Question

A restaurant is improving its processes, and is currently focusing on the final bill presented to customers. The manager samples 50 bills per day for 12 days, and checks to see how many are incorrect using a z-value of 3. [ Select ] Is the billing process consistent (in other words, is it in control), or do special cause events appear to be present? [ Select ] What is the UCL? [ Select ] What is the LCL? Day Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10 Day 11 Day 12 Bills Sampled 50 50 50 50 50 50 50 50 50 50 50 50 Incorrect Bills 1 3 2 1 4 8 2 1 1 3 1 4 You collect two more days of data, again sampling 50 bills per day, and add the results (11 incorrect bills on day 13, and 9 on day 14) to the control chart. [ Select ] How many days are now outside the control limits? [ Select ] What is the UCL? [ Select ] What is the LCL?

please answer all the questions and explain every process, thanks!

Answer 1

Z = 3
SAMPLE SIZE = 50
NUMBER OF SAMPLES TAKEN = 12

P-BAR = TOTAL NUMBER OF DEFECTS / (SAMPLE SIZE * NUMBER OF SAMPLES = 31 / (50 * 12) = 0.0517
STDEV = SQRT((PBAR * (1 - PBAR)) / SAMPLE SIZE = SQRT((0.0517 * (1 - 0.0517)) / 50 = 0.0313

UCL = PBAR + (Z * STDEV) = 0.0517 + (3 * 0.0313) = 0.146
LCL = PBAR - (Z * STDEV) = 0.0517 - (3 * 0.0313) = -0.042, SINCE LCL IS NEGATIVE, LCL = 0

FRACTION DEFECTIVE = OBSERVATION / SAMPLE SIZE

2. NO, THE PROCESS IS NOT IN CONTROL BECAUSE OBSERVATION 6 IS ABOVE THE CONTROL LIMIT.

3. 3 OBSERVATIONS ARE NOW OUT OF CONTROL LIMIT, OBSERVATION 13, 14, AND 6

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