1. [20 points] The amount of fill dispensed by a bottling machine is normally distributed with standard deviation o=2. Suppose that the sample mean is to be computed using a sample of size n. (a) Find the probability that the sample mean will be within 0.5 ounce of the true mean u for n=15, 25, 36, 49. (8 pts) (b) Plot the probabilities you computed in part (a) as a function of n. Use R to generate the plot and make sure to attach your code at the end of the homework. (4 pts) (c) What pattern do you observe for the various values of n? Provide an intuitive explanation along with your answer. (4 pts) (d) What sample size is required if we wish the sample mean to be within 0.5 ounce of the true mean u with probability 0.95. (4 pts) 2. [20 points] Suppose X1, ,Xm is an independent random sample of size m from a population which is normally distributed with mean .111 and variance 0'12. Let Y1, , Y\" be another independent random sample of size n from a population which is normally distributed with mean #2 and variance 0%. Let)? and )7 denote the sample means from these two samples, respectively. Since the two samples are independent of each other, the corresponding sample means are also independent. (a) What are the sampling distributions of)? and 17 ? Give their names and parameters. (5 pts} (b) It is known that, both, the sum and the difference of two independent random variables that follow normal distributions are also normally distributed random variables. Using this information, what are the distributions of J? +17 and X -17 ? Give their names and para meters. [Hint: You just need to find the mean and variance of X+l7 and XY . Note, ifX and Y are two independent random variables, then V(X+Y)=V(X)+V(Y), and we know from property of variance that V(-Y)=V(Y)] (8 pts) (c) Let of =4 and 0'22 =1, and m=32 and n=8. Then, find the probability that X47 is within 1 unit of the difference in true population means ,ul pg. (7 pts) 3. [20 points] In social networks a connection/friendship between two individuals is a random variable Y that takes values 1 or 0 depending upon whether they are connected/friends or not. Let the random variable Y follow a Bernoulli (0.20) distribution. Suppose a sample of 1,000 such pairs of individuals are randomly selected from the network and their friendship status is tested (Le. the value OH is noted}. The sample mean 17 can be used to estimate the population mean. Assume that a sample size of 1,000 is large enough to apply the Central Limit Theorem: (a) Write down the approximate distribution of the sample mean, 17. You should indicate the name of the distribution and what the parameters are. (5 pts) (b) Find x such that approximately P(x S 17 0.20 S x) = 0.95. (7 pts) 4. (c) Find how many pairs of individuals (samples) you need to randomly select from the social network such that the sample mean is within 0.01 of the true parameter with probability 0.95? (8 pts) [18 points] The joint probability density function of X and Y is given by: = x+y 0