36-303: Sampling, Surveys & Society HW02: Due Thursday Feb 11, 2010 in class

\underline{Reminders:}

• Things to turn in:

– Due Tonight (Feb 4) by Midnight: Team Assig. I.2 (revising I.1, especially target pop, frame, mode) – email to brian@stat.cmu.edu

– Due Mon Feb 8: Team Assig. I.3 (choosing which project to do) – email to brian@stat.cmu.edu

∗ Please wait for email feedback from me on I.2 (and instructions on I.3) before starting to work on I.3.

• Things to read:

– Done already: Groves, Ch 1, 2, 4 (first part), 5, 11 (first part); and Lohr Appx B

– For next week (and this hw assignment): Groves Ch 7; after that, parts of 8 and 9

• Clear, careful writing and interpretation of results is an important part of both weekly homeworks and the projects. I always expect neatly typed or neatly handwritten work.

• Always be judicious about including computer output and graphs: show enough that we can clearly see what you are doing, but not so much that we will get lost or bored leafing through your work!

• You are operating under the same ethical guidelines as any intellectual worker: beware of falsification, fabrication and plagiarism. Citing the source (person, book, etc) of material you use usually protects you against plagiarism claims.

Exercises to Turn In (there are 4 excercises):

1. Let X and Y be discrete random variables with finite sample spaces {x_{1}, …, x_{K}} and {y_{1}, …, y_{K}}, and let p_{ij} = P[X = x_{i} and Y = Y_{j}]. Use the definitions of E[ ], Var( ), and summation notation (∑ _{i=1}^{K}), to show

(a) E[aX + bY + c] = aE[X] + bE[Y] + c

(b) Var(aX + bY + c) = a^{2}Var(X) + 2abCov(X, Y) + b^{2}Var(Y)

(c) If X and Y are independent, then E[X|Y = y] = E[X], for any y.

2. Recall the “Randomized response” model, from lecture, for the question:

Flip a coin, but dont tell me whether its heads or tails.

• If heads, answer truthfully: have you ever cheated in a CMU class?

• If tails, answer truthfully: is the last digit of your SSN odd?

Recall from lecture that

π = \frac{λ − (1/2) · (1 − p)}{p}

where p = P[Heads]; π = P[Cheat]; and λ = P[Yes]. Consider a SRS of n students \underline{with replacement}^{1}. Let λ̂ be the fraction of “Yes” answers in the survey, and let π̂ = (λ̂ − \frac{1}{2}(1 − p))/p.

(a) Show that E[π̂] = π.

(b) Express Var(π̂) in terms of Var(λ̂) and show that, as p gets closer and closer to 1, Var(π̂) gets closer and closer to Var(λ̂).

(c) Suppose you use a fair coin, so that p = \frac{1}{2}, and you think the true rate of cheating on campus is around 0.10. How large a sample would you need, so that a 95% confidence interval for π would be only 0.02 wide?

3. Groves, Ch 2, pp. 64–65. #1.

4. For the following, do not provide responses based on politics, social desirability, etc. Instead, provide analytic responses based on the principles outlined in Groves, Ch 7, and/or in class.

(a) Groves, Ch 7, pp. 238–239, #4.

(b) Groves, Ch 7, p. 239, #5.

(c) Groves, Ch 7, p. 240, #6.

1Surveys are seldom conducted this way, but it is easier for the math.