Problem:
One of the stages in the process of making denim cloth at the Southern Mills Company is to spin cotton yarn onto spindles for subsequent use in the weaving process. Occasionally the yarn breaks during the spinning process, and an operator ties it back together. Some number of breaks is considered normal; however, too many breaks may mean that the yarn is of poor quality. In order to monitor this process, the quality-control manager randomly selects a spinning machine each hour and checks the number of breaks during a 15-minute period. Following is a summary of the observations for the past 20 hours:
| Sample |
Number of
Breaks |
Sample |
Number of
Breaks |
| 1 |
8 |
11 |
1 |
| 2 |
10 |
12 |
6 |
| 3 |
9 |
13 |
1 |
| 4 |
8 |
14 |
0 |
| 5 |
6 |
15 |
4 |
| 6 |
10 |
16 |
8 |
| 7 |
4 |
17 |
4 |
| 8 |
10 |
18 |
5 |
| 9 |
9 |
19 |
0 |
| 10 |
8 |
20 |
7 |
Required:
Question 1) What is the average number of yarn breaks for this data? 
Question 2) Construct a c-chart for yarn breaks and answer the following questions.
Question 3) The corresponding c-chart for this process using 3? control limits has an upper control limit of .
Question 4) The corresponding c-chart for this process using 3? control limits has a lower control limit of .
Question 5) Are any points outside the control limits? noyes
Question 6) Which of the following patterns are evident? 8 consecutive points increasing or 8 consecutive points decreasing8 consecutive points on one side of center linenone of these patterns is evident14 consecutive points alternating up and down
Based on your answers to the above questions, is this process in control?
Provide correct solution of the given problem with step by step calculations.