How to solve this set of problems?
1- MdrM's candies, part I Suppose we have a bag of 15!} lufch's candies, and 3|] of them are green, 15 are orange, 45 are yellow, and ll are brown. We randomly select 4!] candies with mplamment from this bag {meaning a candy is put back after it's been selected} and let X denote the number of green candies we get from these 40 draws. a] What is the distribution of X? b) What is the mean and what is the standard deviation of X? Interpret the results. c] What is the pmbability that we get fewer than 12 green candies? 2. MdrM's candles, part [I Now we perform the process in part I for n times and denote the number of green candies we get from the kth process by 1],. Let i be the mean ofxl, 1'2,\" ., In. Are the following statements true or false? Explain. a] The larger :1 is, the smaller the mean of f is. b) The larger :1 is, the smaller the standard deviation of J? is. c] As It increases, 17: gets closer to the mean of X we calculated in lb). 'I'hen answer the following questions. d) If n=3 , what is the probability that if is at least 5? What theorem are you applying here? e] If n=1 1, can you answer the question in d) applying the same theorem? Why or why not? 3. MdrM's candles, part III Now suppose that of all M&M's candies 2% are green and areas are brown, and we have a random sample of 40 candies. [We are not necessarily drawing candies from the bag in Q1 and Q2, but from all the MSrM's candies.) Let 331 denote the proportion of green candies and let z denote the proportion of brown candies in our sample. a] Calculate the mean and standard deviation for 331 and 362. b) Can you apply the central limit theorem to calculate Pf] andi'or P'j? If yes, calculate the result. If no, explain why not. c] If we have a larger sample, say, we have a sample of l candies now. Would you expect 33, to estimate the population proportion of green candies better? Explain. 4. For the following parts, select the R code that would solve the problem. a] Suppose Xmil]. What. is HEP-4)? i.dbinom{4,si2e=?,prob=.2j .pbinom{4,aiae=?,prob=.2} Lpbinomtd,size=T,prob=D.2,lower.tail=FALSEJ W.pbinom{5,size=?,prob=.2,lower.tail=FLSE} b) Suppose X~N[5.3}. Find c such that P(X}c}=.4. i. 1pnnrmi . 4] ii. qnormH{J . d J- *3+5 iii. qnorm {ID . =1} *3+5 iv. {l-qnorm [CI . 4}] *3+5 5. We can use the command rbinom (100, size=3, prob=0. 5) to randomly generate 100 observations from Bin(3,0.5). Run the following code. set . seed (3011) xbar . list=c () for (i in 1:1000) { xbar. list [i]=mean (rbinom (100, size=3, prob=0.5) ) qqnorm (xbar . list) qqline (xbar . list) What do you observe and what theorem can you apply to explain that
