Let x be the mean of a random sample of size 100 from a

Question

1. Let x be the mean of a random sample of size 100 from a random variable that is distributed with mean 200, variance 400, and a distribution that is not the normal distribution.
2. give the mean of x
3. give the variance and standard deviation
4. is x likely to be normally distributed?
5. the dataset has the sample means x and corresponding standard standard deviations s from 1000 random samples of size 4 where X~N (100, 16*2).
6. obtain the descriptive statistics for x. are the mean and standard deviation what you expect? explain.
7. obtain the descriptive statistics for se(x) = s/square root n for these data. is the mean what you expect? explain.
8. compute z=(x-100)/16. explain why z is standard normal distributed. for these data does z appear to be normally distributed?
9. compute t=(x-100)/se(x). explain why t is standard normal distributed. for these data does t appear to be t3 distributed? the t3 distribution has mean 0 and variance 3 and for example, Pr[T3>1.638]=0.10.
10. an insurance company offers insurance to 10,000 people with independent loss distributions that have mean $5000 and standard deviation $20,000. Let x=1/10000 times summation when i=1 and n=10000 denote the average loss per individual.
11. find the mean and standard deviation of x.
12. suppose the insurance company sells insurance that provides complete coverage for $5400, and for simplicity suppose that the insurance company has no costs aside from paying out any insurance claims. is the insurance company likely to make a loss? explain your answer. hint: by the CLT x is normally distributed.
13. suppose we take s simple random sample from X~(miu, population variance).
14. we estimate miu by X1, the first value of X in the sample. Is this estimate unbiased for miu? is this estimator consistent for miu? explain.
15. we estimate miu by x+1/n. is this estimate unbiased for miu? is this estimator consistent for miu? Explain.