574 Chapter 13:Quality Control

Subgroup No. X Subgroup No. X

11 35.8 16 31.6

12 35.8 17 33.0

13 34.0 18 33.2

14 35.0 19 31.8

15 33.8 20 35.6

2. Suppose that a process is in control withμ=14 andσ=2. AnX-control chart
   based on subgroups of size 5 is employed. If a shift in the mean of 2.2 units occurs,
   what is the probability that the next subgroup average will fall outside the control
   limits? On average, how many subgroups will have to be looked at in order to
   detect this shift?


3. IfYhas a chi-square distribution withn−1 degrees of freedom, show that

E[

√

Y]=

√

2

(n/2)

[(n−1)/2]

(Hint: Write

E[

√

Y]=

∫∞

0

√

yfχn (^2) − 1 (y)dy

∫∞
0
√
y
e−y/2y(n−1)/2−^1 dy
2 (n−1)/2
[
(n−1)
2
]

∫∞
0
e−y/2yn/2−^1 dy
2 (n−1)/2
[
(n−1)
2
]
Now make the transformationx=y/2. )

4. Samples of size 5 are taken at regular intervals from a production process, and
   the values of the sample averages and sample standard deviations are calculated.
   Suppose that the sum of theXandSvalues for the first 25 samples are given by
   ∑
   Xi=357.2,

∑

Si=4.88

(a) Assuming control, determine the control limits for anX-control chart.

(b) Suppose that the measurable values of the items produced are supposed to be

within the limits 14.3±.45. Assuming that the process remains in control

with a mean and variance that is approximately equal to the estimates derived,