Explain why a process can be out of control even though all

Question

QUESTION 1 Probability Show all calculations/reasoning

(a) Describe the two basic laws of probability.
(b) What is the expected value and what does it measure? How is it computed for a discrete probability distribution?
(c) Consider the following record of sales of iPhone SEs for the last 25 days.

SALES UNITS	NUMBER OF DAYS
8	5
9	3
10	12
11	3
12	2
TOTAL	25

1. What was the probability of selling 9 or 10 units on any one day?
2. What were the average daily sales?
3. What was the probability of selling 9 or more?
4. What was the probability of selling 11 or less?
(d) There is a cage of 80 unwanted kittens. 36 are female (F) and black (B); 14 are male (M) and black (B); 17 are female and tabby coloured (T); 13 are male and tabby coloured (T).

1. Using Excel construct a table showing gender in the rows and colour in the columns with total of rows and columns. Then paste this Excel table into Word.
2. Now calculate the probability of a kitten being:
(i) Female
(ii) Tabby coloured
(iii) Black given a male
(iv) Tabby coloured given a female
(e) The life of a light bulb is normally distributed with a life of 1025 hours and a standard deviation of 75 hours.
1. What is the probability that the lightbulb will last 930 hours or less?
2. What is the probability that the lightbulb will last 820 hours or less?
3. What is the probability that the lightbulb will last longer than 1040 hours?

QUESTION 2 Research Question, Constructing data table and calculating probabilities

14 marks ­ 5 for 1, 3 for 2, 6 for 3
The following question involves learning/employing research skills in searching out data on the Internet, presenting it in a well­constructed and informative table, and calculating some probabilities showing calculation methods.

1. Search the Internet for the latest figures you can find on the age and main source of household income of the Australian population. Type of household income is broken down into employee income, own unincorporated business income and other income.

2. Then using Excel, prepare a table of population numbers (not percentages) by type of household income (in the columns) and age (in the rows). Break age into about 6 standard groups, eg, 15­24, 25­34, 35­44, 45­54, 55­64 and 65 and over. Insert total of each row and each column. Paste the table into Word as a picture. Give the table a title, and below the table quote the source of the figures.

3. Calculate from the table, showing your calculation methods:
The probability that any person selected at random from the population earns employee income. The probability that any person selected at random from the population is aged between 15 and 34.
The joint probability that any person selected at random from the population has own unincorporated business income and aged between 25 and 44.
The conditional probability that any person selected at random from the population is 35 or under given that the person has employee income.

QUESTION 3 Statistical Decision Making and Quality Control

Show all calculations and reasoning.

a) Colonial Electric is a large company that produces lightbulbs and other electrical appliances. One particular lightbulb is supposed to have an average life of about 1,000 hours before it burns out. Periodically the company will test 5 of these and measure the average time before these burn out. The following table gives the result of 10 such samples:

Sample	1	2	3	4	5	6	7	8	9	10
Mean	979	1087	1080	934	1072	1007	952	986	1063	958
Range	50	94	57	65	135	134	101	98	145	84

1. What is the overall average of these means? What is the average range?
2. What are the upper and lower control limits for a 99.7% control chart for the mean?
3. Does this process appear to be in control? Explain.
4. Explain why a process can be out of control even though all the samples fall within the upper and lower control limits.

b) Claire Underwood has been concerned about Machine 8 at the Collard Factory. To make sure that the machine is operating correctly samples are taken and the average and the range of each sample is calculated. Each sample consists of 12 items produced from the machine. Recently, 12 samples were taken and for each the sample range and average were calculated. The sample range and sample average were 1.1 and 46 for the first sample, 1.31 and 45 for the second sample, 0.91 and 46 for the third sample, and 1.1 and 47 for the fourth sample. After the fourth sample the sample averages increased. For the fifth sample the range was 1.21, and the average was 48; for sample 6, it was 0.82 and 47, for sample 7 it was 0.86 and 50, for sample 8 it was 1.11 and 49. After the eight sample the sample average continued to increase never getting below 50. For sample 9 the range and average were 1.12 and 51; for sample 10 they were 0.99 and 52; for sample 11 they were 0.86 and 50 and for sample 12 they were 1.2 and 52.

Although Claire's boss wasn't overly concerned, Claire was. During installation the supplier had set a value of 47 for the process average and an average range of 1.0. It was Claire's feeling that something was definitely wrong with machine number 8. Do you agree? Explain your view.