A type of camera pixel has three configurations: a model with 128 megapixels, priced at 60 KD, a 256 megapixels model at 80 KD, and a 512 megapixels version with a price tag of 100 KD. The probability of purchasers choosing the 128 megapixels model is 0.35, to choose the 256 megapixels model is 0.5, and to choose the 512 megapixels model is 0.15.

The probability distribution of the cost X is

Model 128 Megapixels 256 Megapixels 512 Megapixels

Cost 60 KD 80 KD 100 KD

Probability 0.35 0.5 0.15

Suppose that on a particular day only 4 cameras are sold. Let X1 = the price from the first sale, X2 = the price of the second sale, X3 = the price of the third sale, and X4 = the price of the fourth sale. Suppose that X1, X2, X3 and X4 are independent, each with the calculated probability distribution. Then any possible 4-tuple (1, 2, 3, 4) constitutes a random sample of size 4 from this distribution. In detailed steps, solve the following:

1) List all possible (1, 2, 3, 4) 4-tuples.

2) Compute the probability of each 4-tuple using the above distribution.

3) Calculate the sample mean and sample variance values of all 4-tuples.

4) Find the probability distribution (sampling distribution) of the sample average for price, and plot the histogram of the calculated sample means.

5) Find the probability distribution (sampling distribution) of the sample variance for price, and plot the histogram of the calculated variances.

6) Calculate the expected value of all sample means and compare it with the theoretical value and explain whether they are similar or not and why.

7) Calculate the variance of all sample means (use software) and compare it with the theoretical value. Explain whether they are similar or not and why.

8) Calculate the expected value of all sample variances and compare it with the population variance. What do you observe about their relationship?

9) Calculate the probability that the sample mean is within one standard error from its expected value.