APMA 1650 Final Exam Due Saturday, August 7 at 9am Eastern The work you hand in via Gradescope should be yours and yours alone. You may use your book and lecture notes, but no other sources....

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APMA 1650 Final Exam Due Saturday, August 7 at 9am Eastern The work you hand in via Gradescope should be yours and yours alone. You may use your book and lecture notes, but no other sources. Calculators and Matlab are fine. 1. Suppose Xavier and Yolanda agree to have children until they have their first girl. Assume each child is a boy or a girl independently with probability 12 and children are born one at a time (no multiple births). (a) What is the expected number of girls they will have? (b) What is the expected number of boys they will have? Now suppose they agree to have children until they either have a girl or have N children, where N is some fixed number. (c) What is the expected number of girls they will have? (d) What is the expected number of boys they will have? 2. Suppose X and Y are independent random variables such that E(X) = µ,E(Y ) = µ, V (X) = 3, V (Y ) = 9. Consider the following estimators for µ: µ̂1 = X 6 + 5Y 6 , µ̂2 = X 2 + Y 3 + 1 (a) What is the bias of µ̂1 and µ̂2? Is either unbiased? (b) What is the variance of µ̂1 and µ̂2? (c) What is the MSE of µ̂1 and µ̂2? Which is smaller if µ = 0? If µ = 30? 3. (a) Suppose the mean score on the SAT math section is 511. What is an upper bound on the probability that a student scores over 700? (b) Suppose you also know the standard deviation on the SAT math section is 120. Can you get a better bound on the probability that a student scores over 700? (c) The College Board works to ensure that the distribution of SAT scores is roughly normal. Assuming this is the case, and given the information from (a) and (b), what is the probability that a student scores over 700 on the SAT math section? 1 Kamille Thai 4. Nutritional information provided by McDonald’s states that each small order of fries contains 71g of food and 220 calories. A sample of 10 orders from McDonald’s restaurants in Massachusetts and Rhode Island averaged 253 calories with standard deviation 54 calories. (a) Is there sufficient evidence to indicate that the average number of calories in a small order of McDonald’s french fries is greater than advertised? Test at the 1% level of significance. (b) What is the p-value of this test? Give an exact value using Matlab or use the book table to give a range. (c) How could you formulate your test as a statement about an upper or lower confidence bound? (d) How would your answers change if you had 50 samples with the same average and standard deviation? 5. Suppose that the random variables X and Y have joint density function: f(x, y) = x+ y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 (a) Verify that this is a valid density. (b) Find the marginal densities of X and Y . (c) Find the covariance of X and Y . (d) Find the correlation coefficient of X and Y . 6. The following (hypothetical) data table concerns vehicles and drivers in the city of Providence: Driver age Sedan SUV All others Total Under 25 – .12 .03 .25 25-39 .15 – .05 – 40+ – .13 .02 .4 Total – – – (a) Fill in the blanks to complete the table. (b) If you choose a vehicle at a random in the city, what is the probability that it will be an SUV? (c) Given that the vehicle you choose is an SUV, what is the probability that it will be driven by someone under 25? 2 (d) Consider the following: (1) sample space, (2) event, (3) random variable, (4) Bayes’ Rule, (5) Law of total probability, (6) expected value, (7) joint proba- bility, (8) marginal probability, (9) conditional probability, (10) independence. Define each term (or state the law / rule) and give an example in the context of this problem. Note that there are many possible answers; the point is for you to make connections, and we will grade accordingly! 3