STAT 400 G. Fellouris Fall 2015 Homework 10 Readings Sections 5.6, 5.7, 5.9 Sections 7.3-7.4 (pg. 326) Exercises 1. Sums of iid random variables are asymptotically normal Let X1 , . . . , Xn be independent and identically distributed (iid) random variables (that 1 have a nite variance). According to the Central Limit Theorem (CLT), Xn = n n Xi i=1 is approximately normal when n is large, in the sense that the cdf of Xn E[X1 ] sd[X1 ]/ n converges as n to , the cdf of a standard normal distribution, N (0, 1). (a) Show that an equivalent way to express the CLT is to say that proximately normal when n is large, in the sense that the cdf of n i=1 Xi is ap- n i=1 Xi nE[X1 ] sd[X1 ] n converges to as n . (b) Use (a) and Ex. 4 in HW 9 in order to show that the following random variables are approximately normal when n is large: i. Y Binom(n, p), ii. Y P ois(n), iii. Y N eg.Binom(n, p), iv. Y Gamma(n, ). (c) Compute approximately the probability that you need to toss a fair coin more than 120 times in order to observe 40 Tails. (d) Exercise 12 from Section 5.6. Remark: Part b(i) shows that the Normal approximation to the Binomial is a special case of the CLT. 1 STAT 400 G. Fellouris Fall 2015 2. Products of positive iid random variables are asymptotically lognormal Let X1 , . . . , Xn positive, independent and identically distributed random variables. Set Tn = n Xi and show that ln Tn is approximately normal when n is large, in the i=1 sense that the cdf of ln T n E[ln X1 ] sd[ln X1 ] n converges to as n . 3. Normal Approximation and Continuity Correction Let S be a Binomial random variable with parameters n = 20 and p. (a) Explain intuitively why a normal approximation to the distribution of S should work better when p = 0.5 than when p = 0.2. (b) Compute P(12 S 14) exactly when p = 0.5 and when p = 0.2. (c) Approximate P (12 S 14) using the continuity correction described in page 209 of the book. 4. An exam consists of 30 multiple choice questions with 4 possible answers, only one of which is correct. Each correct answer gets 4 points, whereas each wrong answer gets -1 points. A student answers all questions completely at random and independently of one another. Let X be the number of his correct answers in the exam. (a) Explain why the distribution of X can be approximated by a normal distribution. What are the parameters of this normal distribution? (b) Compute, exactly and using the normal approximation, the probability that the student answers correctly at most 4 questions. (c) Compute approximately, using the normal approximation, the probability that the student scores more than 10 points. 5. Ex. 10, Section 5.7 6. Ex. 14, Section 5.7 7. Estimating a population proportion Suppose that you take a random sample of size n in order to estimate some population proportion, p. Let pn be the sample proportion. 2 STAT 400 G. Fellouris Fall 2015 (a) Explain what is wrong in the following statements: i. p is unknown, therefore it is a random variable. ii. if n is suciently large, an approximate 95% condence interval for pn is pn 1.96 pn (1n ) p . n iii. the sample proportion pn is normally distributed with mean p and variance p(1 p)/n. (b) How much should you increase the sample size in order to cut the standard deviation of pn , sd(n ), in half? p (c) What is the largest possible value (as a function of n) that sd(n ) can take? p (d) What is the minimum sample size required so that the length of an approximate 95% condence internal for p be at most 3% ? Hint: See pg. 326-327 8. Computing condence intervals for a population proportion A random sample of n people is the US is asked whether they approve of Barack Obama's performance as President. Give an approximate 95% condence interval for the population proportion who approve of Obama's performance when (a) n=10, 5 say yes and 5 say no. (b) n=100, 50 say yes and 50 say no. (c) n=100, 60 say yes and 40 say no. (d) n=100, 90 say yes and 10 say no. (e) n=1000, 600 say yes and 400 say no. (f) n=1 million, 600,000 say yes and 400,000 say no. 9. Ex. 3 in Section 7.3. 10. Ex. 6 in Section 7.3. 3