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10-3 INFERENCE FOR THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN 347

Definition
2
2
If x , ˛x , s 1 , and s are the means and variances of two random samples of sizes n and
2
1
1
2
n , respectively, from two independent normal populations with unknown and unequal
2
variances, an approximate 100(1 )% confidence interval on the difference in
means is
2
1
2 2 2 2
s 1 s 2 s 1 s 2
x x t ˛ x x t ˛ (10-20)
n
1
2
2
1
1
n
2

2,
B 1 n 2
2, B 1 n 2
where v is given by Equation 10-16 and t
2,˛ is the upper
2 percentage point of the
t distribution with v degrees of freedom.
EXERCISES FOR SECTION 10-3
10-17. The diameter of steel rods manufactured on two dif- 10-20. The deflection temperature under load for two dif-
ferent extrusion machines is being investigated. Two random ferent types of plastic pipe is being investigated. Two random
samples of sizes n 1 15 and n 2 17 are selected, and the samples of 15 pipe specimens are tested, and the deflection
2
8.73, s 1 0.35, temperatures observed are as follows (in F):
sample means and sample variances are x 1
2 2 2
x 2 8.68, and s 2 0.40, respectively. Assume that 1 2 Type 1: 206, 188, 205, 187, 194, 193, 207, 185, 189, 213,
and that the data are drawn from a normal distribution. 192, 210, 194, 178, 205.
(a) Is there evidence to support the claim that the two ma- Type 2: 177, 197, 206, 201, 180, 176, 185, 200, 197, 192,
chines produce rods with different mean diameters? Use 198, 188, 189, 203, 192.
0.05 in arriving at this conclusion.
(b) Find the P-value for the t-statistic you calculated in (a) Construct box plots and normal probability plots for the
part (a). two samples. Do these plots provide support of the as-
(c) Construct a 95% confidence interval for the difference in sumptions of normality and equal variances? Write a prac-
mean rod diameter. Interpret this interval. tical interpretation for these plots.
(b) Do the data support the claim that the deflection tempera-
10-18. An article in Fire Technology investigated two dif-
ferent foam expanding agents that can be used in the nozzles ture under load for type 2 pipe exceeds that of type 1? In
of fire-fighting spray equipment. A random sample of five ob- reaching your conclusions, use 0.05.
servations with an aqueous film-forming foam (AFFF) had a (c) Calculate a P-value for the test in part (b).
sample mean of 4.7 and a standard deviation of 0.6. A random (d) Suppose that if the mean deflection temperature for type 2
sample of five observations with alcohol-type concentrates pipe exceeds that of type 1 by as much as 5 F, it is important
(ATC) had a sample mean of 6.9 and a standard deviation 0.8. to detect this difference with probability at least 0.90. Is the
Find a 95% confidence interval on the difference in mean choice of n 1 n 2 15 in part (a) of this problem adequate?
foam expansion of these two agents. Can you draw any con- 10-21. In semiconductor manufacturing, wet chemical etch-
clusions about which agent produces the greatest mean foam ing is often used to remove silicon from the backs of wafers
expansion? Assume that both populations are well represented prior to metalization. The etch rate is an important characteris-
by normal distributions with the same standard deviations. tic in this process and known to follow a normal distribution.
Two different etching solutions have been compared, using two
10-19. Two catalysts may be used in a batch chemical
process. Twelve batches were prepared using catalyst 1, re- random samples of 10 wafers for each solution. The observed
sulting in an average yield of 86 and a sample standard devia- etch rates are as follows (in mils per minute):
tion of 3. Fifteen batches were prepared using catalyst 2, and
they resulted in an average yield of 89 with a standard devia- Solution 1 Solution 2
tion of 2. Assume that yield measurements are approximately 9.9 10.6 10.2 10.0
normally distributed with the same standard deviation. 9.4 10.3 10.6 10.2
(a) Is there evidence to support a claim that catalyst 2 pro- 9.3 10.0 10.7 10.7
duces a higher mean yield than catalyst 1? Use 0.01. 9.6 10.3 10.4 10.4
(b) Find a 95% confidence interval on the difference in mean
yields. 10.2 10.1 10.5 10.3