Statistics
1) An electronics company is about to launch a new product. If the serial number for each piece produced has the following format: LLLNN where L stands for any letter in the English alphabet and N is a number from 0 to 9, please answer the following:
a) What is the name of the counting rule used to find all the possible outcomes?
b) How many different items can be uniquely numbered?
2) A company needs to select an expert group of consultants to provide advice for a given project. How many possible selections consisting of 3 project managers, 3 legal advisors, 3 computer scientists and 2 systems engineers can be selected if the company can pick from 4 project managers, 7 legal advisors, 5 computer scientists and 6 systems engineers? Think carefully about the counting rules involved before attempting to do any calculations.
3) Consider that you are a line manager in your current Corporation. There is a 0.40 probability that you will be promoted this year. There is a 0.65 probability that you will get a promotion or a raise. The probability of getting a promotion and a raise is 0.35. (Use letter P to denote promotion and letter R to denote raise.)
a. If you get a promotion, what is the probability that you will also get a raise?
b. What is the probability that you will get a raise?
Please show all the steps in your calculations
4) Three of the cylinders in an eight-cylinder car are defective and need to be replaced. If three cylinders are selected at random, what is the probability that
a. all defective cylinders are selected?
b. no defective cylinder is selected?
c. at least one defective cylinder is selected?
Consider a population of five items identical in appearance but weighing 10, 14, 16, 18 and 19 grams.
a. Determine the mean and the variance of the population.
b. How many possible samples can we have if we sample without replacement from the above population with a sample size of 2. Please justify your choice of counting rule to determine the number of possible samples.
c. Write down all the possible samples, either in a linear format (i.e. a simple list) or using a tree diagram.
d. Using the ten sample-mean values, estimate the mean of the population and the variance of x.
e. Compute the standard error of the mean.
f. What is the relevant theorem that describes the behaviour of the sample mean when repeated samples are taken? Briefly describe the theorem in your own words.
6) The bottling station in plant A of a fizzy drinks company fills cans with a nominal value of 330 ml. Five days before the annual maintenance of machinery, a sample of 16 cans is taken and the volume of liquid in each one measured. At the 5% level of significance test whether the bottling process is carried out accurately or whether the filling machinery needs calibration. Use both the critical and the p-value approaches. (You will find the data in table 1 below, under question 8. Use only the first column of the table corresponding to plant A.)
7) The company owns another bottling station located in a different plant (plant B). The plant manager claims that the variance of the can volumes (in ml2) of the 2nd bottling process (plant B) is the same as that of the 1st plant. A sample of 10 cans is taken from the bottling station in plant B and the sample standard deviation is found to be 4.001 ml. Using the value of the sample variance of plant A that you have already calculated, test the claim of the plant manager at the 5% level of significance, using the critical value approach. You do not need to use the data from table 1 that corresponds to plant B.
8) Consider the sample data for plants A, B, C & D of the above company, which are tabulated in table 1. Test whether the four means are equal at the 5% level of significance. Perform a test that will compare all 4 means at once. We are not interested in pair-wise comparisons at this stage. Please show all the steps in your calculations.