pr[A OR B] = Pr[A] + Pr[B] Pr[1 or 2 or 3 or 4 or 5 or 6] = 1 What if they are not mutually exclusive? General Addition Rule General Addition Rule Pr[Walks or Flies] = Pr[Walks] + Pr[Flies] - Pr[Walks and Flies] General Addition Rule Pr[A OR B] = Pr[A] + Pr[B] - Pr[A AND B]. Independence Multiplication rule Pr[boy]=0.512 Two events are independent if the occurrence of one gives no information about whether the second will occur. If two events A and B are independent, then Pr[A and B] = Pr[A] x Pr[B] Pr[ (first child is a boy) AND (second child is a boy)] = 0.512! = Equivalent definition: The occurrence of one does not change the probability that the second will occur

5 Multiplication rule If two events A and B are independent, then Pr[A and B] = Pr[A] x Pr[B] OR versus AND OR statements: Involve addition It matters if the events are mutually exclusive AND statements: Involve multiplication It matters if the events are independent Probability trees Sex of two children family Dependent events Fig wasps Variables are not always independent; in fact they are often not Fig wasps Testing independence Are the previous state of the fig and the sex of an egg laid independent? Test the multiplication rule: Pr[A and B]?=? Pr[A] x Pr[B] Pr[fig already has eggs and male]?=? P[fig already has eggs] x Pr[male] Are the previous state of the fig and the sex of an egg laid independent? Pr(male) = = 0.22 Pr(fig already has eggs) = 0.2 Pr(male AND fig already has eggs) = 0.18! " Pr(male) x Pr(fig already has eggs) = 0.22 x 0.2 = So these two events are NOT independent.

6 Conditional probability Short summary The probability of A OR B involves addition. P(A or B) = P(A) + P(B) if the two are mutually exclusive. The probability of A AND B involves multiplication P(A and B) = P(A) P(B) if the two are independent The conditional probability of an event is the probability of that event occurring given that a condition is met. Pr[X Y] P(X Y) means the probability of X if Y is true. It is read as "the probability of X given Y." P(female lays a male egg fig has eggs already) = 0.9. [ ] = Pr Y Pr X Law of total probability: " All valuesof Y [ ]Pr X Y [ ] The probability of a male egg is Pr[male]= Pr(male egg fig has no eggs) Pr(fig has no eggs) + Pr(male egg fig already has eggs) Pr(fig already has eggs) = 0.9 (0.8) (0.2) = 0.22 The general multiplication rule Bayes' theorem Pr[A AND B] = Pr[A] Pr[B A]. Does not require independence between A and B Pr[ A B] = Pr B A [ ]Pr[A] Pr[B]