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:

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Presentation transcript:

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

How confident can we be about this one estimate of the mean?

Estimating error of the mean Hard method: take a few MORE random samples, and get more estimates for the mean Easy method: use the formula:

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

 = 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.

What if we took larger samples? Say, n=20 instead of n=10?

Probability

The Birthday Challenge

Probability The proportion of times the event occurs if we repeat a random trial over and over again under the same conditions Pr[A] –The probability of event A

(cannot both occur simultaneously)

Mutually exclusive

Venn diagram

Mutually exclusive Venn diagram Sample space

Mutually exclusive Venn diagram Sample space Possible outcome Pr[B] proportional to area

Mutually exclusive

Pr(A and B) = 0

Mutually exclusive Visual definition - areas do not overlap in Venn diagram

Not mutually exclusive Pr(A and B)  0 Pr(purple AND square)  0

For example

Probability distribution

Random variable - a measurement that changes from one observation to the next because of chance

Probability distribution for the outcome of a roll of a die Number rolled Frequency

Probability distribution for the sum of a roll of two dice Sum of two dice Frequency

The addition rule

Addition Rule Pr[1 or 2] = ?

Addition Rule Pr[1 or 2] = Pr[1]+Pr[2]

Addition Rule Pr[1 or 2] = Pr[1]+Pr[2]

Addition Rule Pr[1 or 2] = Pr[1]+Pr[2] Sum of areas

The probability of a range For families of 8 children, Pr[Number of boys ≥ 6] = ?

The probability of a range 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.

Addition Rule Pr[1 or 2 or 3 or 4 or 5 or 6] = ?

Addition Rule Pr[1 or 2 or 3 or 4 or 5 or 6] = 1

Probability of Not Pr[NOT rolling a 2] = ?

Probability of Not Pr[NOT rolling a 2] = 1 - Pr[2] = 5/6

Probability of Not Pr[NOT rolling a 2] = 1 - Pr[2] = 5/6 Pr[not A] = 1-Pr[A]

The addition rule

What if they are not mutually exclusive?

General Addition Rule A B Pr[A or B] = ?

General Addition Rule A B Pr[A or B] = ? A B

General Addition Rule A B Pr[A or B] = ? A B

General Addition Rule A B Pr[A or B] = ? A B

General Addition Rule A B A B

Pr[Walks or flies] = ?

General Addition Rule Pr[Walks or flies] = ?

General Addition Rule Pr[Walks or Flies] = Pr[Walks] + Pr[Flies] - Pr[Walks and Flies]

General Addition Rule

Independence

Equivalent definition: The occurrence of one does not change the probability that the second will occur

Multiplication rule If two events A and B are independent, then Pr[A and B] = Pr[A] x Pr[B]

Pr[boy]=0.512

General Addition Rule

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 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]

≠

Conditional probability Pr[X|Y]

Law of total probability:

The general multiplication rule

Does not require independence between A and B

Bayes' theorem

In class exercise

Answer

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