Question

Flaws are counted on a standard steel plate (sheet). Each sheet has the exact same dimensions. If 8 sheets are selected at random and the number of flaws per sheet is as follows, determine the lower and upper control limits.
Sheet Number of Flaws
1 5
3 2
2 6
1 2
3 7
3 0
4 8
0 3
A. LCL = -1.70, UCL = 2.84
B. LCL = 0.00, UCL = 6.24
C. LCL = 0.00, UCL = 2.84
D. LCL = -2.22, UCL = 5.72
E. LCL = 0.00, UCL = 5.72

Answer 1

Answer:

LCL = 0 and UCL = 10.21

there isn't any option like this in the options. It may be because of wrong data is provided in the question. But the way to solve for UCL and LCL is correctly written in this solution.

Explanation:

Total Number of Sheets = 8

Data Given:

Sheet                Number of Flaws

1.                                     5

2.                                    2

3.                                    6

4.                                    2

5.                                    7

6.                                    0

7.                                    8

8.                                    3

And here are the multiple choices to choose from:

A. LCL = -1.70, UCL = 2.84

B. LCL = 0.00, UCL = 6.24

C. LCL = 0.00, UCL = 2.84

D. LCL = -2.22, UCL = 5.72

E. LCL = 0.00, UCL = 5.72

Note: Lower Control Limits or LCL can not be negative so, neglect options A and D straight away. Now, we are left with following options to choose from:

B. LCL = 0.00, UCL = 6.24

C. LCL = 0.00, UCL = 2.84

E. LCL = 0.00, UCL = 5.72

So, our first step is to calculate the total number of flaws:

Total Flaws = 5 + 2 + 6 + 2 + 7 + 0 + 8 + 3 = 33

Total Flaws = 33

Formula to Calculate the Upper Control limit and Lower Control limits are as follows:

UCL = Total Flaws/Total Sheets + 3 x $\sqrt{Total Flaws/Total Sheets}$

Let's denote Total Flaws/Total Sheets with A.

A = Total Flaws/Total Sheets

UCL = A + 3 x $\sqrt{A}$

Similarly, Lower Control Limit or LCL formula is:

LCL = Total Flaws/Total Sheets - 3 x $\sqrt{Total Flaws/Total Sheets}$

So,

Upper Control limit or UCL = 33/8 + 3 x $\sqrt{\frac{33}{8} }$    = 10.21

Lower Control Limit or LCL =  33/8 - 3 x $\sqrt{\frac{33}{8} }$   = -1.96 = 0 (because it can't be zero).

So, according to the that is given and that we have arranged, the answer is:

So, the LCL = 0 and UCL = 10.21

But there isn't any option like this in the options. It may be because of wrong data is provided in the question. But the way to solve for UCL and LCL is correctly written above.