1. If S2 is the variance of a random sample of size n, show that it can be written as

Let xi be the number of fish caught by the i th fisherman in a random sample of six fishermen. Using the fact that Px2 i = 171 and Pxi = 31, find the variance of the data.

2. Suppose X1, X2, ··· , Xn are independent and identically distributed random variables from a normal population with mean µ and variance σ2. The mean and variance of the random sample are respectively X¯ and S2. Show that

Given the random variables Z ~ N (0, 12), V ~ χ2 n are such that Z and V are independent, then the random variable T defined as

is t-distributed with n degrees of freedom, i.e. T ~ tn. Use this fact to prove that

3. With the use of appropriate definitions, show that

(i) If X ~ Fn,m, then X−1 ~ Fm,n.

(ii) If T ~ tn, then T2 ~ F1,n.

4. The Apple company normally has a monthly stock return 1% with a standard deviation of 4%. In a given year, there are 12 months. Assume that the stock return is approximately normal, what is the probability that the average monthly return was more than 2%?

5. Suppose X is a random variable with mean µ and variance σ2 = 1.0 . Suppose also that a random sample of size n is to be taken and x¯ is to be used as an estimate of µ. When the data are taken and the sample mean is measured, we wish it to be within 0.05 unit of the true mean with probability of at least 0.99, i.e.

What is the minimum sample size required?

6. Assume apple company in USA is totally independent with Xiaomi company in Hongkong (their returns are independent). Apple company has mean daily return 0.2% and standard deviation 0.5%, while mean daily return of Xiaomi is 0.4% and standard deviation 0.7%. what is the probability that a random sample of 30 days for apple company has a mean daily return at least 0.1% higher than that of 56 days from Xiaomi company?

7. In a factory, a filling machine is used to fill cartons with a liquid product. The filling machine is required to operate under the specification of 9 ± 1.5 oz. If any carton is produced with weight outside these bounds, it is considered to be defective. It is hoped that at least 99% of cartons will meet these specifications.

(i) With the conditions µ = 9 and σ = 1, what proportion of cartons from the process is defective?

(ii) If changes are made to reduce variability, what must σ be reduced to in order to meet the specifications with probability 0.99?

Assume a normal distribution for the weight.

8. Find k such that P (k

9. If S1 2 and S2 2 represent the variances of independent random samples of size n1 = 25 and n2 = 31, taken from normal populations with variances σ1 2 = 10 and σ2 2 = 15 respectively, find