A Canadian manufacturer identified a critical diameter on a crank bore that needed to be maintained within a close tolerance for the product to be successful. Samples of size 4 were taken every hour. The values of the differences (measurement - specification), in tenthousandths of an inch, are given in Table 1.4.

(a) Calculate the central line for an $X$-bar chart for the 24 hourly sample means. The centerline is $\overline{\bar{x}}=(4.25-3.00-\cdots-1.50+3.25) / 24$.

(b) Is the average of all the numbers in the table, 4 for each hour, the same as the average of the 24 hourly averages? Should it be?

(c) A computer calculation gives the control limits

\[\begin{aligned}& \mathrm{LCL}=-4.48 \\& \mathrm{UCL}=7.88\end{aligned}\]

Construct the $X$-bar chart. Identify hours where the process was out of control.

Hour	I	2	3	4	5		6	7	8	9	10	I I	12
	10	-6	-1	-8	-14		-6	-1	8	-1	5	2	5
	3	1	-3	-3	-5		-2	-6	-3	7	6	1	3
	6	-4	0	-7	-6		-1	-1	9	1	3	1	10
	-2	-3	-7	-2	2		-6	7	11	7	2	4	4
$\bar{x} \quad 4.25$		-3.00	-2.75	-5.00	-5.75	$5-3.7$	$75-0$	0.256	6.253	3.50	4.00	2.00	5.50
Hour	13	14	15	16	17	18	19	20	21		22	23	24
	5	6	-5	-8	2	7	8	5	8	$3-1$	-5	-2	-1
	9	6	4	-5	8	7	13	4	1	1	7	-4	5
	9	8	-5	1	-4	5	6	7	0	)	1	-7	9
	7	10	-2	0	1	3	6	10	-6		2	7	0
$\bar{x}$	7.50	7.50	-2.00	-3.00	1.75	5.50	8.25	6.50	0.75	$5 \quad 1.2$	$25-$	-1.50	3.25

Data From Table 1.4

Transcribed Image Text:

Table 1.4 The differences (measurement - specification), in ten- thousandths of an inch Hour 2 3 4 5 6 7 8 9 10 II 12 10 -6 -1 3 1 -3 -8 -14 -6 -1 8 -1 5 2 -5 -2 -6 -3 7 6 1 6 -4 0 -7 -6 -1 -1 9 1 3 1 -2 -3 -7 -2 2 -6 7 11 7 2 4 5304 Hour 13 14 15 6 -5 -8 6 4 -5 x4.25 -3.00 -2.75 -5.00 -5.75 -3.75 -0.25 6.25 3.50 4.00 2.00 5.50 59 16 17 18 19 20 21 22 28 77 8 54 23 24 8 -5 -2 -1 13 1 7 -4 9 8 -5 1 -4 5 6 7 0 1 -7 7 10 -2 0 1 3 6 10 -6 2 7 590 5 9 0 x 7.50 7.50 -2.00 -3.00 1.75 5.50 8.25 6.50 0.75 1.25 -1.50 3.25