Stat 135_Problem Set 2_ Fall 2020 UC Berkeley PDF

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This is the Stat 135_Problem Set 2_ Fall 2020 UC Berkeley...

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Stat 135: Concepts of Statistics, UC Berkeley, Fall 2020

Problem Set 2 Instructor: Prof. Yun S. Song

Due: 10 PM PDT (UTC-7) September 11, 2020 (on Gradescope)

Reading: Rice, 7.1-7.3 Show all your work to receive full credit. 1. Suppose X1 ∼ Normal(µ1 , σ 2 ), X2 ∼ Normal(µ2 , σ2 ) are independent random variables. Show that X1 + X2 and X1 − X2 are independent. (Hint: Use one of the lemmas presented in class.) 2. Consider an i.i.d. sample X1 , . . . , Xn ∼ Uniform[−θ, θ], where θ > 0 is unknown, and suppose a and b are some constants satisfying 0 < a, b < 1. (a) What is the minimum sample size n needed to guarantee that P(| max{X1 , . . . , Xn } − θ| ≤ aθ) ≥ b, for all possible θ > 0? (b) What happens to the answer as a tends to 0? What happens to the answer as b tends to 1? Are these results consistent with your intuition? Explain why. 3. Suppose two independent surveys were conducted to estimate a population mean µ. Let µ ˆ 1 and 2 2 and Var(ˆ µ2 ) = σ2 . Suppose µ ˆ 2 denote the two corresponding estimators, and let Var(ˆ µ1 ) = σ 1 both µ ˆ 1 and µ ˆ 2 are unbiased. For some a and b, the two estimators can be combined to produce an improved estimator of the form µ ˆ = aˆ µ1 + bˆ µ2 . Subject to the condition of unbiasedness, find a and b such that Var(ˆ µ) is minimized. 4. It can be difficult to obtain accurate survey answers to sensitive questions such as “Have you ever used heroin?” or “Have you ever cheated on an exam?” Randomized response is a method to deal with such situations. A respondent spins an arrow on a wheel or draws a ball from an urn containing balls of two colors to determine which of two questions to respond to: (1) “I have characteristic A,” or (2) “I do not have characteristic A.” The interviewer does not know which question is being responded to but merely records a yes or a no. The hope is that an interviewee is more likely to answer truthfully if he or she realizes that the interviewer does not know which question is being responded to. Let R be the proportion of a sample answering Yes. Let q be the probability that question 1 is responded to (q is known from the structure of the randomizing device), and let f be the fraction of the population that has characteristic A. Let r be the probability that a respondent answers Yes. (a) Show that r = (2q − 1)f + (1 − q ). [Hint: P (yes) = P (yes given question 1)∗P (question 1)+P (yes given question 2)∗P (question 2).] (b) If r were known, how could f be determined? (c) Show that E(R) = r, and propose an estimate, F , for f . Show that the estimate is unbiased.

(d) Ignoring the finite population correction, show that Var(R) =

r(1 − r ) n

where n is the sample size (e) Find an expression for Var(F ) 5. Refer to Example D in Section 7.3.3 of Rice. Suppose that a second survey is done of another condominium project of 10,000 units. The sample size is 400, and the proportion planning to sell in this sample is .18. (a) What is the standard error of this estimate? Give a 95% confidence interval. (b) Suppose we use the notation pˆ1 = .12 and pˆ2 = .18 to refer to the proportions in the two samples. Let dˆ = pˆ1 − pˆ2 be an estimate of the difference, d, of the two population proportions p1 and p2 . Using the fact that pˆ1 and pˆ1 are independent random variables, find expressions for the variance and standard error of dˆ (c) Because pˆ1 and pˆ2 are approximately normally distributed, so is ˆd. Use this fact to construct 99%, 95%, and 90% confidence intervals for d. Is there clear evidence that p1 is really different from p2 ?

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