3/29/11 review for exam 2 1 STOR 155 Introductory Statistics Review for Midterm Exam 2 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL3/29/11 review for exam 2 2 Midterm Exam 2 • When: Thursday 3/31 • Where: In class • Cover: Sec. 4.1 – 4.5, 5.1 – 5.2 • Format: – about 30 questions (multiple choice) – closed-book, calculator needed – scantron3/29/11 review for exam 2 3 Review of Chapter 4 • Sample spaces, events, union, intersection, complement, disjoint events • Probability rules: addition, multiplication • Random variables, probability distributions, means, variances, rules for means and variances • Conditional probability, independence • Venn and tree diagrams3/29/11 review for exam 2 4 Probability Rules • For any event A, P(A) = 1 - P( not A). • For any two events A and B, P(A B) = P(A) + P(B) - P(A B). • If A, B, C, … are disjoint, then P(A or B or C or ...) = P(A) + P(B) + P(C) + … • Conditional probability: • For any two events A and B, P(A and B) = P(A) P(B | A) = P(B) P(A | B) . )()()|(BPBAPBAP3/29/11 review for exam 2 5 Independence • Two events A and B are said to be independent if and only if any of the following identities holds (one is enough) – P(A | B) = P(A) – P(B | A) = P(B) – P(A B) = P(A) P(B) … Use whichever convenient …3/29/11 review for exam 2 6 combination of rules • For any two events A and B, P(B) = P(B A) + P(B {not A}) = P(B | A) P(A) + P(B | not A) P( not A) • Draw a Venn diagram or a tree diagram to see why. Bayes rule: Don’t memorize it, but derive it !3/29/11 review for exam 2 7 Some Tips • Define events related to the problem of interest; • Calculate the probabilities by using correct rules; • Use Venn or tree diagramsReview of Chapter 5 Q & A related to sampling distributions: • Which statistics: count X, proportion , or mean ? • What sample size n ? What role to play ? • What distribution: binomial or normal ? Which table to use ? • What are the means and standard deviations for those statistics ? 3/29/11 review for exam 2 8 pˆX3/29/11 review for exam 2 9 Sources for Review • lecture notes, textbook • summarize the homework problems you did • the practice exam Suggestion: problem-oriented review !3/29/11 review for exam 2 10 Problem 1 The height of American women aged 18 – 24 can be well approximated by a normal distribution with mean = 64.3 in. and s.d. = 2.4 in. • Five women in the age group are randomly selected, what’s the probability that at least one of them are taller than 66 inches? • Answer: 0.745Problem 1 continued • How many women need to be sampled such that the sample average has its s.d. smaller than 0.2 ? Answer: n > 144 • If n women are sampled, how likely it is for the sample average to be greater than the population mean ? P( > ) = 0.5 regardless of what values n and take ! Why ? 3/29/11 review for exam 2 11 X3/29/11 review for exam 2 12 • An investor has allocated an equal amount of money in two investments. • Find the expected return of the portfolio (0.21) • If the returns on the two investments are independent, find the standard deviation of the portfolio. (0.2358) Mean returnStandard dev.Investment 115% 25%Investment 227% 40% Problem 23/29/11 review for exam 2 13 Problem 2 (continued) Redo those parts with the following extensions: • An unequal allocation of the wealth (1/3 , 2/3) over the two investments • Assuming a correlation – 0.1 between returns for the two investments3/29/11 review for exam 2 14 Problem 3 • A lab test yields 2 possible results: positive or negative. 99% of people with a particular disease will produce a positive result. But 2% of people without the disease will also produce a positive result. Suppose that 0.1% of the population actually has the disease. • What is the probability that a person chosen at random will have the disease, given that the person’s blood yields a positive result? • (4.7%)3/29/11 review for exam 2 15 Problem 4 • A Tar Heel basketball player is a 80% free throw shooter. • Suppose he will shoot 5 free throws during each practice. • X: number of free throws he makes during practice. • Find the mean and variance of X. • (4 , 0.8)3/29/11 review for exam 2 16 Problem 4 (continued) • Given that the player has made more than 2 free throws, what is the probability that his 4th and 5th shots are both good ? Hint: This is a conditional probability P(A | B) = P(A and B) / P(B), identify A and B, and consider the 1st three throws and the last two throws as disjoint blocks so you can use independence
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