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Solve these questions in R 1.Suppose we have a population of 10,000 elements, each with a unique label from the set [1,2,3,.,10000].
Solve these questions in R
1.Suppose we have a population of 10,000 elements, each with a unique label from the set [1,2,3,...,10000].
(a) Generate a sample of 500 labels from this population using simple random sampling.
(b) Generate a sample of 500 labels from this population using i.i.d. sampling.
2.Generate a sample of 30 from an N 10 2 distribution and a sample of 1 from an N 30 2 distribution. Combine these together to make a single sample of 31
(a) Produce a boxplot of these data.
(b) What do you notice about this plot?
(c) Based on the boxplot, what characteristic do you think would be appropriate to measure the location and spread of the distribution? Explain why.
3. A likelihood function is given by exp(-(theta-1)^2/2)+3*exp (-(theta-2)^2/2) for theta belongs to R^1 Numerically approximate the MLE by evaluating this function at 1000 equispaced points in (-10,10]. Also plot the likelihood function.
4.Suppose the proportion of left-handed individuals in a population is . Based on a simple random sample of 20, you observe four left-handed individuals.
(a) Assuming the sample size is small relative to the population size, plot the log- likelihood function and determine the MLE.
(b) If instead the population size is only 50, then plot the log-likelihood function and determine the MLE. (Hint: Remember that the number of left-handed individuals fol- lows a hypergeometric distribution. This forces theta to be of the form i/50 for some integer i between 4 and 34. From a tabulation of the log-likelihood, you can obtain the MLE.)
5.Generate 10^4 samples of size n 5 from the N (0,1) distribution. For each of these samples, calculate the interval (xbar - s/(5^0.5), xbar + s/(5^0.5) where s is the sample standard deviation, and compute the proportion of times this interval contains Miu . Repeat this simulation with n = 10 and 100 and compare your results.
6.For the data:
3.27 -1.24 3.97 2.25 3.47 -0.09 7.45 6.20 3.74 4.12
1.42 2.75 -1.48 4.97 8.00 3.26 0.15 -3.64 4.88 4.55 ,
use the plug-in MLE to estimate F3 for an N2 distribution. Use bootstrapping to estimate the MSE of this estimate for m = 10^3 and m = 10^4.
