Applied Statistics and Probability for Engineers

EXAMPLE 10-13 A company manufactures impellers for use in jet-turbine engines. One of the operations

involves grinding a particular surface finish on a titanium alloy component. Two different

grinding processes can be used, and both processes can produce parts at identical mean sur-

face roughness. The manufacturing engineer would like to select the process having the

least variability in surface roughness. A random sample of n 1 11 parts from the first

process results in a sample standard deviation s 1 5.1 microinches, and a random sample

of n 2 16 parts from the second process results in a sample standard deviation of s 2 4.7

microinches. We will find a 90% confidence interval on the ratio of the two standard devi-

ations,

Assuming that the two processes are independent and that surface roughness is normally

distributed, we can use Equation 10-31 as follows:

or upon completing the implied calculations and taking square roots,

Notice that we have used Equation 10-28 to find f0.95,15,10 1 f0.05,10,15 1 2.540.39.

Since this confidence interval includes unity, we cannot claim that the standard deviations of

surface roughness for the two processes are different at the 90% level of confidence.

EXERCISES FOR SECTION 10-5

0.678

 1

 2 1.887

1 5.1 22

1 4.7 22

0.39

^21

^22



1 5.1 22

1 4.7 22

2.85

s^21

s^22

f0.95,15,10

^21

^22



s^21

s^22

f0.05,15,10

 1
 2.

360 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

If and are the sample variances of random samples of sizes n 1 and n 2 , respec-

tively, from two independent normal populations with unknown variances and

then a 100(1)% confidence interval on the ratio is

(10-31)

where and are the upper and lower 2 percentage

points of the Fdistribution with n 2 1 numerator and n 1 1 denominator degrees

of freedom, respectively. A confidence interval on the ratio of the standard deviations

can be obtained by taking square roots in Equation 10-31.

f 2,n 2 1,n 1  1 f 1  2,n 2 1,n 1  1

s^21

s^22

f 1  2,n 2 1,n 1  1 

^21

^22



s^21

s^22

f 2,n 2 1,n 1  1

^21 ^22

^21  22 ,

s^21 s^22

Definition

10-45. For an Fdistribution, find the following:

(a)f0.25, 5, 10 (b)f0.10, 24, 9

(c)f0.05, 8, 15 (d)f0.75, 5, 10

(e)f0.90, 24, 9 (f )f0.95, 8, 15

10-46. For an Fdistribution, find the following:

(a)f0.25, 7, 15 (b)f0.10, 10, 12

(c)f0.01, 20, 10 (d)f0.75, 7, 15

(e)f0.90, 10, 12 (f)f0.99, 20, 10

10-47. Two chemical companies can supply a raw material.

The concentration of a particular element in this material is

important. The mean concentration for both suppliers is the

same, but we suspect that the variability in concentration may

c 10 .qxd 5/16/02 1:31 PM Page 360 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: